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University of Amsterdam
Faculty of Economics and Business
Master of Science in Econometrics Thesis
Dispersion TradingA signal-based trading approach
Author: Daan Olivier Rotsteege
Student number: 10259384
Date: August 14, 2015
Specialisation: Financial Econometrics
Supervisor: Prof. dr. C. G. H. Diks
Second reader: Prof. dr. H. P. Boswijk
Abstract
The aim of this thesis is to investigate the characteristics and trading opportunities of the
implied volatility spread between CAC 40 index options and its corresponding portfolio of
single stock options. Dispersion trading is a trading strategy based on monetising this implied
volatility dispersion, by creating a hedging portfolio with options or third generation volatility
products. The focus in this thesis is on signal trading strategies which make use of
combinations of options and weighting schemes created by principal component analysis
(PCA) and differential evolution and combinatorial search (DECS), where the latter weighting
scheme is optimised with market impact constraints. Herein, a specific dispersion trade is
entered into based on market signals about a collection of volatility smiles and (forecasted)
implied correlations. It is found that the profitability of a highly active naive dispersion
trading strategy is very sensitive to extreme market events; signal trading can reduce this
market exposure while at the same time increase profits.
i
Acknowledgement
My gratitude and appreciation goes out to my internal supervisor and co-director of the
Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Prof. dr. C.G.H. Diks,
for his invaluable advice, selflessness and pleasant conversations. Furthermore, I would like to
thank Prof. dr. H.P. Boswijk, for his questions and comments in the final period of my study.
ii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
1 Introduction 1
2 Theory 4
2.1 Dispersion trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 The concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Optimal dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Market neutrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.4 Tracking P&L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Options as hedging strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Why options? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 Price and value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Combinations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.4 The volatility surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Swaps as hedging strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.1 Why swaps? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.2 Price and value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3.3 Volatility dispersion trading and correlation trading . . . . . . . . . . . . 19
2.4 Volatility and correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Portfolio variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.2 Implied correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Tracking portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Methodology 24
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Tracking portfolio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 PCA analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 Differential evolution and combinatorial search . . . . . . . . . . . . . . . 26
iii
3.3 Implied volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Index options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.2 Stock options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.1 Naive strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4.2 Position forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4.3 Combination forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.4 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Tracking P&L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5.1 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Data 39
5 Evaluation 41
5.1 Preliminary results and analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.2 Naive dispersion trading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.3 Position signal dispersion trading . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 Combination signal dispersion trading . . . . . . . . . . . . . . . . . . . . . . . . 54
5.5 Mixing signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6 Robustness checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.6.1 Do the strategy returns have finite variance? . . . . . . . . . . . . . . . . 57
5.6.2 Is dispersion trading profitable under a transaction costs scenario? . . . . 57
5.6.3 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
6 Conclusion 60
Appendix A 65
Appendix B 78
iv
List of Figures
2.1 Example of S&P 500 and random generated tracking portfolio implied volatility. 6
2.2 Evolution of the Greeks for an ATM straddle and an OTM strangle. . . . . . . . 16
2.3 Evolution of the Greeks for the Variance Swap. . . . . . . . . . . . . . . . . . . . 18
4.1 The market conditions of the CAC 40 index during the trading period. . . . . . . 40
5.1 PCA method tracking portfolio characteristics. . . . . . . . . . . . . . . . . . . . 42
5.2 DECS method tracking portfolio characteristics. . . . . . . . . . . . . . . . . . . 43
5.3 The historical volatility surface of the CAC 40 index over the period 01-01-2010
to 31-05-2010, using both put and call options. . . . . . . . . . . . . . . . . . . . 45
5.4 The cumulative returns of the delta-hedged naive trading strategies against the
CAC 40 index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5 Straddle implied correlation with DECS. . . . . . . . . . . . . . . . . . . . . . . . 53
A.1 Implied correlation DECS tracking portfolio (strangle). . . . . . . . . . . . . . . . 65
A.2 Implied correlations PCA tracking portfolio. . . . . . . . . . . . . . . . . . . . . . 66
A.3 Implied volatilities for the straddle combination. . . . . . . . . . . . . . . . . . . 67
A.4 Cumulative return DECS signal trading (delta-hedged). . . . . . . . . . . . . . . 68
A.5 Cumulative return PCA signal trading (delta-hedged). . . . . . . . . . . . . . . . 69
A.6 Daily returns of the delta-hedged naive and combination strategies. . . . . . . . . 70
A.7 Daily returns of the delta-hedged EGARCH strategies. . . . . . . . . . . . . . . . 71
A.8 Daily returns of the delta-hedged Bollinger Bands strategies. . . . . . . . . . . . 72
A.9 Daily historical returns of the CAC 40 index and some constituents. . . . . . . . 73
A.10 Implied volatility CAC 40 for different strike prices over the period 01-01-2010
to 31-05-2010. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
v
List of Tables
5.1 Garch-type modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 Non delta-hedged strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Delta-hedged strategies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.4 Straddle implied volatility spread. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5 Delta-hedged strategies under transaction costs. . . . . . . . . . . . . . . . . . . . 59
A.1 Augmented Dickey-Fuller test return series. . . . . . . . . . . . . . . . . . . . . . 75
A.2 Characteristics of the implied volatilities. . . . . . . . . . . . . . . . . . . . . . . 76
A.3 Characteristics of the implied volatility spread. . . . . . . . . . . . . . . . . . . . 77
B.1 Parameters of the DECS optimisation. . . . . . . . . . . . . . . . . . . . . . . . . 78
vi
Chapter 1
Introduction
In the period after the Global Financial crisis, the correlations between stocks increased to new
historical records (Kolanovic, 2010). As a result, the trading strategy called Dispersion Trading
gained renewed interest by sophisticated hedge funds and proprietary trading desks (Marshall,
2009), but remained limited in the academic world. Prior to this period, the strategy has been
discussed in some business papers and reports, especially because of implied correlation spikes,
that occurred on account of global events (e.g. London terrorist bombings and the 9/11 terrorist
attacks), in which some hedge funds unwound a short position on the high correlation observed
in these indecisive financial markets. At the same time, the studies of Bakshi et al. (2003),
Bakshi and Kapadia (2003) and Bollen and Whaley (2004) contributed to empirical evidence
that generally index options are traded against a premium compared to their theoretical Black-
Scholes prices, while individual stock options do not appear to be overpriced.
Dispersion trading is a trading strategy which aims to profit from ostensible risk premiums
in implied volatilities and is closely related to correlation trading. Because the value of an
index is equal to a weighted average of the underlying stocks prices, by the Law of One Price
the implied volatility derived from index options should also be equal to the implied volatility
derived from options on the corresponding portfolio of stocks. Thus the mispricing suggests that
index volatility is more rich and the volatility of the constituents is cheaper. Several papers
have investigated this implication (e.g., Deng, 2008; Bakshi and Kapadia, 2003) and as a result
two main hypotheses are made. The first argument is a risk-based hypothesis, which states
that index options are more expensive because the market volatility risk premium is smaller for
stock options compared to index options and that index options hedge a certain correlation risk
(Driessen et al., 2005). This is confirmed by Bakshi et al. (2003), who address the differential
pricing of index and single stock options to the different skewness of the risk-neutral distribution
of the underlying asset.
On the other side of the academic literature one assigns the expensive index options to
market inefficiencies. Bollen and Whaley (2004) state that the net buying pressure drives the
index option prices out of parity. Herein it is suggested that as a market maker builds up a
larger position in a given option, the volatility risk exposure of his portfolio, i.e. vega, also
1
increases. As a result, hedging costs will increase and the market maker is forced to demand a
higher price for the option, which leads to an increase in implied volatility. This hypothesis is
complemented by Garleanu et al. (2006) and Lakonishok et al. (2007), who both argue that the
demand pattern of stock options is different from index options.
Some major changes in the US options market around the beginning of 2000, such as the
launch of the International Securities Exchange (ISE) and an overall market reduction in the bid-
ask spreads made way for a natural experiment to investigate both hypotheses. By the launch
of the ISE as the first electronic options exchange in the US, costs for taking advantage of any
differential pricing of indices and associating stocks reduced. Therefore following the demand
and supply-based argument, it would be expected that the market became more efficient after
this change and hence the profitability of dispersion trading reduced. Deng (2008) shows that
dispersion trading was profitable in the five year period prior these structural changes but that
in the same timespan after the 2000 break point average monthly returns decreased significantly,
from 24% to -0.03%.
Marshall (2009) evaluates the efficiency of US options in pricing volatility in the period of
2005-2007. Using a modification of the Markowitz variance equation to estimate the volatility
of the portfolio of stocks underlying the index, she was able to show the existence of a volatility
premium implicit in index options on the S&P 500 index. Even when a transaction costs
scenario was taken into account, there were a significant number of days with potential volatility
dispersion trading opportunities. The results of Marshall are of great importance because it
proves the existence of volatility dispersion in the US option market in this specific period,
however the results do not imply a trading strategy and can merely be used as a signal of
potential arbitrage.
Identification of dispersion trading opportunities can be done in various ways, but the most
elegant way is by looking at the implied correlation of the index, which is an average correlation
measure derived from the implied volatilities of index options and individual options. Another
measure is the volatility dispersion statistic. Although the identification methods give approx-
imately the same signal, a more vital choice of the strategy is how the volatility discrepancy
is monetised. Typically, a dispersion trade can be entered by taking positions in plain vanilla
options or variance/volatility swaps. Hereby, one takes a short position in the overpriced el-
ement and a long position on the cheaper element of the strategy. The advantage of using
swaps is that delta-hedging is not labour intensive and that they give direct exposure to the
variance/volatility of the underlying, however since the financial crisis of 2008 the liquidity of
these swaps on individual stocks has decreased (Martin, 2013). Using plain vanilla options
for dispersion trading usually involves taking positions in straddles and strangles, this because
at-the-money straddles or out-of-the-money strangles have a delta exposure close to zero and
the strategy is for this reason hedged against large market fluctuations (Deng, 2008, p. 2).
Because hedge funds prefer to conceal a profit-making strategy, it is unknown to what extent
dispersion trading strategies are used and whether it is possible to make a realistic excess profit
2
based on dispersion trading. Furthermore, it is the author’s personal belief that there are merely
academic studies that take a pragmatic approach in the research to a realistic and frequently
trading dispersion strategy. Probably the most realistic strategies are developed by Deng (2008)
and Magnusson (2013). Although both find some significant trading opportunities and extended
the general knowledge on this topic, the strategies are still elementary and open for evolution.
For this reason this thesis aims to construct and evaluate a close to real life signal dispersion
trading strategy on the CAC 40 index, where options are used to take advantage of the relative
differences in implied volatilities of the index and the constituents.
There are many crucial elements in developing a successful and feasible quantitative trading
strategy like dispersion trading, e.g. the choice of weighting schemes, positions and position
limits, hedging the Greeks and the market impact of a trade. However, due to the scope of this
thesis not all factors determining the profit and loss (P&L) of the strategy can be dealt with.
The crucial question of this research is whether there are dispersion trading opportunities on
the CAC 40 index with a naive dispersion trading strategy. On the way to close this question,
it will be examined whether a weighting scheme based on a tracking portfolio constructed by
evolutionary heuristics performs different than a tracking portfolio based on a linear dimension
reduction method. The two naive trading strategies, one for each optimisation method, will
then be adjusted to become more dynamic and realistic by allowing for entry signals, position
signals, daily delta-hedging and an approximation for transaction costs.
The remainder of this thesis is organised as follows. Chapter 2 provides all the indispensable
knowledge to conduct the research and lays the foundation for the subsequent chapters. Sub-
sequently in Chapter 3 the methodology is presented. Chapter 4 describes the data used for
the empirical analysis. Chapter 5 presents the preliminary results and analyses of the training
period, whereupon the results of the trading period are portrayed and evaluated. Chapter 6
concludes.
3
Chapter 2
Theory
A fortunate dispersion trade is established from a proper interplay between the volatilities of
the assets underlying the option contracts. Therefore, in order to build a profitable dispersion
strategy, a thorough knowledge of the possible financial derivatives used in a hedging portfolio
must be created, together with their interactions. Consequently the goal of this chapter is to
lay a strong theoretical foundation for the remainder of this thesis.
2.1 Dispersion trading
Before the more technical side of this chapter is touched, a short description of dispersion trading
is given in this section.
2.1.1 The concept
Next to the return, probably the best-known and used concept in the financial world is the
volatility of an asset, i.e. the standard deviation of the return series of an asset, hence a measure
for the variability of the price. This is partly due the fact that over the last decades the
demand for options has been booming and because of the emergence of more complex investment
products, including structured products. On these financial derivatives volatility means the
conditional standard deviation of the underlying asset’s return and some of these products’
payoffs are solely based on this volatility measure. Dispersion trading is a trading strategy that
aims to profit from the discrepancy in implied volatilities between different products and hence
notwithstanding its elegant name, it is a reasonably simple concept.
Dispersion trades can be set up using (combinations of) options and third generation volatil-
ity products (e.g. volatility/variance swaps) and in general there are two reasons to enter a
dispersion trade. Naturally the first reason is because of statistical arbitrage opportunities,
the second reason is to hedge correlation products. As will be explained in Subsection 2.1.2 a
long position in a dispersion trade, i.e. a long position in the volatility of the components of an
index and a short position on the volatility of the index itself, can practically be seen as a short
position in correlation. However strictly speaking, as shown by Jacquier and Slaoui (2007),
4
implied correlation from a dispersion trade with variance swaps tends to exceed the strike of
a correlation swap. This result is important because financial institutions over the years have
sold structured products such as mountain range options1 and consequently by selling these
products a financial institution exposes itself with a short position in correlation. Therefore
by taking a short position in a dispersion trade it neutralises the exposure in short correlation
(FDAXhunter, 2004).
The concept is clarified with an example. Lets assume that an investor lives in a simple
world with no transaction costs, which comprises of four stocks A, B, C and D, together with
a weighted average index of these stocks X. He observes that the implied volatility of the index
has a premium compared to the implied volatility of a same weighted portfolio of stocks A, B,
C and D (derived from option prices). The goal of the investor is to create a hedged position
which takes advantage of the relative value differences in the implied volatilities of the options,
hence he decides to take a long volatility dispersion position, i.e.:
• A long position on the volatility of stocks A, B, C and D
• A short position on the volatility of the index X
The investor thus initiates a short position in index options and a long position in options on
the stocks A,B,C and D. Profits are realised in the following events (Marshall, 2008):
1. Implied volatilities return into equilibrium
2. The options expire and more is earned on the long position than the costs on the short
position
As a matter of course, the first case is fairly straightforward because the investor observes
that there is a disequilibrium in implied volatilities and buys the relatively cheap options on
the constituents and sells the relatively expensive options on the index. When the implied
volatilities converges back into equilibrium the investor makes a profit on (1) the long leg, (2)
the short leg or in the most advantageous situation (3) a combination thereof.
If the disequilibrium between implied volatilities does not soften, the investor makes only a
profit at the time of maturity when the long leg is worth more than the negative of the short
leg (from the dispersion investor’s point of view). This situation is most likely to happen when
during the period where the investor has a long dispersion position active in the market, there
is minimal volatility on the index X and maximal volatility on the components of the index
(stocks A,B, C and D). The next subsection 2.1.2 deals further with this issue.
1Options where the payoffs depend on the performance of a basket of underlying securities, e.g. Everest-,
Atlas- and Himalaya options.
5
2.1.2 Optimal dispersion
In the previous example it was mentioned that in the case where the spread between the two
implied volatilities does not mean-revert to its long-term mean zero, the most likely way that
a long dispersion position would end up in the money is with minimal volatility on the index
and maximal volatility on its constituents. This is only possible whenever the stocks comprising
the index appear to be uncorrelated, meaning that the move of one stock is canceled out by
the move of another stock with the result that the index stays close to put. The investor may
wish to delta-hedge his volatility position on the individual stocks, however as the index hardly
moves, no delta-hedging is required on the short position. Altogether, the investor makes a
theta related profit on the index and a gamma profit on the individual stocks.
Although one can earn a lot of profit on a dispersion position with optimal dispersion on
its constituents, it can be hazardous. When the stocks have perfect correlation more money is
lost on the short position than earned in theta and this is exactly the reason why dispersion
trading is closely related to the concept to correlation trading. A long position in a dispersion
trade can be seen as a bet on low correlation, i.e. a short position in correlation, and vice versa
for a short dispersion position.
Jan10 Apr10 Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12 Jul12 Oct120.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.55
Date
Impl
ied
Vol
.
S&P 500 Model−Free (VIX)Simulated Constituents
Long Dispersion Trade Long Dispersion Trade
Figure 2.1: Example of S&P 500 and random generated tracking portfolio implied volatility.
6
2.1.3 Market neutrality
A market neutral strategy is a popular strategy taken by hedge funds and proprietary trading
desks. Herein, a trader does not bet on broad market movements but rather the goal is to profit
from a relative mispricing which exhibits in the market. This is done by taking a long position
in the relatively cheap security and a short position in the relatively expensive security and
therefore the strategy is hedged against specific market movements, hence the theoretical beta
of such a strategy is equal to zero.
Dispersion trading in a world with no additional trading costs (e.g. transaction costs and
market impact), where a trade is possible on all the constituents of the index, is a market
neutral strategy. In the end, the market index can be seen as a weighted average of single
stocks and thus both the long and the short leg are characterised with the same risk. However,
because volatility pricing discrepancies in the market are small and consequently payoffs due
to volatility dispersion are marginal, profits fade away after adjusting for trading costs. If one
tackles this problem by taking a position on a portfolio which mimics the index instead of a
weighted average of all constituents, a correlation risk between the tracking portfolio and the
market index arises. Hence dispersion trading in its authentic form and in a perfect world is a
market neutral strategy, in reality it is statistical arbitrage.
The weights of the volatility positions on the single stocks can be determined based on the
preferences of the trader with respect to the portfolio’s market risk exposure, i.e. the Greeks, and
the financial products used. One method is already described: using the weights of a tracking
portfolio which mimics the characteristics of the index. Alternative weighting strategies can
for example be based on vega-neutrality, gamma-neutrality or theta-neutrality. In the case of
vega/gamma neutral weights, the vega/gamma of the index equals the sum of the single stock
vegas/gammas. If one aims for theta-neutrality, a short position in vega and gamma is entered
into.
2.1.4 Tracking P&L
When initiating a dispersion trade one needs to decide whether the aspire is to enter into a
self-financing portfolio, which means that the market value of the short leg offsets the value of
the long leg and accordingly the market value of the portfolio is equal to zero at inception. An
imaginable way to open a self-financing portfolio is to first enter the short position and then
directly go into the long position with the proceeds of the short leg. The advantage of a self-
financing portfolio is that no initial investment needs to be made. Nevertheless more leverage
is created because the value of the long leg is adjusted to the short leg at the starting point.
Another issue of the P&L of a strategy is the way of calculating the total simple return when
a short position is present in the portfolio. A short sell can be translated as selling a financial
product which is not owned by the seller, but borrowed from someone else in exchange for a
borrow fee and an obligatory repayment of the financial asset at some future time. Thus a short
seller has a financial liability in the future while receiving money at the start, meaning that the
7
return on a short position can be calculated as the negative return of a long position. When a
self-financing portfolio is initiated then the total simple return of the portfolio can be calculated
as the ratio between the total value of the portfolio at maturity divided by the proceeds of the
short position at the start. In all other cases the simple returns need to be reweighted to the
size of the positions.
2.2 Options as hedging strategy
In this section the various aspects of options are explained and the way how they can be used
in a dispersion hedging strategy.
2.2.1 Why options?
An option gives the holder the right but not the obligation to buy or sell an underlying asset at
a specified strike price on or before a specified date, called the maturity date (Etheridge, 2008).
At first sight, this definition suggests that options are some kind of extension of forwards/futures
contracts, however, whereas it costs nothing to enter into a forward/futures contract, an option
has a price because it gives the right and not the obligation to buy or sell the underlying asset.
In the global financial world there are many different types of options and depending on its
terms, some are sold on OTC markets and others on exchanges. Options in their simplest form
are plain vanilla options and the more complex options are called exotic options. The most
commonly traded options are European and American options; both are categorised as plain
vanilla options and depending on its terms and conditions they are sold on OTC markets and on
exchanges. The difference between American and European options is that American options
may be exercised before the maturity date, whereas European options can only be exercised at
maturity, i.e. when the contract expires. In general it applies that options on indices are of the
European variant and single stock options are American, however some exchanges also provide
European options for stocks.
The advantage of using plain vanilla options in a volatility dispersion strategy is that ex-
changes (usually) offer a lot of different standardised options, i.e. standardised strike prices and
maturities, on the same underlying asset. Because an exchange continuously publishes publicly
option prices, it enables itself to attract many independent buyers to carry out a trade, which
intensifies the volume and lowers the margins. Hence these options have fairly liquid markets
(Etheridge, 2008). Another advantage is that a trader can combine options with different strike
prices and maturities in a portfolio to hedge certain market factor exposure or to minimise
undesirable future outcomes.
Although using options has its advantage in a dispersion trading strategy wherein a trader
takes a portfolio in index options and in single stock options, this strategy has also its downsides.
In particular this strategy is path-dependent and the volatility risk exposure of this portfolio
can become unhedged as the market environment changes, moreover delta-hedging is required
8
continuously. Especially variance/volatility swaps are a solution for the path-dependent issues
of this strategy, see Subsection 2.3.1.
2.2.2 Price and value
In this subsection the price of a European option is given, the Black-Scholes pricing formula.
The price of an American option is not considered here because an explicit formula only exist
in a few special cases, which means that in general this option must be priced with numerical
methods, such as with the Binomial Option Pricing model or with Monte Carlo estimation
(Etheridge, 2008).
The price of a European option with constant volatility
Consider a market with a riskless cash bond, Btt≥0 and a risky stock with stochastic process
Stt≥0. It is assumed that the riskless borrowing rate is constant and that
dBt = rBtdt with B0 = 1, (2.1)
dSt = µStdt+ σStdWt, (2.2)
where Wtt≥0 is (P, Ftt≥0) Brownian motion. Thus it is assumed that Stt≥0 is a geometric
Brownian motion with constant drift.
Now one may define the discounted price process Stt≥0, where St = Bt−1St, from here it
can be derived that
dSt = (µ− r)Stdt+ σStdWt.
The process is defined as
Xt = Wt + σ−1(µ− r)t,
and hence
dXt = dWt + σ−1(µ− r)dt,
dSt = σStdXt.
By Girsanov’s theorem, under the risk-neutral measure Q, Xt follows a standard Brownian
motion and therefore St a martingale. Now, by expressing the value of a European call or put
option as Vt = F (t, St) and Vt = Bt−1Vt = e−rtVt and defining F such that V = F (t, St),
applying Ito’s formula to V and using the zero drift condition for a martingale under Q,
∂F
∂t(t, x) = −1
2
∂2F
∂2x(t, x)σ2x2.
The following equation is obtained, which is the Black-Scholes PDE:
−rF (t, x) +∂F
∂t(t, x) + rx
∂F
∂x(t, x) +
1
2
∂2F
∂x2(t, x)x2σ2 = 0. (2.3)
9
The Black-Scholes PDE has an explicit solution for European options, the Black-Scholes pricing
formula, and at time t ∈ (0, T ), the value of this option, Vt, whose payoff at maturity is
VT = f(ST ) with strike price K and θ = (T − t) is given by
Vt = e−rθ∫ ∞−∞
f
(Stexp
((r − 1
2σ2)θ + σz
√θ
))· 1√
2πexp
(−z2
2
)dz. (2.4)
Now if we denote the price of a European call option as C(t, St;K) and a European put option
as P (t, St;K) at time t ∈ (0, T ) on a non-dividend paying stock with price St, using the same
notation it can be shown that
C(t, St) = StΦ(d1)−Ke−rθΦ(d2), (2.5)
P (t, St) = Ke−rθΦ(−d2)− StΦ(−d1), (2.6)
d1 =log(StK
)+(r + σ2
2
)θ
σ√θ
, (2.7)
d2 = d1 − σ√θ, (2.8)
Φ(z) =1√2π
∫ z
−∞e−z
2/2dz. (2.9)
The price of a European option with time-varying volatility
The same process is assumed as in (2.1) and (2.2), only σ is replaced by σt, where the latter
satisfies that∫ T0 σ2t dt is finite with P-probability one. Again Girsanov’s Theorem is used to find
a risk-neutral measure, Q, under which Wtt≥0 is a standard Brownian motion, where
Wt = Wt +
∫ t
0γsds,
γt = (µ− r)/σt.
The discounted stock price process Stt≥0 is characterised by the stochastic differential equation
dSt = (µ− r − σtγt)Stdt+ σtStdWt,
and Stt≥0 is a Q-martingale when the following boundedness assumptions are satisfied:
EP[exp
(1
2
∫ T
0γ2t dt
)]<∞,
EQ[exp
(1
2
∫ T
0σ2t dt
)]<∞.
By defining the (Q, Ftt≥0)-martingale Mtt≥0, where Mt = EQ [B−1T CT |Ft], and by showing
that any claim CT can be replicated by φt units of stocks and ψ = Mt−φtSt units of cash-bonds
at time t, the fair value of the claim is, Vt = EQ [e−r(T−t)CT |Ft]. Because σt only depends on
(t, St), using the Feynman-Kac Stochastic Representation Theorem, the price can be expressed
as a solution to (2.3), with σ2 = σ2(t, x). This means that in the Black-Scholes pricing formula
σ2 is replaced by 1T−t
∫ Tt σ2sds.
10
The P&L of a delta-hedged portfolio
A trader may wish to combine certain put and call option in a portfolio to minimise the risk
and the exposure to the Greeks, however before these combinations are treated it is pleasing
to investigate the P&L for a single delta-hedged option with time-varying volatility. Assuming
the same process as in (2.1) and (2.2), and replacing σ by σt, a delta hedged portfolio Πt at
time t ∈ (0, T ), implies that one has two opposite positions in a derivative and the associating
underlying asset. It is only enthralling to consider one of the two possible cases and therefore
assume that this portfolio Πt consists of a short position in the asset, and a long position in an
option with value Vt. The value of this portfolio changes over period τ ∈ R+ with
Πt+τ −Πt = Vt+τ − Vt −∫ t+τ
t
∂Cu∂Su
dSu −∫ t+τ
tr
(Cu −
∂Cu∂Su
)Sudu,
∆Π = ∆Vt − δt∆St + (δtSt − Vt)r∆t,(2.10)
and with δt = ∂Ct∂St
. However to obtain a more insightful expression of the P&L over the period
τ , assumed infinitely small, the second-order Taylor expansion of dVt is taken. The next steps
are based on the derivations of Forde (2003) and Jacquier and Slaoui (2007);
dVt =∂V
∂t(t, St)dt+
∂V
∂St(t, St)dSt +
∂V
∂σt(t, St)dσt
+1
2
(∂2V
∂S2t
(t, St)(dSt)2 +
∂2V
∂σ2t(t, St)(dσt)
2 + 2∂2V
∂St∂σt(t, St)dStdσt
),
when there exists some risk-neutral measure, P, such that the Black-Scholes implied volatility,
σt, has a drift.
By rewriting the Black-Scholes PDE and replacing the unknown time-varying volatility for
the implied volatility, one can find an expression for rVtdt. After substituting in Eq. (2.10), we
obtain
dΠt =∂V
∂t(t, St)dt+
∂V
∂St(t, St)dSt +
∂V
∂σt(t, St)dσt
+1
2
(∂2V
∂S2t
(t, St)(dSt)2 +
∂2V
∂σ2t(t, St)(dσt)
2 + 2∂2V
∂St∂σt(t, St)dStdσt
)− δtdSt + rδtStdt−
(∂V
∂t(t, St) +
1
2
∂2V
∂S2t
(t, St)S2t σ
2t + rSt
∂V
∂St(t, St)
)dt.
This can be rewritten in terms of the Greeks Γ (Gamma), ν (Vega), Vanna and Vomma as
dΠt =1
2ΓS2
t
[(dStSt
)2
− σ2t dt
]+ νdσt +
1
2V omma · (dσt)2
+ V anna · σtStρζdt,
(2.11)
where ζ is the volatility of volatility (vol-of-vol) and ρ is the correlation between the price of
11
the asset and the volatility. The Greeks are defined as
Γ =∂δt∂St
=∂2V
∂2St(t, St), (2.12)
ν =∂V
∂σt(t, St), (2.13)
V omma =∂2V
∂σ2t(t, St), (2.14)
V anna =∂δt∂σt
=∂2V
∂St∂σt(t, St). (2.15)
Hence the P&L of a long delta-hedged dispersion strategy is found by summing the individual
stock P&Ls and subtracting the index P&L, i.e. dΠLDt =
∑ni=1 dΠi,t − dΠI,t. Also, this delta-
hedged portfolio of single options can readily be extended to the P&L of combinations of options,
such as straddles and strangles.
It can be shown that under the Black-Scholes framework, i.e. constant volatility, Eq. (2.11)
can be simplified to
dΠt =∂V
∂t(t, St)
[(dSt
Stσ√dt
)2
− 1
], (2.16)
where ∂V/∂t is theta and dSt/(Stσ√dt) can be interpreted as a standardised move of the
underlying asset’s price over a specific time. If we now consider an index, I, together with its n
constituent stocks, with σi the volatility of the ith stock, wi its corresponding weight in index
I, pi the number of shares of stock i and ρij the correlation between the ith and the jth stock,
i, j ∈ (1, . . . , n), Eq. (2.16) in terms of the index is given by
dΠI,t =∂V
∂t(t, SI,t)
( dSI,t
SI,tσI√dt
)2
− 1
. (2.17)
Writing zI,t = dSI,t/(SI,tσI√dt) and zi,t = dSi,t/(Si,tσi
√dt) for the standardised move of the
index and single stocks, respectively, then we can derive that
zI,t =dSI,t
SI,tσI√dt
=
∑ni=1 pidSi,t
σI√dt∑n
j=1 pjSj,t
=1
σI∑n
j=1 pjSj,t·n∑i=1
σipiSi,tdSi,t
σiSi,t√dt
=1
σI∑n
j=1 pjSj,t·n∑i=1
σipiSi,tzi,t
=
n∑i=1
wiσiσI
zi,t.
(2.18)
Thus this implies that the P&L of the delta-hedged index option, Eq. (2.17), can be written in
12
terms of its constituents as
dΠI,t =∂V
∂t(t, SI,t)
[z2I,t − 1
]=∂V
∂t(t, SI,t)
( n∑i=1
wizi,tσiσI
)2
− 1
=
1
σ2I
∂V
∂t(t, SI,t)
n∑i=1
w2i σ
2i z
2i,t +
n∑i=1,j 6=i
wiσiwjσjzi,tzj,t − σ2I
=
1
σ2I
∂V
∂t(t, SI,t)
n∑i=1
w2i σ
2i z
2i,t +
n∑i=1,j 6=i
wiσiwjσjzi,tzj,t −
n∑i=1
w2i σ
2i +
n∑i=1,j 6=i
wiσiwjσjρi,j
=
1
σ2I
∂V
∂t(t, SI,t)
n∑i=1
w2i σ
2i
(z2i,t − 1
)+
n∑i=1,j 6=i
wiσiwjσj (zi,tzj,t − ρi,j)
.(2.19)
Therefore, the P&L of a long dispersion trade under the Black-Scholes framework is given by
dΠLDt =
n∑i=1
dΠi,t − dΠI,t
=n∑i=1
(z2i,t − 1
) [∂V∂t
(t, Si,t)− w2i σ
2i
1
σ2I
∂V
∂t(t, SI,t)
]
− 1
σ2I
∂V
∂t(t, SI,t) ·
n∑i=1,j 6=i
wiσiwjσj (zi,tzj,t − ρi,j) .
(2.20)
2.2.3 Combinations
A combination is an option strategy wherein a position is taken on both call and put options on
the same underlying stock (Hull, 2012). The best-known combinations are strangles, straddles,
strips and straps, but because the latter two are a bet on a specific market movement, they are
not optimal for a dispersion trading strategy, which is market neutral in its purest form. For
this reason only the straddle and the strangle are discussed below.
Straddle
This strategy involves a long position in both a European call and put option on the same
underlying asset with the same strike price and time to maturity. The payoff is V-shaped,
which means that a trader limits its downside risk by accepting a loss when the underlying
asset does not move much in either direction. However a significant profit is made when at
maturity the underlying ends up with a large distance from its initial value. Thus someone who
enters into a straddle is uncertain in which way the underlying asset is going to move. It is a
straightforward observation that a reverse position in a straddle (i.e. a short position) is very
risky because the loss arising from a large move in the underlying asset is unlimited.
Like the fact that the Black-Scholes model gives under certain parameter conditions the
price of a European put or call option, the Greeks of European options do also have an exact
expression. Therefore by using simple calculus rules for taking derivatives, the Greeks of a
13
straddle can be found from the Black-Scholes formula (Eq. (2.4)). If we denote the value of a
straddle with Πt at time t, then
∆ =∂Πt
∂St= 2Φ(d1)− 1, (2.21)
Γ =∂2Πt
∂S2t
=2φ(d1)
Stσ√T − t
, (2.22)
Θ =∂Πt
∂t= −Stφ(d1)σ√
T − t− rKe−r(T−t)(2Φ(d2)− 1), (2.23)
ν =∂Πt
∂σ= 2Stφ(d1)
√T − t, (2.24)
where φ(z) is the first derivative of Φ(z).
Strangle
In a strangle an investor goes long in a European call and put option with the same time to
maturity, however the difference with a straddle is the fact that the strike prices of the two
options differ. A strangle yields less downside risk than a straddle and as a consequence the
underlying asset must move more intense to make a profit. The Greeks for a strangle can
be found in an analogous way as for the straddle and by comparing an at-the-money (ATM)
straddle with an out-of-the-money (OTM) strangle (the most common combinations), although
both have a very small initial delta exposure, the OTM strangle has less delta exposure than the
ATM straddle for small movements of the underlying and is more preferred in a delta optimal
point of view. The Greeks of a straddle are defined as
∆ =∂Πt
∂St= Φ(d1
c) + Φ(d1p)− 1, (2.25)
Γ =∂2Πt
∂S2t
=φ(d1
c) + φ(d1p)
Stσ√T − t
, (2.26)
Θ =∂Πt
∂t= −Stσ(φ(d1
c) + φ(d1p))
2√T − t
− rKe−r(T−t)(Φ(d2c)− Φ(−d2p)), (2.27)
ν =∂Πt
∂σ= St(φ(d1
c) + φ(d1p))√T − t, (2.28)
where the subscript denotes whether the variable is evaluated with respect to the call or the
put option.
P&L combinations
In Subsection 2.2.2 the P&L of a delta-hedged long dispersion strategy was presented, assuming
that the underlying stock followed a geometric Brownian motion with constant drift and (time-
varying) volatility and with a constant riskless borrowing rate. Naturally, this solution can
be extended to the combinations considered in this subsection by summing the (reweighted)
individual P&Ls of the options within the portfolio.
14
2.2.4 The volatility surface
The Black-Scholes pricing model assumes that the price process of the option’s underlying asset
is ruled by a geometrical Brownian motion, which theoretically implies that options on the
same asset should trade at the same implied volatility regardless of the time to maturity and
the strike price. This assumption is not observed in real financial markets however; in reality
there is empirical evidence that the assets return distribution exhibits excess kurtosis and is
skewed compared to the lognormal distribution (Hull, 2012). Hence, implied volatilities differ
between options on the same underlying but with a different strike price (volatility smile) and
with distinct time to maturities (term structure of volatility).
Different kinds of assets display different kinds of behaviour in their prices. For example,
the asset class equity (stocks) shows in general the so-called leverage effect, where a negative
price shock (e.g. stock market crash) has a larger effect on the future volatility than a large
positive price shock. As a large negative stock return leads to a decrease in equity value for
the company, its leverage increases, i.e. the debt-to-equity ratio increases, and hence a larger
return on equity is expected.2 But if this effect is indeed a common stylised fact for stocks,
then the implied volatility can be seen as a decreasing convex function of the strike price and
therefore this type of volatility smile is also known as the volatility skew. Another example where
there is no constant implied volatility from options as a function of strike prices are exchange
rates. Typically the time path of an exchange rate is rough and exhibits jumps, furthermore the
volatility shows time varying properties and consequently extreme outcomes are more likely to
occur. Hence, generally the implied volatility is an increasing convex function of the absolute
distance between the current exchange rate and the strike prices.
When short-dated volatilities are low it is expected that the volatility will increase in the
future and vice versa. Combining this effect with the volatility smile is called the volatility
surface, i.e. the implied volatility as a function of both the time to maturity as the strike prices
of an option. A ramification of the existence of this volatility surface is that the formulas of the
Greeks derived from the Black-Scholes pricing model and given in the previous subsection are
no longer correct. For example by taking the volatility surface into account, the delta of a call
option is given by
∆ =∂C
∂S+
∂C
∂σimp
∂σimp∂S
.
Because normally an option does not yield a constant implied volatility as a function of the
standardised strike prices (K/S) (Etheridge, 2008),
∂C
∂σimp
∂σimp∂S
6= 0,
and is in most of the cases positive for equity options. Nonetheless, the changes in the volatility
surface observed in the market are usually small and the Greeks of the Black-Scholes model can
be used as a reasonable approximation (Hull, 2012).
2Financial Econometrics, University of Amsterdam, catalogue number 6414M0007Y.
15
80 85 90 95 100 105 110 115 120−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Price Underlying
Delta
4 dtm
12 dtm
30 dtm
(a) Delta straddle K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 120−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Price Underlying
Delta
4 dtm
12 dtm
30 dtm
(b) Delta strangle K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 1200
0.05
0.1
0.15
0.2
0.25
Price Underlying
Gam
ma
4 dtm
12 dtm
30 dtm
(c) Gamma straddle K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 1200
0.02
0.04
0.06
0.08
0.1
0.12
Price Underlying
Gam
ma
4 dtm
12 dtm
30 dtm
(d) Gamma strangle K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 1200
5
10
15
20
25
30
Price Underlying
Vega
4 dtm
12 dtm
30 dtm
(e) Vega straddle K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 1200
5
10
15
20
25
Price Underlying
Vega
4 dtm
12 dtm
30 dtm
(f) Vega strangle K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 120−100
−80
−60
−40
−20
0
20
Price Underlying
Theta
4 dtm
12 dtm
30 dtm
(g) Theta straddle K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 120−60
−50
−40
−30
−20
−10
0
10
Price Underlying
Theta
4 dtm
12 dtm
30 dtm
(h) Theta strangle K=100, σ=0.3, r=0.75%
Figure 2.2: Evolution of the Greeks for an ATM straddle and an OTM strangle.
16
2.3 Swaps as hedging strategy
In this section the different components of specific swaps are explained, and how they can be
used in a dispersion hedging strategy.
2.3.1 Why swaps?
Next to options, other financial derivatives which have good properties for volatility dispersion
trading are variance/volatility swaps. A volatility swap is an OTC product similar to a forward
contract where one can speculate on the amount of realised volatility of an asset over a specific
prespecified period. The payoff of this swap is the difference between the realised volatility of
the asset and a fixed amount of volatility determined at the beginning, multiplied by a notional
principal. The variance swap is analogous to the volatility swap, only the variance is used instead
of the volatility. Both products are designed to give a direct exposure to the volatility/variance
of an asset for hedging and risk-management purposes, however because the payoff of a variance
swap can typically be replicated by a portfolio of vanilla options (see Subsection 2.3.2) they are
easier to value and more popular (liquid) than volatility swaps (Carr and Lee, 2009). On the
other hand, the advantage of a volatility swap is that the payoff is a linear function of the
realised volatility of an asset and hence they give direct exposure to vega.
Taking a long position in an option always has strictly positive costs, except in the special
case that an option is worthless. Initiating a position in a volatility/variance swap however, has
zero costs because the fixed amount in these swaps represents the expected value of the realised
volatility/variance under the risk-neutral distribution of the underlying. Another important
difference between options and these specific swaps is that the latter instruments are a pure
play on the realised volatility, meaning that delta-hedging is not labour intensive. On the
contrary, the delta in a strategy with options is path-dependent and must be hedged in theory
continuously. However Martin (2013) explains that the market for variance swaps collapsed
during the Global Financial crisis because the prices of most of the underlying assets showed
discontinuous jumps, and variance swaps are not able to be replicated with options in this
situation. Moreover, the market for variance swaps has not recovered since then.
2.3.2 Price and value
In this subsection the theoretical strike price of a variance swap is given together with the
Greeks, the volatility swap is not considered here because the payoff can not be replicated by a
portfolio of options and hence is hard to value.
A variance swap is an agreement to exchange realised variance
1
T
n∑i=1
(log
StiSti−1
)2
, (2.29)
where ti = iδt, i = (0, . . . , n) and δt = T/n, for a predefined variance strike K (i.e. fixed
17
variance) at some future time T . In the limit, δt → 0, this implies that the payoff, V, of a
variance swap with notional principal N is given by
V = N
(1
T
∫ T
0σt
2dt−K). (2.30)
It is conventional to set N = Nσ/(2K), where Nσ is the vega notional of a volatility swap.
Demeterfi et al. (1999) show that under nice3 behaviour of the underlying asset, the price
of a variance swap can be replicated by an infinite number of put options with strike prices
Kput ∈ [0, S∗] and an infinite number of call options with strike prices Kcall ∈ [S∗,∞). The fair
fixed variance K is equal to
K =2
T
(rT −
(S0S∗erT − 1
)− log
S∗S0
+ erT∫ S∗
0
1
K2P (S0,K, T )dK + erT
∫ ∞S∗
1
K2C(S0,K, T )dK
),
(2.31)
where S∗ is a parameter which defines the boundary between call and put options. It can be
shown that in the case that this boundary parameter is equal to the fair forward value of the
underlying asset price, i.e. S∗ = S0erT , Eq. (2.31) can be simplified to
K =2erT
T
(∫ S∗
0
1
K2P (S0,K, T )dK +
∫ ∞S∗
1
K2C(S0,K, T )dK
). (2.32)
The greeks of a variance swap are given by Hardle and Silyakova (2010) as
∆ = 2T−1(S∗−1 − St−1
), (2.33)
Γ = 2St−2T−1, (2.34)
Θ = −σ2T−1, (2.35)
ν = 2σ(T − t)T−1. (2.36)
80 85 90 95 100 105 110 115 120−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Price Underlying
Delta
4 dtm
12 dtm
30 dtm
(a) Delta Variance Swap K=100, σ=0.3, r=0.75%
80 85 90 95 100 105 110 115 1200
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Price Underlying
Gam
ma
4 dtm
12 dtm
30 dtm
(b) Gamma Variance Swap K=100, σ=0.3, r=0.75%
Figure 2.3: Evolution of the Greeks for the Variance Swap.
3No discontinuous jumps.
18
2.3.3 Volatility dispersion trading and correlation trading
Volatility and variance swaps provide pure exposure to volatility with low sensitivity to the
direction of the underlying asset, i.e. low delta and gamma risk. Therefore, a dispersion trade
initialised with variance swaps can disclose some important properties of the relationship be-
tween dispersion trading and correlation trading. The payoff at time T of a long dispersion
trade using variance swaps is given by
ΠLD =
(n∑i=1
Ni
2Ki(σ2i −K2
i )
)− NI
2KI(σ2I −K2
I )
=1
2
(n∑i=1
Niσ2i
Ki−NIσ
2I
KI
)+
1
2
(NIKI −
n∑i=1
NiKi
),
(2.37)
where NI is the vega notional of a volatility swap on the index and Ni is the vega notional of
a volatility swap on stock component i ∈ (1, . . . , n). As will be explained in the next section,
ρ = σI/ (∑n
i=1wiσi), can be seen as a proxy for the average correlation of a market index.
When this statistic is substituted in the last line of Eq. (2.37), it is obtained that
ΠLD =1
2
(n∑i=1
Niσ2i
Ki−NI ρ
2 (∑n
i=1wiσi)2
KI
)+
1
2
(NIKI −
n∑i=1
NiKi
). (2.38)
Differentiating Eq. (2.38) with respect to ρ gives
∂ΠLD
∂ρ= −
NI ρ (∑n
i=1wiσi)2
KI≤ 0, (2.39)
and hence a long volatility dispersion trade corresponds to short selling correlation.
Eq. (2.38) is also differentiated with respect to the single stock volatility,
∂ΠLD
∂σj=NjσjKj
−wjNI ρ
2∑n
i=1wiσiKI
. (2.40)
If it is assumed that that the sum of the single stock vega notional is equal to the negative of
the index vega notional, i.e. the dispersion trade is vega neutral, and wi = Ni/NI is the weight
for vega-neutrality for stock component i ∈ (1, . . . , n), Eq. (2.40) can be simplified as
∂ΠLD
∂σj= Nj
(σjKj−ρ2∑n
i=1wiσiKI
). (2.41)
When the proxy for average correlation is evaluated with the implied volatilities, denoted by ρ,
and assuming t = 0 such that the value of the variance swap equals zero, the latter equation
can be written as
∂ΠLD
∂σj= Nj
(σjKj−ρ2∑n
i=1wiσiρ∑n
i=1wiKi
)= Nj
(εj −
ρ2ζ
ρ
),
(2.42)
19
where εj = σj/Kj is the ratio of realised and implied volatility for stock component j ∈(1, . . . , n), and ζ = (
∑ni=1wiσi) (
∑ni=1wiKi)
−1. Hence the payoff is non-decreasing in the
volatility of stock j if εj ρ ≥ ρ2ζ. In the special case that the implied volatility risk premium
is roughly a constant proportion of the implied volatility at t = 0, i.e. v = (Ki − σi)/Ki, ∀i ∈(1, . . . , n), hence the bias in the implied volatility is the same for each stock and index, we can
rewrite Eq. (2.42) as
∂ΠLD
∂σj= Nj
((1− v)Kj
Kj−ρ2∑n
i=1wi(1− v)Ki
ρ∑n
i=1wiKi
)= Nj(1− v)
(1− ρ2
ρ
).
(2.43)
The right-hand side of the latter equation is positive when ρ2 < ρ, which is most likely the case
because it is reasonable to assume that the implied correlation is close to the realised correlation.
Furthermore, the only non-trivial way in which Eq. (2.43) is equal to zero is when ρ2 = ρ = 1.
It may be clear that in general the single stock volatility exposure is non-zero and thus a long
volatility dispersion trade is not equal to a perfect short correlation trade.
2.4 Volatility and correlation
Determining the price of a basket of options is not an effortless exercise. In general there
does not exists an explicit expression for the value of a weighted sum of options due to the
correlation between the price movements of the underlying assets. Unless these underlying
assets are perfectly correlated, an index option typically costs less than the basket of options
on each of the individual assets within the index. The concepts of volatility and correlation of
assets will be explored in this section.
2.4.1 Portfolio variance
The variance of a portfolio consisting out of n securities, with σi the volatility of the ith security,
wi its corresponding weight in the portfolio and ρij the correlation between the ith and the jth
security, using the modern portfolio theory of Markowitz (1952) can be written as
σ2p =n∑i=1
w2i σ
2i +
n∑i=1,j 6=i
wiwjσiσjρij . (2.44)
Because the correlation between two securities, ρij , is in absolute value between zero and one,
the maximum variance of the portfolio is attained when all underlying securities are perfectly
positively correlated and thus ρij = 1 ∀i, j. Hence the variance of a portfolio is reduced by
including mutually uncorrelated securities and this embodies the concept of diversification and
the reason why index options generally do not have the same price as the corresponding weighted
sum of single security options.
The standard deviation of a portfolio can be computed by taking the square root of the
portfolio variance (2.44). However, in the case of a market portfolio, which can be seen as a
20
portfolio where the unsystematic risk is diversified away and hence only exhibits systematic
risk, a simpler approach can be used as shown by Marshall (2009). By using the properties of
beta, which is a measure of systematic risk in equity markets,4 she shows that the volatility of
a portfolio containing only systematic risk can be written as5
σm =n∑i=1
wiσiρi,m, (2.45)
and this is called the modified Markowitz equation.
2.4.2 Implied correlation
The implied correlation is an average correlation measure and adequately indicates the difference
between the index implied volatility and the weighted average implied volatility of the basket of
underlying assets. For an index which is not necessarily a market index the average correlation
can be found by solving Eq. (2.44) for a constant correlation parameter ρ, hence it is given by
ρ =σ2p −
∑ni=1w
2i σ
2i∑n
i=1,j 6=iwiwjσiσj. (2.46)
One obtains the implied correlation by evaluating Eq. (2.46) with the implied volatilities. More-
over, using the the modified Markowitz equation (2.45), a good proxy for the average correlation
of a market index is given by
ρ =σm∑n
i=1wiσi, (2.47)
and the implied correlation is approximated by the evaluation of the latter equality with implied
volatilities.
It was found that a vega-neutral dispersion trade is not equal to a pure correlation trade. Having
defined the implied correlation it is it is insightful to approximate this spread using Eq. (2.44)
and an index variance swap with one variance unit (i.e. N = 1),
σ2p − σ2p =
n∑i=1
w2i σ
2i −
n∑i=1
w2i σ
2i +
n∑i=1,j 6=i
wiwjσiσjρ−n∑
i=1,j 6=iwiwj σiσj ρ
=n∑i=1
w2i
(σ2i − σ2i
)+
n∑i=1,j 6=i
wiwj (σiσjρ− σiσj ρ)
=
n∑i=1
w2i
(σ2i − σ2i
)+
n∑i=1,j 6=i
wiwj (σiσj (ρ− ρ)− (σiσj − σiσj) ρ)
=
n∑i=1
w2i
(σ2i − σ2i
)+
n∑i=1,j 6=i
wiwjσiσj (ρ− ρ)−n∑
i=1,j 6=iwiwj (σiσj − σiσj) ρ,
(2.48)
4This simplification is not generally applicable to other markets than equity markets by the definition of beta.5The subscript of the portfolio has changed from p to m compared to Eq. (2.44) to pinpoint that the portfolio
only contains the systematic market risk.
21
where σi and σi are the realised and implied volatility for i ∈ (1, . . . , n), respectively, and ρ
and ρ are the average correlations evaluated with the realised and implied volatilities. Now the
latter equation can be rewritten as
n∑i=1,j 6=i
wiwj (σiσj − σiσj) ρ =
n∑i=1
w2i
(σ2i − σ2i
)+
n∑i=1,j 6=i
wiwjσiσj (ρ− ρ)− (σ2p − σ2p). (2.49)
The right-hand side of Eq. (2.49) can be interpreted as the payoff of a short index variance
swap, long single stock variance swaps and a correlation swap. In other words, the payoff of a
specific long dispersion trade, i.e. a short position in correlation, and a long correlation swap.
Hence the left-hand side defines the considered spread.
The implied correlation is a measure for the market’s expectations of future correlation and it
reflects the changes in the relative premium between index and stock options. Also, it indirectly
expresses the implied volatility spread, but with the advantage that it is independent of the
current level of volatility. It can therefore be used to identify opportunities in which a mispricing
of implied volatility has created a disparity between the implied volatility of the index and its
components. Another measure of identifying remunerative volatility dispersion trades is found
by setting ρij in Eq. (2.44) equal to one, in this way one derives an upper bound for the
variance and volatility of a portfolio. The difference between the square root of both sides of
this expression can be seen as the volatility dispersion statistic,
D = σp −n∑i=1
wiσi. (2.50)
The implied correlation and the volatility dispersion statistic can both be used to describe the
dispersion trading opportunities. A long dispersion position corresponds roughly to a short
position in correlation and vice versa. The volatility dispersion statistic directly specifies an
upper bound for the volatility spread and can trivially be used for identifying a dispersion
trade, however it is a distance measure between two implied volatilities and therefore contains
less information than the implied correlation.
2.5 Tracking portfolio
Until now it was assumed that a dispersion trade was done by taking positions in derivatives
on both the index as well as the corresponding basket of constituents. However, if discrepancies
in implied volatility exist, they are marginal and a dispersion trade on all the constituents
of an index is not profitable due to the transactions costs from initiating and hedging the
positions. Referring to Subsection 2.1.3, one possible solution to this problem is creating a
tracking portfolio which mimics the characteristics of the index with a minimum amount of
securities and hence this comes down to a trade-off between transaction costs and a correlation
risk of the tracking portfolio with the market index. Because a tracking portfolio is based on
the history of returns of the constituents of an index, it is backward looking and hence choosing
22
the amount of securities in the tracking portfolio too small can lead to uncorrelated behaviour
of the short and long leg in a dispersion trade and therefore uncontrolled payoffs.
In the process of constructing a tracking portfolio one needs to decide which securities to
include and which weight they should have. Because index tracking is a low-cost alternative to
active portfolio management and financial institutions even publicly offer index tracking funds,
such as ETFs, the problem of creating a tracking portfolio is well investigated by academics
and practitioners alike. The complexity of this problem can widely vary, depending on the
given restrictions and objectives. However in the general case, without a prespecified number
of securities in the tracking portfolio, this corresponds to a conjunction of a combinatorial and
a continuous numerical problem, where both problems need to be approached simultaneously
(Krink et al., 2009).
A relatively simple approach to creating a tracking portfolio is based on principal component
analysis (PCA), herein one decomposes the sample covariance matrix of the returns into pairs
of eigenvalues and eigenvectors ordered by importance. Hence the ith principal component of a
return vector r is the linear combination yi = wi′r that maximises V ar(yi) with the constraints
that wi′wi = 1 and Cov(yi, yj) = 0 ∀i 6= j (Tsay, 2010). Then the first n principal components
are chosen such that the cumulative proportion of variance of these principal components is large
enough.6 More advanced methods such as DECS7 (Krink et al., 2009) use search heuristics to
encounter simultaneously the combinatorial problem of choosing the number of securities.
6e.g. more than 90%.7Differential evolution and combinatorial search.
23
Chapter 3
Methodology
A dispersion trading strategy can be implemented and tested in various ways, yielding different
results and conclusions on the P&L of a dispersion trade. In this chapter it is explained what
methods are used for this research and how they can be blended into a complete dispersion
strategy; furthermore, testing methods for the results are described.
3.1 Overview
Practically initiating a dispersion trade comes down to two steps. The first step is selecting a
weighting scheme. In this thesis a tracking portfolio is used which is expected to display the
same characteristics as the market index over the period where a dispersion trade is active.
Thereafter, the second step is to formalise, execute and maintain a trading strategy based on a
specific information set available, which could be no information whatsoever and consequently
a naive trading strategy is entered into, or the information set could contain several signals
of the market available on that date. In most of the existant literature on dispersion trading,
the focus is on whether the market shows volatility dispersion, and if so, naive positions in
financial derivatives are used to take advantage of this discrepancy in the market. However,
there hardly exists any academic research on how market signals can be used in determining the
position in a financial product and the purpose of this thesis is to contribute to the knowledge
of dispersion trading in this direction. Furthermore it is of interest to know whether different
optimisation methods for a tracking portfolio yield significant different results. In this chapter
each component of the research method followed is discussed separately.
24
3.2 Tracking portfolio
In this study two tracking portfolio optimisation methods are used: the PCA method and the
DECS method. Both were already touched upon in Section 2.5, but in this section they are
explained in further detail.
3.2.1 PCA analysis
Principal component analysis (PCA) is a statistical method to explain the structure of the
covariance matrix or equivalently the correlation matrix of a multidimensional random variable
with a few linear combinations of the components of this random variable. Likewise, it is a
popular tool for dimension reduction of a multidimensional random variable without significant
loss of information (Tsay, 2010). The main procedure of the PCA method is decomposing the
covariance-/correlation matrix into its eigenvalues and eigenvectors, where the eigenvalues and
corresponding eigenvectors are ordered by their importance. The principal components are then
defined as the product of the eigenvectors and the multidimensional random variable minus its
mean vector (Su, 2005).
To be more specific, consider a k-dimensional return vector r and wi = (wi,1, ..., wi,k)′ a
k-dimensional real-valued vector, i = 1, ..., k, satisfying (Tsay, 2010):
E(r) = µ and V(r) = Σr,
yi = wi′r i = 1, ..., k,
Cov(yi, yj) = wi′Σrwj , i, j = 1, ..., k.
The idea of PCA is to find linear combinations wi such that yi has maximal variance and yi and
yj are uncorrelated for i 6= j. But since the covariance matrix is positive definite, it has a spectral
decomposition1 and therefore wi = ei for i = 1, ..., k, where ei is the ith normalised eigenvector
corresponding to the ith eigenvalue λi of Σr, ordered with respect to their importance.
Practically, for this study this means that at the moment of time when a tracking portfolio is
created, one needs to decompose the covariance matrix into its eigenvalues ordered in significance
and the associating eigenvectors. The first m principal components are then chosen such that the
cumulative variance proportion2 is greater than or equal to a pre-specified percentage. Within
these m principal components, the N∗ most prevailing stocks of the index are chosen to form
the tracking portfolio by evaluating their cumulative squared correlation (Su, 2005)
N∗∑j=1
q2i,j =
∑N∗
j=1 λjγ2i,j
σ2i,
where γi,j is the (i, j)th element of the ordered eigenvector matrix of Σr.
1Σr = PΛP ′, where Λ and P are the diagonal eigenvalue matrix and the corresponding eigenvector matrix
respectively, with the eigenvalues in descending order.2i.e. the sum of the first m eigenvalues divided by the total sum of all the eigenvalues, because in this case
V(yi) = wi′Σrwi = wi
′PΛP ′wi = ei′PΛP ′ei = λi.
25
3.2.2 Differential evolution and combinatorial search
Although the simplicity and the low computational complexity are the main advantages of
principal component analysis, it is very restrictive in the sense that it does not tackle the index
tracking problem in a simultaneous search for a selection of optimal assets in combination with
associating optimal weights. This is important because in determining whether a selection of
assets is optimal in mimicking the index characteristics, the result depends on the positions in the
assets taken and vice versa. Moreover, non-linear constraints such as minimum and maximum
holding positions in a single asset can not be solved with PCA. Because in a dispersion trading
strategy the tracking portfolio is a critical element, it is of great interest to investigate whether
a more sophisticated method for constructing a tracking portfolio would yield significantly
different results from the PCA method.
Because of the duality in the optimisation problem, quadratic programming can not be used.
An alternative is therefore to use search heuristics, which iteratively searches for a superior solu-
tion within a problem. The main advantage of using search heuristics is that various constraints
can easily be implemented, and that optimisation is based on a single evaluation criterion, such
as a distance measure.3 The disadvantage of search heuristics is that a problem may require
many iterations and that its rate of convergence and consequently the accuracy of the solution
is poor. There are many notorious examples of search heuristic optimisation methods, e.g.
particle swarm optimisation, genetic algorithms, simulated annealing and differential evolution,
and although most of them are inspired by biological and sociological motivations (Abraham et
al., 2008), they can resolve many different problems. However as shown by Kennedy and Eber-
hart (1995), differential evolution has very good performance in continuous numerical problems
compared to the others, and Krink et al. (2009) complements on this field by proposing a hybrid
model for index tracking, namely the differential evolution and combinatorial search (DECS)
algorithm. This method combines differential evolution with combinatorial search to determine
the optimal subset of assets in a tracking portfolio. Because Krink et al. show that it can deal
with non-continuous numerical problems and moreover the focus of this method is to construct
a tracking portfolio of an index, DECS will be used in this study as the competitor of PCA.
DECS is a variant on differential evolution, where the latter is a population based search heuris-
tic. This means that it generates an initial population, P, of possible solutions which it itera-
tively refines by the following procedure:
1. For each element of the population P(j), three other elements: P(x), P(y) and P(z), are
selected randomly. Subsequently a new candidate solution, c, is created by a combination
of the three random selected candidate solutions of the population, together with scaling
factor, f :
c = f · (P(x) - P(y)) + P(z).
3e.g. the tracking error.
26
2. c is substituted by a recombination between P(j) and c, where each component, o, of the
recombination is equal to P(j, o) with probability 1-cf and c(o) with probability cf , cf
is the crossover factor.
3. The new candidate solution c∗ is substituted in the population for P(j), if c∗ has better
fitness in the criterion function.
This general procedure is in the DECS extended with a position swapping procedure in which
with a certain probability an asset with a zero weight allocation is swapped with an asset
with a non-zero weight allocation. Furthermore it is supplemented with constraint violation
handlings.4
The index tracking problem which is considered in this thesis as a test against PCA and solved
with DECS is given by (Krink et al., 2009)
minimisew
f0(w) =
√√√√ 1
T
T∑t=1
(RPt −RBt )2,
subject to
n∑i=1
wi = 1,
|wi| ≤ 1, i = 1, . . . , n,
εiδ(wi) ≤ wi ≤ ξiδ(wi), δ(wi) =
1 if wi > 0
0 elsei = 1, . . . , n,
L ≤n∑i=1
δ(wi) ≤ K, i = 1, . . . , n,∑i:wi>Lb
wi ≤ Ub,
n∑i=1
|∆wi| ≤M,
where:t = 1, . . . , T Time period considered.
Rxt Return of either the benchmark (B) or the tracking portfolio (P ).
w Real valued vector of asset weights in tracking portfolio, n-dimensional.
εi, ξi Lower and upper bounds for single asset weights.
L,K Lower and upper bounds for the total assets in tracking portfolio.
Lb Lower threshold for classifying asset weights as large.
Ub Maximum proportion of large asset weights in tracking portfolio.
M Maximum deviation from previous weight allocation.
Thus the tracking error is the criterion function in which the population is evaluated and
4e.g. a penalty function for constraint violations.
27
iteratively improved based on realistic constraints on the asset allocations, to limit market
impact and transaction costs. The PCA method can not tackle these constraints. The specific
values of the parameters are given in Appendix B.
3.3 Implied volatility
The volatility of a stock is not directly observable by the market data. If high-frequency data is
used in estimating the volatility of a stock, e.g. intraday data, the prices are generally measured
with market microstructure noise,5 leading to an MA(1) effect in the returns and hence the
estimated realised volatility over one trading-day is inaccurate.6 However in option markets, if
one assumes that the prices are ruled by an econometric model such as the Black-Scholes model,
then using the price of an option one is able to solve the volatility parameter from the model,
i.e. the implied volatility. The advantage of implied volatility compared to the realised volatility
is that it is forward-looking, assuming that information is processed in the market immediately.
A disadvantage of using implied volatility is that a mathematical option pricing model is used
and therefore a specific process of the underlying asset is assumed, the implied volatility may
therefore differ from the actual volatility. In this section it is explained which pricing model is
used for index and single stock options and how the implied volatility is derived from such a
model.
3.3.1 Index options
Most index options are of the European type. Referring to Section 2.2, European options
are plain-vanilla options and relatively easy to value because their pay-off at maturity is not
path dependent. Furthermore assuming that the underlying asset is governed by a geometric
Brownian motion this option can be priced with the Black-Scholes pricing formula. Because
the price of an option is increasing in the volatility of the underlying, having observed the
price of an option one is able to back out the unique implied volatility from the Black-Scholes
model by an iterative search procedure. If put-call parity7 is satisfied then the implied volatility
derived from European put and call options on the same underlying index, with the same time
to maturity and strike price, is the same (Hull, 2012). However by the volatility smile and term
structure generally observed in equity options, the implied volatility differs for options on the
same index but with different strike prices and maturities.
Because an index can be seen as a portfolio of a certain number of stocks, the main difficulty
in pricing index options is the dividend estimate of the index. The underlying stocks pay
different amount of dividends on different dates and only the dividend payments within the
option’s life must be considered and weighted accordingly. A relatively simple method to deal
5e.g. bid-ask spread, discreteness of price changes etc.6Financial Econometrics, University of Amsterdam, catalogue number 6414M0007Y.7p+ S0e
−qT = c+Ke−rT , where c and p are European call and put prices, q is the dividend yield, K is the
strike price and T is the time to maturity.
28
with index dividends is to substitute the index price level by the risk-free rate adjusted forward
prices of the index with the same time to maturity as the index options. Next to the dividend
issues, a risk-free interest rate with approximately the same time to maturity as the option must
be used because of the yield curve.8
3.3.2 Stock options
In contrast to index options, single stock options are American options and do usually not have
an explicit expression for the price. There do exists a number of pricing methods which can
be used to price these options, yet in this thesis the Cox-Ross-Rubinstein binomial tree model
(1979) is used. This method involves dividing the option’s life in small discrete time intervals,
wherein the underlying asset’s price can either go up or go down with a certain probability and
moreover the risk-neutral probability of an up (down) movement is the same on each upward
(downward) branch of the tree. Pricing an option with the binomial tree model is done by
backward-induction, more specifically: by first computing the binomial tree of stock prices, the
payoff function of the European option at time of maturity, T , is known and hence the delta of
the option at time T -1 can be calculated and therefore the price of the European option at time
T -1 under risk-neutral valuation. Using the same principle iteratively one can approximate the
price of the option at time 0, where the rate of convergence depends on the amount of periods on
the binomial tree. However, American options may be exercised prior to maturity and therefore
one can approximate the price of these options in the same manner as the European options by
additionally checking at each node of the binomial tree whether the payoff from exercising the
option is greater than the value of the option derived by backward-induction.
Likewise index options, issues when pricing single stock options stem from dividend payments
made by the stock in the period where the option is active in the market. Nonetheless, the
binomial tree can efficiently be adjusted both in the case of a continuous dividend yield as
discrete dividend payments on a stock. In the situation of a continuous dividend yield, the risk-
neutral probability of an up (down) movement in the binomial tree is adjusted, and when discrete
dividend payments are made on a stock, one can adjust the prices of the stock downwards in
the binomial tree with the dividend amount on the ex-dividend node (Etheridge, 2008). The
issues arising from the yield curve can be solved in the same way as index options by taking a
risk-free interest rate with approximately the same time to maturity as the option considered.
Because the holder of an American option has the chance but not the obligation to exercise
prior to maturity, as a consequence the value of an American option is always greater than or
equal to a European option. However, it is never optimal to early exercise an American option
with a zero interest rate and also not an American call option when the underlying asset is a
non-dividend-paying stock.
8The concept that the risk-free interest rate differs for different contract lengths.
29
3.4 Strategies
In this research options are chosen as the financial product to monetise the potential implied
volatility dispersion in the market, it is outside the scope of this study to evaluate market
signal trading strategies for more financial products such as variance swaps. A naive trading
strategy is constructed based on: (1) the straddle and (2) the strangle, and extended towards
more realistic signal-trading approaches based on the implied volatility spread, the (forecasted)
implied correlation and the volatility smile. Furthermore the effects of daily delta-hedging
and transaction costs on the P&L of the strategies are considered. This section describes the
strategies in the same order.
3.4.1 Naive strategy
The naive trading strategy is fueled by the existing theory that in general index options trade
against a premium compared to individual stock options. It is based on the following steps,
which are clarified thereafter:
1. At each first day of the month a tracking portfolio is constructed by either PCA or DECS.
2. At the end of each trading day a long dispersion position is initiated with either an OTM
strangle or an ATM straddle and with exactly 30-days to maturity. Thus a combination
is bought on the index and sold on each individual stock of the latest tracking portfolio,
with the same weights as in the tracking portfolio.
3. The positions are held until the maturity date of the option and hence in the core of the
strategy there are 30 active dispersion positions.
Tracking portfolio
The tracking portfolio is constructed with PCA or DECS on the first day of the month, based
on the 1-year historical covariance matrix of the daily log returns with as goal to mimic the
characteristics of the index in the remainder of the month as good as possible. The choice
of the history length is rather arbitrary and it is out of the scope of this study to test the
specific effects of using other covariance matrices because the vital question is whether there
is a significant difference in trading opportunities between the PCA and the DECS method.
However, a 1-year historical covariance matrix can be seen as compromise between accuracy
and the use of relevant information.
Combinations
As combinations of plain vanilla options are initiated at the end of each trading day within
the trading period, it is hard to isolate the effect of volatility dispersion on the P&L of these
different positions because market environments change during a trading day. The first step to
uniformity is to stabilise the volatility smile by the use of standardised strike prices (K/S) to
30
select which options to enter into within a combination (Hull, 2012), hence a theoretical ATM
straddle contains of a put and call option with a standardised strike price of 1 at inception.
Furthermore, in this research the OTM strangle is set equal to a put and call option with
standardised strike price of 0.95 and 1.05, respectively, which means that both options are
out-of-the-money with a strike price that differs 5% from the equity value at inception.
In practice, however, the effect of volatility dispersion is still not isolated by the use of
standardised strike prices because options trade with a very small probability exactly ATM or
5% OTM. For example options are only quoted in minimum strike price increments, such as:
e55, e60, e65, etc. This issue is overcome by using linear interpolation between options which
strike prices are on the moment of initiating the closest above and below the theoretical strike
price, such that a synthetic option is constructed with the exact theoretical strike price.
The other element of the volatility surface, the term structure, also needs to be addressed
because as explained in Subsection 2.2.4 an option’s price is usually increasing in the time to
maturity and consequently in order to isolate the volatility dispersion effect, the term structure
needs to be held constant. The same method is applied as the CBOE9 does for their VIX
index, which is an implied volatility index of the S&P 500 index, and the term structure of
the options is held constant at 30 days to maturity. In this procedure, the CBOE combines an
option with more than 30 days to maturity with an option with less than 30 days to maturity
to approximate an option with exactly 30 days to maturity.10 However options with less than
6 trading days to maturity are never used because market factors, e.g. demand and supply,
can stretch significantly the option prices between distinct strike prices on the same underlying
security. In the latter case the CBOE extrapolates between options which expire in the two
consecutive dates following the nearest expiring date. As a consequence, for an option to hold
the term structure constant one needs two options with the same strike price but with different
maturities.
All in all, to construct a synthetic option with a constant term structure and a standard-
ised strike price, four different options are needed. This means that to initiate a synthetic
combination, which can either be a strangle or a straddle, a trade in eight different options is
required.
Restrictions
Every dispersion trade entered at the end of a trading day is self-financing (see Subsection
2.1.4), moreover the initial value of the short leg, hence the long leg, is set equal to e100. In
this manner, the payoffs of the daily dispersion trades are directly comparable and not scaled
in the index level. Also, it is not allowed to trade both a strangle and a straddle in the same
naive dispersion trading strategy.
9Chicago Board Options Exchange.10The month wherein the option with the least days to maturity expires is called the front-month and the other
the back-month.
31
3.4.2 Position forecasting
As is indicated by the name, the naive trading strategy is not very realistic in the sense that it
simply assumes that the index options are overpriced compared to single stock options. However,
in the case the reverse occurs, the strategy tends to lose money and it is the inverse trade that
should be taken. A potential improvement on the naive strategy is therefore a strategy in which
the position in a dispersion trade is conditional on an implied correlation forecast from option
prices. The outline of this new strategy is as follows:
1. At each first day of the month a tracking portfolio is constructed by either PCA or DECS.
2. At the end of each trading day, the implied correlation is forecasted based on Bollinger
Bands or EGARCH models.
3. If future correlation is expected to grow significantly over the next 30 days, a short dis-
persion trade is entered into (i.e. long correlation position). In all other situations a long
dispersion trade is initiated.
4. The positions are held until the maturity date of the option and hence in the core of the
strategy there are 30 active dispersion positions.
This strategy is an extension on the naive trading strategy and hence the discussion from Sub-
section 3.4.1 stays valid unless stated otherwise.
EGARCH correlation forecast
An elegant way to forecast the future implied correlation is by a GARCH type of model. How-
ever, as explained in Subsection 2.2.4, the asset class equity exhibits generally the so-called
leverage effect and as a consequence the news impact curve11 is probably not symmetric. The
exponential GARCH (EGARCH) model is a GARCH variant which allows for leverage effects.
The procedure that is applied to estimate the implied correlation over the next 30 days is
the following. At the end of each trading day an EGARCH(1,1) model without ARMA terms12
is estimated for the index and the individual stocks, using daily log returns over the 5 years
prior this date.
rt = µ+ at,
at = σtεt,
log σ2t = (1− α1)α0 + θεt−1 + γ[|εt−1| − E(|εt−1|)] + α1 log σ2t−1,
(3.1)
where εt ∼ N(0, 1) and hence E(|εt|) =√
2/π. Subsequently, the volatility over the next 30
11The effect of shock, at, on σt+12, keeping σt
2 and the past fixed (Financial Econometrics, University of
Amsterdam, catalogue number 6414M0007Y).12Herein it is used that the conditional mean of daily returns is approximately zero (Hull, 2012).
32
days is forecasted as
FVt =30∑h=1
σ2t (h), (3.2)
σ2t (h) = σ2α1t (h− 1)exp((1− α1)α0)E[exp(g(εt+h−1))], (3.3)
E[exp(g(εt))] =1√2πe−γ√
2/π
∫ ∞−∞
eθε+γ|ε|−12ε2dε, (3.4)
g(εt) = θεt + γ[|εt| − E(|εt|)]. (3.5)
The 30 days volatility forecasts of the index and the components of the tracking portfolio
are then plugged into Eq. (2.47), i.e. the implied correlation using the modified Markowitz
equation. This less cumbersome version of the implied correlation can be used when a portfolio
is structured to replicate the benchmark index (Marshall, 2008), hence it is applicable to the
index tracking portfolio.
A short dispersion position is entered if the future correlation (FC) is expected to grow
significantly. Assuming that the implied correlation between the index and the tracking portfolio
is a bounded process with mean-reversion characteristics, the rolling 30-day historical standard
deviation of the implied correlation is denoted by
MSt =
√√√√ 1
30
29∑i=0
(ICt−i − ICt,t−29)2, (3.6)
where ICt and ICt,t−29 are the implied correlation and the rolling 30-day mean implied corre-
lation at time t respectively. If and only if the inequality
FCt > ICt +MSt (3.7)
is fulfilled a short dispersion trade is entered at the end of day t, and a long dispersion trade
otherwise. Assuming the tendency for an index implied volatility premium, i.e. the existence
of an implied correlation premium, the forecasted correlation needs to be at least one rolling
standard deviation higher than the implied correlation.
Bollinger Band forecast
Another way to forecast the future implied correlation is with Bollinger Bands, i.e. a simple
MA model with standard deviation bands. Likewise the previous trading rule it is assumed that
the spread between the implied volatilities of the index and the tracking portfolio will converge
back to an arbitrary long-term mean, entailing the same properties for the implied correlation.
Consequently, if the implied correlation from Eq. (2.47) is found to be too far down from its
rolling mean, a short dispersion trade is entered. In this study the two standard deviation, 30-
day period Bollinger Band for the implied correlation is used and a short position in a dispersion
trade is entered at the end of day t if and only if
ICt < ICt,t−29 − 2 ·MSt, (3.8)
and a long dispersion trade otherwise.
33
3.4.3 Combination forecasting
Until now it was not allowed to trade both an OTM strangle and an ATM straddle in the
same strategy, hence it was indispensable to trade the same combination over the entire trading
period. From this point this restriction is loosened because although the strangle and the
straddle are very similar, they have different properties which can be used to enhance the P&L
of a dispersion trade. Because a dispersion trading strategy is market-neutral in its purest form,
the difference in delta properties between the strangle and straddle do not matter if the tracking
portfolio has a good fit with the benchmark index. However, a switching strategy based on the
volatility smile may augment the payoff from a dispersion trade by buying volatility where it is
cheap and selling volatility where it is expensive. The outline of this strategy is as follows:
1. At each first day of the month a tracking portfolio is constructed by either PCA or DECS.
2. At the end of each trading day a long dispersion trade is initiated with a portfolio of
strangles and straddles, where the latter is based on the slope of the individual volatility
smiles of the index and the components of the tracking portfolio.
3. The positions are held until the maturity date of the option and hence in the core of the
strategy there are 30 active dispersion positions.
This strategy is an extension on the naive trading strategy and hence the discussion from Sub-
section 3.4.1 stays valid unless stated otherwise.
Trading the slope
Denote the standardised strike price of an option by m = K/S; the slope of the volatility smile
between m1 and m2, m1 ≥ m2 at time t can then be defined as
βt =σm1(t)− σm2(t)
m1 −m2. (3.9)
Also define
∆βt = βt − βt−1, (3.10)
which is simply the daily change in the slope Eq. (3.9). If the volatility smile pattern is believed
to describe a stable long-term relationship between the implied volatility and the standardised
strike price, then it is a reasonable assumption that (3.10) is a mean-reverting process (this
assumption is studied in chapter 5 of this thesis). Suppose in this case that from the end of day
t to t+1 the slope of the volatility smile between m1,m2 increases: βt+1 > βt, then
σm1(t+ 1)− σm2(t+ 1) > σm1(t)− σm2(t). (3.11)
This means that the distance between the implied volatilities increases from day t to t+1.
For this study the only points on the horizontal axis of the volatility smile which are of interest
are 0.95, 1 and 1.05, because of the uniformity in the standardised strike prices of options in
34
the OTM strangle and ATM straddle. Hence the slopes of interest are
β∗t =σ1.00(t)− σ0.95(t)
0.05, (3.12)
β∗∗t =σ1.05(t)− σ1.00(t)
0.05. (3.13)
An OTM strangle position is entered on the index or on one of the components of the tracking
portfolio if the left slope of the volatility smile, i.e. Eq. (3.12), is attractive low for put options
with a constant maturity of 30 days. The reason that the slope for OTM put options is used
to determine the combination is that the slope of the volatility smile is generally much less
pronounced for OTM equity call options than for OTM equity put options (see Subsection 2.2.4).
To qualify the slope as sufficient attractive, the rolling 30-day historical standard deviation for
the slope βt is defined as
MSt =
√√√√ 1
30
29∑i=0
(βt−i − βt,t−29)2, (3.14)
where βt,t−29 is the rolling 30-day mean of the slope βt at time t. Then an OTM strangle
position is taken on the index or on one of the components of the tracking portfolio if and only
if both of the inequalities
β∗t > β∗t,t−29 +MS∗t for put-options,
β∗∗t < β∗∗t,t−29 for call-options
(3.15)
are satisfied, and an ATM straddle otherwise.
Thus the volatility smile flattens compared to its rolling 30-day average, where the left slope
is at least one standard deviation larger than its moving average.
3.4.4 Remarks
Up until now, none of the three strategies used a form of delta hedging to reduce the exposure
to price changes of the underlying stock or index. As explained in Subsection 2.2.3, both the
ATM straddle and the OTM strangle are delta neutral at the moment of inception, nevertheless
as the market environment changes a (small) delta exposure on these combinations will arise
and need to be hedged. Therefore a daily delta hedge procedure will supplement the three
strategies, i.e. each individual option within a portfolio is daily delta hedged. Because dynamic
hedging of the Greeks is in itself a complicated process and not the essence of this research,
other forms of hedging will not be considered.
Discrepancies in the implied volatility between an index and the underlying components are
only a violation of the Law of One Price when transaction costs have been taken into account.
The main components of the transaction costs are: the market impact, the commission and
the bid-ask spread (Marshall, 2008). Because the market impact and the commission can be
reduced,13 the bid-ask spread is the principal element of the transaction costs. The effect of the
bid-ask spread on the P&L of the strategies is modeled in this study.
13Large hedge funds can negotiate commissions and computer algorithms can buy securities little by little to
prevent shocks.
35
3.5 Tracking P&L
This section summarises the methodologies used in this study to evaluate the daily portfolio
returns, together with its concerns.
3.5.1 Evaluation
In the first place the returns need to be defined. As explained in Subsection 3.4.1, each day a
self-financing dispersion trade is initiated and held until the maturity time, i.e. 30 days, where
the value on the short leg and hence the long leg at inception is e100. In the core of the strategy
the portfolio exists out of 30 different active dispersion trades14 and therefore a total value of
e3000 is invested in both the short and the long positions of the operating portfolio. The daily
P&L is measured by a margin-account principle and the daily returns are obtained by the daily
gains or losses on the margin account divided by the size of the total amount of short positions,
i.e. rst = πt−πt−1
πst, where πt is the total value of the margin-account and πst is the size of the short
leg at time t. The reason why simple returns are used instead of log returns is that it is expected
that the returns of dispersion trading are high-peaked and fat-tailed because the high degree of
leverage created within the portfolio,15 hence log returns can become disproportionally negative
for days with significant losses, magnifying the kurtosis and therefore leading to an unrealistic
importance on the overall performance measures.
Evaluation of the return series is done by the Sharpe Ratio (SR). This metric is named after
Sharpe (1966) and defines the expected excess return of a portfolio with the risk-free interest
rate to its return volatility, hence it is a measure for the efficiency of a portfolio. It displays
to what extent the excess return received on a portfolio is exchanged for additional risk. The
ex-ante SR is defined as
SR =E[rt]− rf√
V[rt], (3.16)
where E[rt] and V[rt] are the expected value and the variance of the return series rtTt=1
respectively and rf is the constant risk-free interest rate over the sample period t ∈ (1, . . . , T ).
Thus the SR assumes that the risk-free interest rate is an appropriate comparable return
for the strategy return series. However, in this study the risk-free interest rate should be
substituted for an adequate benchmark asset return series in order for the comparison to be
fair and not obtaining artificially inflated values of the SRs. Because dispersion trading is a
statistical arbitrage strategy which aims to profit from market discrepancies, the index return
series is used as benchmark in the definition of the SR. However, since the index is time-varying,
in Eq. (3.16) the numerator is changed to the expected value of the excess return series and the
denumerator is changed to the standard deviation of the excess return series.
14On the first day of the trading period one dispersion trade is entered, on the next day another etc., hence
after 30 days the portfolios exists out of 30 different dispersion trades.15A high form of leverage is created because the use of self-financing positions and the unlimited downside
potential of short selling combinations of options.
36
In order to estimate the SR for an observed return series, rtTt=1, the population moments are
replaced by their sample moments. Based on these estimates one obtains an approximation
for the daily SR, however it is common to calculate this ratio on a yearly basis. The daily
SR is therefore annualised by multiplying it with√
252, however this is an approximation and
without taking into account for any serial correlation in the daily financial returns, it is most
likely incorrect. Furthermore, the SR assumes that the return series of interest is approximately
normally distributed. As explained earlier, due to the fact that dispersion trading is based on
utilising the discrepancies in implied volatilities between an index and the underlying compo-
nents and because the high amount of leverage in the portfolio, it is likely that the daily return
series are serial correlated and exhibit excess kurtosis. One solution to this problem is to replace
the regular sample standard deviation in Eq. (3.16) by the standard deviation corresponding to
the long-run variance, or Newey-West (1987) heteroskedasticity and autocorrelation consistent
(HAC) standard deviation, given by
σ2 = γ0 + 2L∑j=1
wj γj , (3.17)
wj = 1− j
1 + Lfor j ∈ (1, . . . , L), (3.18)
where γ0 is the sample estimate of the variance of the (excess) return series rtTt=1, γj is the lag-
j sample covariance series, and wj are the Bartlett weights for j ∈ (1, . . . , L) and t ∈ (1, . . . , T ).
Several papers have investigated this long-run variance estimator and under certain conditions
specified by Giraitis et al. (2003), Berkes et al. (2005) show that this estimator is almost sure
consistent in the case of weak dependence and in the case of long-memory. However, the
number of lags, L, must be high enough to capture the most significant lag-dependency within
the process. In this study the lags are chosen based on the rule-of-thumb approach by Greene
(2002), which states that L ≈ T 0.25.
To further investigate the daily return series of the strategies, various tests are implemented
to examine the characteristics of the data. Autocorrelation is tested by regressing the return
series on a constant after which a Breusch-Godfrey LM test is applied on the first two lags, hence
this LM-statistic has a χ2-distribution with two degrees of freedom. Also, the non-normality
in the daily returns is validated with a Lilliefors test and the CAPM model is estimated to
test the market-neutrality assumption, i.e. β = 0. From these tests it can be invoked whether
the Central Limit Theorem can be applied to the daily return series such that the regular
t-statistic can be applied to test whether the average daily returns are significantly different
from zero. Nonetheless, (stationary) bootstrapping is used to approximate the distributions of
several statistics by Monte Carlo simulations.
It is imaginable, although this is already minimised by the use of simple returns, that the
return series show signs of extreme excess kurtosis and long-memory such that the possibility
exists that the asymptotic variance is not defined. This is tested based on the extreme value
theorem, by estimating the tail index, 1/ξ, of a return series with the Hill (1975) estimator.
37
When a return series is assumed to be weakly dependent, the Hill estimator can directly be
applied to this time series (Tsay, 2010). In the case of autocorrelation the Hill estimator is
applied to the residuals of the specified AR(p) process. Denoting by z(i)Ti=1 the descending
order statistics of the series, ZtTt=1; the number of order statistics, u, to estimate the tail index
with are chosen such that the fraction k/T of ZtTt=1 exceeds u, with k/T > 1 − q. Herein,
q ∈ (0, 1) and T are the total number of observations in this study. The estimate of the tail
index is defined as
1
ξ=
11k
∑kj=1 [log(zj)− log(zk+1)]
. (3.19)
By taking the negative of the series ZtTt=1, the tail index of the other side of the distribution
can be estimated in the same manner. The limitations of the Hill estimator are that it is
applicable to the Frechet distribution only and that the outcomes are strongly depend on the
choice of k. The asymptotic distribution is given by Hall (1982) as
√kα(k)− αα(k)
a∼ N(0, 1), (3.20)
where α = 1/ξ. Thus by using the critical values of the standard normal distribution, the
assumption of finite variance may be tested with, H0 : α = 2, against the alternative that
Ha : α < 2.
38
Chapter 4
Data
Despite the fact that many data sources are available for financial historical data, not many
of these can be used to evaluate trading strategies based on derivatives. Financial derivatives
require that additional data is used apart from the regular bid-ask spreads, e.g. liquidity in-
formation, time to maturity and the price of the underlying security. Because in this study
synthetic options are used to utilise the potential discrepancies in implied volatilities between
an index and the underlying components, many different options are needed to construct a
dispersion trade and as a consequence an intricate data structure is required to perform a valid
analysis.
The data is acquired from Thomson Reuters Datastream Advance. Daily closing prices of
the CAC 40 (Cotation Assistee en Continu) index and it constituents over the period of the 1st
of January 2006 until the 31st of July 2012 were downloaded, together with their information
about dividends and splits, outstanding shares, market capitalisation, turnover and futures
contracts. Furthermore, end of the day prices and characteristics of all options on the CAC 40
index and all stock options of companies listed on the CAC 40 index from the 1st of January
2010 until the 31st of July 2012 were obtained over the latter period. The 1-month Euribor
rate is used for the risk-free interest rate.
The CAC 40 is a French capitalisation weighted benchmark index that reflects the perfor-
mance of the 40 most significant and actively traded stocks of the Euronext Paris. Hence the
constituents of the CAC 40 index are large-cap stocks and are generally more liquid and less
volatile than small-cap or even mid-cap stocks. Because the CAC 40 is a fairly small index,
which is likely to mirror the state of the economics as a whole and furthermore since no aca-
demic research is published in the field of dispersion trading on this index, it is used in this
study.
The obtained financial data was checked for missing data points and eventual irregularities,
after which steps were taken to repair or to remove these specific data points. First the stock data
over the entire sample period was examined to find non-trading days, such as public holidays.
These days were removed from the data set for both the stock and option prices. Subsequently,
the prices of options were imported in Matlab and algorithmic investigated on missing data
39
points, i.e. gaps between daily option prices. The issue of absent data was addressed by linear
interpolation between the two days adjacent to the missing data point. Stock delisting is not
an issue for this research because options on this security continue to trade until they expire.
The dataset can be divided in two mutually exclusive periods: a training period from 01-01-
2006 to 31-05-2010 and a trading period from 01-06-2010 to 31-07-2012.1 The training period
is used to investigate the properties of the data and the parameters of the tracking portfolio
optimisation methods. Subsequently the trading period is employed to run the different trading
strategies and to evaluate their performances. The reason that the trading period only consists
of two and a half years of data is that the required option data is really voluminous; millions of
option prices were downloaded over this period.
The trading period consists out of 523 trading days and it experiences the following market
cycles:2
1. Bull market - June 2010 to February 2011
2. Bear market - March 2011 to November 2011
3. Bull market - December 2011 to February 2012
4. Bear market - April 2012 to July 2012
It is therefore interesting to investigate how the strategies perform under different market condi-
tions and whether the strategies show features of market neutrality or exhibit a serious market
risk.
Figure 4.1: The market conditions of the CAC 40 index during the trading period.
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12 Jul122600
2800
3000
3200
3400
3600
3800
4000
4200
Date
Index
Bull Market Bear Market Bull Market Bear Market
1Because synthetic options are created from options with different maturities, it is only possible to trade until
mid of June 2012.2Following the definition of a market change of at least 20% over a minimum period of 2 months.
40
Chapter 5
Evaluation
Hitherto, the concepts of dispersion trading were explained and several potential dispersion
trading strategies were epitomised. In this chapter, however, the theory on volatility dispersion
trading is investigated and evaluated on the French Bellwether index, the CAC 40. First the data
of the training period are analysed based on the historical return characteristics of the securities,
the volatility surface implicit in option market prices and the parameters of the PCA and DECS
method are tuned. Thereafter in the trading period, results of the sundry dispersion trading
strategies are apprised with a back-test and evaluated based on their performance and validity.
Possible considerations within the evaluation process of the dispersion trading strategies are
also touched upon.
5.1 Preliminary results and analysis
In this section the data of the training period are described and evaluated on its characteristics.
Furthermore, the parameters of the tracking portfolio optimisation methods are trained and
stated for the subsequent trading period.
The tracking portfolio
The PCA method is a statistical method to explain the structure of the covariance matrix of
a multidimensional random variable with a few linear combinations of the univariate elements.
For this study a covariance matrix is created using log returns of the constituents of the CAC 40
index on the first trading day of January 2010, i.e. 02-01-2010, based on a history of 252 trading
days. The first m principal components are chosen such that the cumulative variance proportion
is greater than or equal to 90%, and together with the individual cumulative squared correlations
the question remains on how much index components are chosen to form the tracking portfolio
with.1 Based on the estimates of the PCA method, the out-of-sample tracking-error (TE) of
the replicating portfolio with the benchmark index and the out-of-sample Pearson correlation
coefficient are iteratively calculated for an increasing gathering of index components2 over the
1See Subsections 3.2.1 and 3.4.1 for a more absolute explanation.2With a maximum of 18 components, because the benchmark must be replicated with a few stocks only.
41
remaining training period. The results of both statistics can be found in Fig. 5.1, whence
it can be concluded that the TE is an approximately decreasing function of the amount of
components in the tracking portfolio and is relatively stable from eight stock representatives.
The out-of-sample Pearson correlation coefficient does not smoothly converge when the number
of components in the tracking portfolio is increased, yet it swiftly alternates between 0.880 and
0.965, with little to no improvements made from a tracking portfolio with eight components.
0 5 10 15 200.5
1
1.5
2
2.5
3
3.5
4x 10
−3
Components tracking portfolio
TE
(a) Tracking-error PCA method
0 5 10 15 200.88
0.89
0.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
Components tracking portfolio
Co
rre
latio
n w
ith
be
nch
ma
rk
(b) Pearson correlation PCA method
Figure 5.1: PCA method tracking portfolio characteristics.
The DECS method requires many more parameter specifications than the PCA method, which
all have impact on the rate of convergence of the search heuristics. Because it is not the aim of
this study to investigate the optimal parameters for this data set, the parameters are chosen as
specified by Krink et al. (2009) for the Nikkei 225 price index over the period 2005-2007. Since
empirical evidence has shown, along with the rise of fear indices,3 that in general major indices
move together, it is reasonable to assume that the optimal parameters of Krink et al. will at least
be a reasonable proxy for the optimal parameters of the CAC 40 index. The most important
parameter when the DECS is compared to the PCA is the choice of the maximum absolute sum
of differences between two consecutive weighting schemes, hence this parameter sets limits for
the market impact and partly the transaction costs.4 There was no such stipulation conceivable
in the PCA and it is set equal to M = 0.2 in the DECS. The other parameters can be found in
Appendix B, Table B.1.
The complete model can now be used to optimise the out-of-sample tracking-error and
the Pearson correlation coefficient to the number of components in the tracking portfolio and
check whether eight stock representatives adequately delineate the characteristics of the CAC
40 index over the training period. Because the computational time vastly increases with the
number of iterations inside the DECS method, a total number of 2500 iterations are used with
an associating duration of 15 minutes and 16 seconds for a single run. From the results in Fig.
3e.g. VIX, an implied volatility index of S&P 500 index options.4Referring to Subsection 3.2.2, this parameter inequality is defined as
∑ni=1 |∆wi| ≤M , hence it is not allowed
to change weights too much.
42
5.2, it is lucid that the functions are much smoother in the components of the tracking portfolio
than the upshot from Fig. 5.1. The Pearson correlation coefficient converges above 0.990 and the
maximum benefit of adding an additional stock to the tracking portfolio is indeed reached after
eight components, whereas the tracking-error is ameliorated with twelve components instead
of eight. However, due to the sake of a comparison analysis between trading strategies formed
with either PCA or DECS, it is important to have the same number of index components in
each tracking portfolio and hence eight stocks are used in this study for both methods.5
0 5 10 15 202
3
4
5
6
7
8x 10
−3
Components tracking portfolio
TE
(a) Tracking-error DECS method
0 5 10 15 200.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
Components tracking portfolio
Co
rre
latio
n w
ith
be
nch
ma
rk
(b) Pearson correlation DECS method
Figure 5.2: DECS method tracking portfolio characteristics.
Data inspection
It is essential to have as accurate as possible knowledge about the characteristics of the CAC 40
index, its constituents and the options written on these underlying securities, before any option
strategy is evaluated based on the theory and methods explained in the previous chapters. For
example, the signal trading strategies are build upon the assumptions that individual stock
returns exhibit the so-called leverage effect and that the implied volatility as a function of the
strike price is rather askew than a smile. Moreover, this assumption is the primary reason that
in this study synthetic options are created to avoid, or at least abate, that differences in the
strategy return are not solely related to the implied volatility premiums in the market.
Inspection of the daily log returns of the CAC 40 index and the four6 most conspicuous
stocks of the two tracking portfolios reveal that all exhibit a positive excess kurtosis and that
the market index has a smaller standard deviation than the four stocks, skewness is not an issue.
The Jarque-Bera test statistic is highly significant for each return time series at the 5% level,
and hence the null-hypothesis of normally distributed log returns is rejected. A more formal
test for normality, the Lilliefors test, also rejects the normality assumption. It is therefore
unlikely that the return series in this data set have a normal distribution, and as a consequence
5Please recall from Subsection 3.2.2 that the DECS does not specify a clear-cut amount of stocks in a tracking
portfolio, but rather a minimum and a maximum. This minimum is equal to one.6Total, EDF, Vinci and Sanofi.
43
Table 5.1: Garch-type modelling.
ARMA GARCH ARCH-LM test (2 lags) EGARCH
Equity (p, q) (m,n) χ22 θ (z-stat)
CAC 40 - (1,1) 1.88 −0.17b (-6.77)
Total - (1,1) 0.49 −0.11b (-3.64)
Sanofi - (1,1) 1.10 −0.07a (-2.18)
Vinci - (1,1) 1.55 −0.15b (-6.93)
EDF (1,0) (1,1) 0.94 -0.05 (-1.59)
a = significant at the 5% levelb = significant at the 1% level
the assumptions of the Black-Scholes pricing formula are not satisfied, allowing the volatility
surface to be non-flat. The histograms and the statistics of the five equities can be found in
Appendix A, Fig. A.9.
In Subsection 3.4.2 it was assumed that the equities in this study exhibit an asymmetric
news impact curve, this proposition is tested on the five equities considered before. For each of
the return series an ARMA-GARCH model is estimated, using Bollerslev-Wooldridge standard
errors, and investigated whether the NIC is indeed asymmetric and whether the model can
approximated with no ARMA terms. The results are summarised in Table 5.1. The first
column indicates whether the daily log returns show significant autocorrelation by means of
the fitted ARMA model; only the stock EDF has an AR term next to a constant. Because it
is too much effort to construct a specific ARMA model for each stock in order to forecast the
implied correlation, from this results it might be concluded that it is appropriate to model the
daily stock returns with no ARMA terms. The second column indicates which GARCH-type
model is able to replicate the volatility clustering dynamics of the returns; using the ARCH-LM
test results in the third column, the GARCH(1,1) model is a suitable choice. The last column
indicates the coefficient of the lagged standardised residuals in the EGARCH(1,1) model, which
is negative and significant different from zero when the return series contain any leverage effects.
Only the stock EDF displays an insignificant coefficient and concluding from this criteria has
no asymmetric NIC.7
Assuming that the future data resembles the dynamics of the sample of equities considered
in the training period, it stipulates that the EGARCH(1,1) model with no ARMA terms can
be used to model the future and forecast the implied correlation of the index and the tracking
portfolio. Moreover by the leverage effect it is expected that implied volatility of a stock is
a convex function of the strike prices, where the volatility used to price low-strike options is
significantly higher than high-strike options (Hull, 2012). Hence the volatility skew offers a
chance to decide whether a straddle or strangle is favorable.
7More formal tests for the asymmetry of the NIC, such as regression models for the squared standardised
residuals were omitted in this study, because it is not of primary interest.
44
From the daily option data of the CAC 40 index and its constituents over the period 01-01-2010
to 31-05-2010 it is possible to acquire the historical volatility smile implicit in option prices.
Ordinarily the volatility smile is constructed using the market option prices of a single moment
and consequently a lot of information is missing on the volatility smile, which are necessarily
interpolated. However, using option prices over a historical time span to create the volatility
smile allows to depict a more stable relation between the implied volatility and the strike prices.
Notwithstanding, one needs to take into account that a historical volatility smile is backward
looking and hence the conventional trade-off between accuracy and irrelevant information is
applicable. Yet for this study it is a captivating exercise to study the dynamics of the historical
volatility smile and it is constructed for the CAC 40 index using both put and call options and
given in Fig. 5.3. Following Whaley (1993), options less than 6 trading days to maturity are not
used because the effects of market factors on option prices; leading to a total number of 34847
data points on the volatility smile. It can be concluded that the historical implied volatility as
function of the strike prices is a merger between askew and a parabola (smile). For the purpose
of clarity, the implied volatility is given as a function of the maturity time for various strike
prices in Appendix A, Fig. A.10.
Figure 5.3: The historical volatility surface of the CAC 40 index over the period 01-01-2010 to
31-05-2010, using both put and call options.
2000
3000
4000
5000
6000
050
100150
200250
300
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Kdtm
Imp
l. v
ol
45
5.2 Naive dispersion trading
In this section the results of the naive dispersion trading strategy within the trading period are
discussed. As indicated in Subsection 3.4.1, at the end of each trading day a long dispersion
position is entered with synthetic options on the CAC 40 index and on the most recent tracking
portfolio, hence on each day in the core of the strategy the portfolio exists out of 30 dispersion
positions, i.e. 540 active synthetic options or 2160 active realistic options.8 Because of this
vast amount of options, the portfolio can have very high delta and gamma exposure as the
maturity is imminent and/or when the correlation between the index and the tracking portfolio
is deteriorated. Both the strategies with and without daily delta-hedging are discussed in this
section and the results are placed in Tables 5.2 and 5.3.
Naive strategy without delta-hedging
As can be seen from Table 5.2, over the 523 trading days, the naive dispersion trading strategies
based on the PCA method and the DECS method depict similar results, yet the DECS outper-
forms the PCA in each strategy. The DECS has an average daily return of 0.31% and 0.38%
with heteroscedasticity and autocorrelation consistent (HAC) standard deviations of 4.84% and
12.73% for the straddle and the strangle combination, respectively, whereas the PCA strategy
yields an average daily return of 0.10% and -0.06% with HAC standard deviations of 6.14%
and 15.33% for the same order of combinations. The autocorrelation is explicitly tested for the
four strategies by regressing the daily return series on a constant, whereafter a Breusch-Godfrey
LM test for two lags is performed on the residuals. Although the latter test cannot reject the
null-hypothesis of no autocorrelation in the first two lags at the 5% significance level, HAC
standard deviations are justified because the estimates are consistent provided that any serial
dependence in the time series dies away sufficiently fast (Cameron and Trivedi, 2009).
To test whether the average daily returns are significantly different from zero a normality
test on the four naive return series is implemented. Because the Kolmogorov-Smirnov test
requires the specification of the parameters of the normal distribution against which the data
is tested on, the return series are tested with a Lilliefors test. At the 1% significance level, the
normality test does not support the null-hypothesis that the daily returns for the naive trading
strategies are normally distributed. Also, by the assumption of serial dependence in the return
series beyond the second lag, the Central Limit Theorem cannot be applied to the average daily
returns of the trading strategies and the regular t-statistic is not valid. In order to test whether
the average daily returns are significant different from zero a stationary bootstrap procedure is
employed to generate 9999 simulations of the average daily returns upon which the bootstrap
estimate of the standard error is calculated. Herein it is used that the stationarity assumption
is satisfied by the daily return series (Appendix A, Table A.1). Based on the 5% critical value
of the t-statistic, i.e. t=1.960, the average daily returns of the four naive trading strategies are
not significantly different from zero. Nevertheless, the bootstrapped t-statistics of the DECS
8Each synthetic option is created from four different options on the same underlying asset.
46
Tab
le5.
2:N
ond
elta
-hed
ged
stra
tegi
es.
Th
ista
ble
rep
orts
the
aver
age
dai
lyre
turn
sof
the
non
del
ta-h
edged
naiv
est
rate
gy,
the
non
del
ta-h
edged
com
bin
ati
on
-an
dp
osi
tion
fore
cast
stra
tegie
san
dth
e
non
del
ta-h
edge
dm
ixin
gst
rate
gies
,in
com
bin
atio
nw
ith
HA
Cst
an
dard
dev
iati
on
s,t-
stats
from
ast
ati
on
ary
boot
stra
pp
roce
du
re,
the
Lil
lief
ors
test
stati
stic
s
andp-v
alu
es,
Bre
usc
h-G
od
frey
test
stat
isti
csan
dp-v
alu
es,
an
nu
ali
sed
SR
sw
ith
stan
dard
dev
iati
on
sfr
om
ast
ati
on
ary
boots
trap
,an
dth
eC
AP
Mm
od
el
coeffi
cien
tes
tim
ates
andt-
stat
isti
cs.
Th
ep
rob
abil
ity
ofth
ese
quen
ceev
ent,L
i=m,m
=1,
2,...,
inth
eafo
rem
enti
on
edst
ati
on
ary
boots
trap
iseq
ual
to
(1−p)m
−1p,
wh
erep
=T
−1/3,
andT
isth
enu
mb
erof
ob
serv
ati
on
sin
the
retu
rnse
ries
.
HA
CB
oots
trap
Lil
lief
ors
test
Bre
usc
h-G
od
frey
test
CA
PM
CA
PM
Str
ateg
yC
omb
inat
ion
Typ
eM
ean
Std
.D
ev.
t-st
at
LF
(p-v
alu
e)χ2 2
(p-v
alu
e)S
R(S
td.
Dev
.)α
(t-s
tat)
β(t
-sta
t)
Nai
veS
trad
dle
DE
CS
0.31
%4.
84%
1.5
30.
11b
(0.0
0)
0.4
4(0
.80)
1.1
86
(0.6
8)
0.0
033
(1.7
2)
1.1233b
(9.1
7)
Str
add
leP
CA
0.10
%6.
14%
0.3
90.1
2b
(0.0
0)
3.1
1(0
.21)
0.3
47
(0.6
9)
0.0
013
(0.5
4)
1.6561b
(10.5
4)
Str
angl
eD
EC
S0.
38%
12.7
3%0.7
00.1
9b
(0.0
0)
4.5
9(0
.10)
0.5
32
(0.8
0)
0.0
046
(1.0
2)
4.2512b
(14.6
3)
Str
angl
eP
CA
-0.0
6%15
.33%
-0.0
90.1
6b
(0.0
0)
0.9
3(0
.63)
-0.0
50
(0.7
4)
0.0
004
(0.0
7)
5.3
155b
(15.5
7)
Com
bin
atio
n-
DE
CS
0.33
%5.
29%
1.4
90.
11b
(0.0
0)
0.7
0(0
.71)
1.1
40
(0.7
0)
0.0
035
(1.7
0)
1.3292b
(10.1
4)
-P
CA
0.11
%6.
57%
0.3
70.1
2b
(0.0
0)
3.0
8(0
.21)
0.3
28
(0.6
9)
0.0
014
(0.5
4)
1.8980b
(11.5
2)
EG
AR
CH
Str
add
leD
EC
S0.
38%
3.75
%2.
49a
0.11b
(0.0
0)
3.2
0(0
.20)
1.7
23
(0.6
0)
0.0039a
(2.2
3)
0.5550b
(4.9
7)
Str
add
leP
CA
0.25
%4.
28%
1.3
80.1
1b
(0.0
0)
23.
80b
(0.0
0)
1.0
39
(0.5
6)
0.0
027
(1.1
8)
0.7
718b
(5.4
0)
Str
angl
eD
EC
S0.
73%
7.71
%2.
39a
0.17b
(0.0
0)
10.
18b
(0.0
1)
1.6
61
(0.6
5)
0.0
077a
(1.9
8)
0.2370b
(8.9
8)
Str
angl
eP
CA
0.41
%8.
84%
1.1
30.
14b
(0.0
0)
21.
23b
(0.0
0)
0.8
23
(0.5
6)
0.0
046
(1.0
0)
2.7
542b
(9.3
2)
BB
sS
trad
dle
DE
CS
0.31
%4.
08%
1.8
50.1
2b
(0.0
0)
1.6
7(0
.43)
1.3
95
(0.6
2)
0.0
033
(1.8
8)
0.9947b
(8.8
9)
Str
add
leP
CA
0.18
%4.
99%
0.8
30.1
2b
(0.0
0)
11.
72b
(0.0
0)
0.6
91
(0.6
3)
0.0
021
(0.9
3)
1.4
902b
(10.6
2)
Str
angl
eD
EC
S0.
49%
9.58
%1.2
20.1
9b
(0.0
0)
1.9
2(0
.38)
0.9
05
(0.6
9)
0.0
055
(1.3
9)
3.6575b
(14.3
6)
Str
angl
eP
CA
0.16
%11
.37%
0.3
30.
17b
(0.0
0)
3.7
5(0
.15)
0.2
71
(0.6
5)
0.0
025
(0.5
4)
4.6132b
(15.6
6)
Mix
-D
EC
S0.
45%
4.05
%2.8
3b
0.11b
(0.0
0)
2.9
2(0
.23)
1.8
21
(0.5
8)
0.0045a
(2.3
7)
0.3712b
(3.0
4)
(EG
AR
CH
)-
PC
A0.
31%
4.50
%1.6
60.1
1b
(0.0
0)
23.
31b
(0.0
0)
1.1
60
(0.5
4)
0.0
031
(1.3
1)
0.6
074b
(3.9
7)
Mix
-D
EC
S0.
32%
4.51
%1.7
50.
13b
(0.0
0)
0.7
7(0
.68)
1.3
10
(0.6
3)
0.0
034
(1.8
2)
1.2102b
(10.1
0)
(BB
s)-
PC
A0.
17%
5.42
%0.7
40.1
2b
(0.0
0)
10.
21b
(0.0
1)
0.6
23
(0.6
3)
0.0
020
(0.8
9)
1.7
514b
(11.9
4)
a=
sign
ifica
nt
atth
e5%
leve
lb
=si
gnifi
cant
atth
e1%
leve
l
47
Tab
le5.
3:D
elta
-hed
ged
stra
tegi
es.
Th
ista
ble
rep
orts
the
aver
age
dai
lyre
turn
sof
the
dai
lyd
elta
-hed
ged
naiv
est
rate
gy,
the
dail
yd
elta
-hed
ged
com
bin
ati
on
-an
dp
osi
tion
fore
cast
stra
tegie
s
and
the
dai
lyd
elta
-hed
ged
mix
ing
stra
tegi
es,
inco
mb
inati
on
wit
hH
AC
stan
dard
dev
iati
on
s,t-
stats
from
ast
ati
on
ary
boots
trap
pro
ced
ure
,th
eL
illi
efors
test
stat
isti
csan
dp-v
alu
es,
Bre
usc
h-G
od
frey
test
stat
isti
csan
dp-v
alu
es,
an
nu
ali
sed
SR
sw
ith
stan
dard
dev
iati
ons
from
ast
ati
on
ary
boots
trap
,an
dth
eC
AP
M
mod
elco
effici
ent
esti
mat
esan
dt-
stat
isti
cs.
Th
ep
rob
abil
ity
of
the
sequ
ence
even
t,L
i=m,m
=1,2,...
,in
the
afo
rem
enti
on
edst
ati
on
ary
boots
trap
is
equ
alto
(1−p)m
−1p,
wh
erep
=T
−1/3,
andT
isth
enu
mb
erof
ob
serv
ati
on
sin
the
retu
rnse
ries
.
HA
CB
oots
trap
Lil
lief
ors
test
Bre
usc
h-G
od
frey
test
CA
PM
CA
PM
Str
ateg
yC
omb
inat
ion
Typ
eM
ean
Std
.D
ev.
t-st
at
LF
(p-v
alu
e)χ2 2
(p-v
alu
e)S
R(S
td.
Dev
.)α
(t-s
tat)
β(t
-sta
t)
Nai
veS
trad
dle
DE
CS
0.31
%4.
61%
1.5
80.
09b
(0.0
0)
1.8
4(0
.40)
1.2
07
(0.6
1)
0.0
031
(1.5
2)
0.1
892
(1.4
7)
Str
add
leP
CA
0.12
%5.
77%
0.4
90.1
0b
(0.0
0)
0.0
2(0
.99)
0.4
19
(0.6
3)
0.0
013
(0.4
9)
0.3
062
(1.8
7)
Str
angl
eD
EC
S0.
37%
12.3
5%0.6
90.1
7b
(0.0
0)
8.6
3a
(0.0
1)
0.5
23
(0.7
9)
0.0
037
(0.7
5)
0.3
028
(0.9
5)
Str
angl
eP
CA
-0.0
3%14
.69%
-0.0
40.1
6b
(0.0
0)
4.5
1(0
.10)
-0.0
13
(0.7
4)
-0.0
002
(-0.0
3)
0.5
051
(1.3
2)
Com
bin
atio
n-
DE
CS
0.32
%5.
06%
1.5
30.
10b
(0.0
0)
1.5
6(0
.46)
1.1
46
(0.6
2)
0.0
032
(1.4
6)
0.2
357
(1.6
9)
-P
CA
0.12
%6.
21%
0.4
60.1
1b
(0.0
0)
0.1
5(0
.93)
0.3
95
(0.6
3)
0.0
013
(0.4
7)
0.3560a
(2.0
4)
EG
AR
CH
Str
add
leD
EC
S0.
37%
3.59
%2.
55a
0.09b
(0.0
0)
1.0
6(0
.59)
1.7
33
(0.5
8)
0.0
037a
(2.1
5)
0.1
158
(1.0
6)
Str
add
leP
CA
0.26
%4.
04%
1.5
70.0
9b
(0.0
0)
10.
33b
(0.0
1)
1.1
33
(0.5
3)
0.0
026
(1.2
2)
0.1
573
(1.1
3)
Str
angl
eD
EC
S0.
72%
7.54
%2.
45a
0.13b
(0.0
0)
6.2
9a
(0.0
4)
1.6
46
(0.6
2)
0.0
072
(1.8
9)
0.1
415
(0.5
8)
Str
angl
eP
CA
0.45
%8.
53%
1.2
70.
14b
(0.0
0)
13.
82b
(0.0
0)
0.9
14
(0.5
4)
0.0
045
(0.9
8)
0.2
247
(0.7
7)
BB
sS
trad
dle
DE
CS
0.32
%3.
91%
2.02a
0.10b
(0.0
0)
1.7
5(0
.42)
1.4
64
(0.5
7)
0.0
032
(1.7
5)
0.1
633
(1.3
9)
Str
add
leP
CA
0.20
%4.
67%
1.0
00.1
0b
(0.0
0)
3.6
0(0
.17)
0.7
99
(0.5
6)
0.0
020
(0.8
8)
0.2
641
(1.7
9)
Str
angl
eD
EC
S0.
49%
9.24
%1.2
90.1
6b
(0.0
0)
0.3
4(0
.84)
0.9
39
(0.6
5)
0.0
050
(1.1
5)
0.2
689
(0.9
8)
Str
angl
eP
CA
0.21
%10
.79%
0.4
50.1
6b
(0.0
0)
2.2
0(0
.33)
0.3
56
(0.6
2)
0.0
022
(0.4
2)
0.4
431
(1.3
5)
Mix
-D
EC
S0.
43%
3.91
%2.7
9b
0.10b
(0.0
0)
1.0
2(0
.60)
1.7
91
(0.5
7)
0.0
042a
(2.2
8)
0.0
992
(0.8
3)
(EG
AR
CH
)-
PC
A0.
31%
4.30
%1.7
80.0
9b
(0.0
0)
9.1
4a
(0.0
1)
1.2
24
(0.5
3)
0.0
031
(1.3
5)
0.1
391
(0.9
4)
Mix
-D
EC
S0.
33%
4.35
%1.9
10.
12b
(0.0
0)
0.0
0(1
.00)
1.3
66
(0.5
7)
0.0
033
(1.6
5)
0.2
113
(1.6
5)
(BB
s)-
PC
A0.
20%
5.11
%0.9
10.1
1b
(0.0
0)
2.0
1(0
.37)
0.7
25
(0.5
7)
0.0
020
(0.8
1)
0.3148a
(2.0
0)
a=
sign
ifica
nt
atth
e5%
leve
lb
=si
gnifi
cant
atth
e1%
leve
l
48
method are of much greater magnitude than the t-statistics of the PCA method and the strangle
combination performs in both cases worse than the straddle combination. The same conclusions
can be derived from the annualised Sharpe Ratios (SRs), where the mean excess return is divided
by the HAC standard deviation and multiplied by√
252 to approximate the ratio on yearly basis.
The DECS method displays SRs which are equal to 1.186 and 0.532 for the straddle and strangle
combination respectively, whereas the PCA method yield SRs of 0.347 and -0.050 for the same
sequence of combinations. However, by again using the bootstrap, the standard deviations of
the SRs are large with an average of 0.68 and 0.77 for the straddle and strangle, respectively.
Dispersion trading in its purest form is market neutral and this assumption can be tested
with the CAPM model which explains cross-sectional variation in the expected excess return
by the covariance with the market return, i.e. the variation in beta.9 Consequently, when
the strategy is market neutral the beta is equal to zero. However, the beta’s of the four
naive strategies are positive and significantly different from zero at the 1% level, rejecting the
assumption that the strategies are market neutral. The last observation is reasonable because
in the four naive strategies considered so far, a daily delta-hedging procedure was not employed
and situations where the value of the portfolio is dependent on the direction in which the market
moves can occur with a greater probability.
Naive strategy with delta-hedging
The results from the naive dispersion trading strategies with a daily-delta hedging procedure
supplementing the portfolio of options are shown in Table 5.3. The average daily returns
change little compared to the previously described naive strategies, however, the HAC standard
deviations show an average decrease in value of 0.41%. As a result, the bootstrapped t-statistics
and most of the SRs show a slight increase as well. The Breusch-Godfrey LM test for two
lags displays significant autocorrelation at the 1% level for the daily returns of the DECS
method using strangles as combinations, nonetheless assuming that the assumptions of the
HAC standard deviations are satisfied this is not an issue.
The most conspicuous result from daily delta-hedging the portfolio of options is that the
value of beta in the CAPM model diminishes towards zero for the four naive strategies, leaving
none of the beta’s significantly different from zero at the 5% level and hence strengthening
the theory that dispersion trading in its purest form is market neutral. Thus using a relatively
simple form of dynamic delta-hedging reduces the market risk significantly and makes dispersion
trading less delicate, yet the standard deviations of the daily return series and the bootstrapped
SRs are substantial and as a consequence the probability of large losses are high.
In Fig. 5.4 the margin account of the four delta-hedged strategies are plotted together with
the CAC 40 index over the trading period, whence still a moderate to strong relationship between
the directional movements of the index with the P&L of the simple trading strategies can be
observed. It is clear that the DECS method is less volatile than the PCA method, additionally,
the difference in performance between the straddle and the strangle is more notable. In contrast
9Financial Econometrics, University of Amsterdam, catalogue number 6414M0007Y.
49
Figure 5.4: The cumulative returns of the delta-hedged naive trading strategies against the
CAC 40 index.
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr122600
2800
3000
3200
3400
3600
3800
4000
4200
Date
Ind
ex
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12−4000
−2000
0
2000
4000
6000
8000
10000
12000
Cu
mula
tive
Retu
rn
CAC 40
DECS Straddle
DECS Strangle
PCA Straddle
PCA Strangle
to the straddle combinations, a naive trading strategy using strangles is rather sensitive to
extreme price movements of the CAC 40. Because the price of a strangle is generally lower
than a straddle, especially an OTM strangle compared with an ATM straddle, the strangle has
greater leverage than the straddle. Moreover, because the value of the short and long leg at
the time of inception are e100, more strangle positions are initiated than straddle positions
causing more force on the changes of the portfolio’s value when an underlying asset’s price has
an extreme move in either direction.
5.3 Position signal dispersion trading
Having observed the general properties and performance of the naive trading strategies, more
sophisticated trading methods can hopefully reduce the amount of market risk and bolster
dispersion trading towards a market neutral strategy. First the results of a procedure using the
implied correlation as a market signal are presented, subsequently the volatility smile is used
as a signal to determine whether a strangle or a straddle position is initiated. This chapter is
concluded by a strategy which is mixture between the previous two.
Position forecasting
As explained in Subsection 3.4.2, whether a short or a long dispersion trade is initiated at
the end of an arbitrary day within the trading period depends on the forecast of the implied
correlation. Based on the existing theory on dispersion trading commonly the implied volatility
of the index exhibits a premium compared to the implied volatility of a tracking portfolio. Hence
50
a short dispersion position is only entered into when the implied correlation is expected to grow
significantly in the 30 days after inception based on the EGARCH(1,1) model or the Bollinger
Bands (BBs), otherwise a long dispersion trade is made.
The results are presented in Tables 5.2 and 5.3. Both forecast procedures show significant
improvements for the daily-hedged procedures as well as the non-hedged procedures for the two
tracking portfolios compared with the naive trading strategies. However two specific conclusions
can be derived from these tables. In the first place the DECS tracking portfolio outperforms the
PCA tracking portfolio again in all strategy variants. Secondly, the EGARCH forecast approach
shows better results than the Bollinger Bands on all figures.
For example, looking only at the delta-hedged EGARCH strategy, the DECS method has
an average daily return of 0.38% and 0.73% for the straddle and strangle, respectively, and the
PCA method, ceteris paribus, has an average daily return of 0.25% and 0.41% for the same order
of combinations. The associated standard deviations (HAC) firmly reduce for the DECS and
the PCA, and the bootstrapped t-statistics imply that the daily mean returns are significant
different from zero for the DECS method. As a consequence of the previous the annualised
SRs increase wherein three out of the four delta-hedged strategies became larger than one; the
simulated standard deviations decrease between 0.1 and 0.2. Moreover, the CAPM estimates
support the market neutrality assumption because of insignificant beta’s at the 5% level and at
the same time increased values of alpha. From the total of 523 initiated dispersion positions,
34 positions were short in the straddle and 31 in the strangle.
The results of the delta-hedged EGARCH strategies can be generalised to the delta-hedged
Bollinger Band strategies, whereby the latter family of strategies has a lower average daily
return combined with a higher variability and less short positions10 initiated over the trading
period. Likewise the naive trading strategies, a non-hedged procedure causes more exposure to
market movements in either direction.
Statistical inference
Up until now it was assumed that the implied correlation is a bounded process with mean-
reversion characteristics, however, this hypothesis was not tested whatsoever. The properties of
the implied correlation and therefore the implied volatility over the trading period are examined
next.
As a result of the pairwise correlation between stock returns in the tracking portfolio, the
implied volatility of the tracking portfolio on a given day is dependent on the specific term
where the historical correlations are based upon. The straddle implied volatility spread, i.e. the
difference between index implied volatilities (MIV) and tracking portfolio implied volatilities
(TIV), is presented in Table 5.4 for three different historical terms and for both the DECS and
the PCA tracking portfolios over the entire trading period. It can be observed that using a
historical term of 30 days increases the implied volatility spread and the standard deviation
10In total 27 short positions were initiated both for the straddle and the strangle variant with Bollinger Bands.
51
Table 5.4: Straddle implied volatility spread.
DECS tracking portfolio PCA tracking portfolio
30 Days 126 Days 252 Days 30 Days 126 Days 252 Days
Mean 0.003 -0.001 -0.001 0.024 0.020 0.020
Median 0.004 -0.002 -0.001 0.021 0.018 0.018
St. dev. 0.025 0.020 0.019 0.024 0.021 0.025
t-stat 0.75 -0.41 -0.47 7.35b 6.72b 5.52b
ADF p-value 0.015 0.012 0.001 0.003 0.004 0.011
b = significant at the 1% level
of the daily differences between the implied volatilities, there is no unambiguous advantage for
using a historical period of 126 or 252 trading days. Because of the autocorrelation between
the daily differences of the implied volatility of the index and tracking portfolio, the average
implied volatility spreads are bootstrapped and 9999 simulations are generated to test whether
the average differences between the MIV and the TIV are significant different from zero. Again
the stationary bootstrap is applied, and at the 5% level (t = 1.960) the average differences are
solely not significant for the DECS method. The strangle shows similar results (Appendix A,
Table A.3).
It is furthermore remarkable that the average daily differences between the implied volatil-
ities are slightly negative for the DECS when correlations are based on the longer historical
periods, meaning that the theoretical premium in the index implied volatility no longer exists.
Because the DECS method has a very accurate out-of-sample correlation with the index, i.e.
low correlation risk, it is conceivable that historical dispersion trading opportunities has been
arbitraged away.
To test the relation between the index implied volatility and the implied volatility spread,
the Spearman’s rank correlation coefficient is estimated for both tracking portfolios. Herein,
the implied volatilities are based on the historical period of 126 days. It is fascinating that for
the DECS tracking portfolio rho is estimated as -0.149 with a t-statistic of -3.44 for the straddle
combination and as -0.182 with a t-statistic of -4.23 for the strangle combination. While for the
PCA tracking portfolio the relation is positive and larger in value with a rho (t-statistic) of 0.547
(14.91) and 0.513 (13.63) for the straddle and strangle combinations, respectively. Although
the sign of Spearman’s rank correlation coefficient is different between the two portfolios, both
are significant at the 5% level.
The characteristics of the implied volatilities of the index and the tracking portfolio itself
are showed in Table A.2 of Appendix A, wherefrom it can be concluded that implied volatili-
ties constructed with correlations over the last 126 days prior the specific date have the best
correlation with the implied volatilities of the index. The difference in size between the implied
volatility of the DECS and the PCA tracking portfolio become more clear as well. The implied
volatility of the DECS is on average 267 out of the 523 days larger than the index implied
52
Figure 5.5: Straddle implied correlation with DECS.
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Date
Implie
d C
orr
ela
tion D
EC
S S
traddle
Implied Correlation
MA
MA plus 2 std
MA minus 2 std
volatility, whereas the one of the PCA tracking portfolio is on average 89 out of the 523 days
larger than the index implied volatility. The implied volatility based on a historical period of
126 days is plotted for the straddle strategy against the index in Fig. A.3 of Appendix A; the
strangle shows roughly the same features and its plot is therefore omitted.
The properties of the implied correlations are directly linked to the properties of the implied
volatilities of the index and the tracking portfolio. For the DECS straddle strategy the implied
correlation is shown in Fig. 5.5, together with its 30-day period Bollinger Bands. Because the
correlation coefficient is in absolute value always between zero and one, an implied correlation
above one indicates pure arbitrage when possible measurement errors and transaction costs are
neglected. This can be seen from
ρ > 1⇒ σ2I >n∑i=1
w2i σ
2i +
n∑i=1,j 6=i
wiwjσiσj >n∑i=1
w2i σ
2i +
n∑i=1,j 6=i
wiwjσiσjρi,j = σ2p. (5.1)
For the DECS straddle strategy this is the situation around June 2011, just before the CAC 40
rapidly decreases in value due to fears of contagion of the European sovereign debt crisis to other
mediterranean countries. The mean-reversion assumption is tested with an augmented Dickey-
Fuller test for the four possible implied correlation time-series (based on the naive strategies),
with an intercept included in the equations. The ADF t-statistics are highly significant with a
p-value of 0.001 for both the DECS- straddle and strangle strategy and a p-value of 0.009 and
0.011 for the PCA- straddle and strangle strategy, respectively. Hence at the 5% significance
level the null-hypothesis of a unit-root is rejected for all four implied correlations, and therefore
it may be assumed that these time series are stationary.
53
5.4 Combination signal dispersion trading
In contrast to the position signal dispersion trading, where two different forecast methodolo-
gies were exploited to forecast the implied correlation of both the strangle and the straddle
separately, the forecast procedure in this section is characterised by only two variants. The
combinations are forecasted by the volatility smile and hence there are neither pure straddle
nor pure strangle strategies anymore, the difference in the two strategies lies whether the DECS
or the PCA tracking portfolio is used. As daily delta-hedging is an improvement on a dispersion
trade in the sense of the overall market exposure, the strategies discussed in this section are
delta-hedged unless stated otherwise.
Combination forecasting
The results of combination forecasting as an extension on the naive strategies are presented in
the same tables as the results of the previous section. The average daily return of the DECS
method is 0.32% with a HAC standard deviation of 5.06%, the PCA method displays an average
daily return of 0.12% and a HAC standard deviation of 6.21%. The bootstrapped t-statistics for
the average daily returns are not significant different from zero at the 5% level and the normality
test rejects the null-hypothesis that the daily returns are normally distributed. Furthermore
the CAPM model does not underpin the market-neutrality assumption for the PCA method
because beta is positive and significant different from zero at the 5% level. The annualised
SRs are equal to 1.146 for the DECS and 0.395 for the PCA tracking portfolio, and their
bootstrapped standard deviations are around 0.62. Hence these results imply that trading the
volatility smile does not yield an unilateral improvement over the results of the naive trading
strategies. A total of 812 strangle combinations were initiated in the trading period, from which
the index was responsible for 16 strangles instead of straddles.
Statistical inference
In the first place it needs to be checked whether the theoretical assumption of a mean-reverting
slope process of the volatility smile is satisfied by the data of the trading period. It is too
much effort to investigate the slope of each stock which was represent in the tracking portfolio,
for this reason the slopes of the volatility smile implicit in options on the CAC 40 index and
on Total11 are inspected (Eqs. (3.12) and (3.13)). It is conspicuous that the left- and right
slope processes of the CAC 40 index and Total have weak to moderate pearson correlation
coefficients with values of 0.23 and 0.42 for the left and right slope processes, respectively. By
an augmented Dickey-Fuller test, the ADF t-statistics of the slope processes for the CAC 40
index are significant at the 1% level with p-values of 0.002 and 0.006 for the left- and right
slope process, respectively. The ADF t-statistics of Total’s slope processes have even lower
p-values of 0.000 for both the left and the right slope of the volatility smile. Consequently the
null-hypothesis of the existence of an unit root is rejected for the considered time series.
11Total was ranked ostentatious in both the DECS- and PCA tracking portfolio.
54
Concluding solely from the figures of combination forecasting, this strategy is probably not
an enhancement on the naive strategy. This raises the question whether trading the slope is
profitable at all. To answer this question the regression model
∆GAINSt = Ω0 + Ω1MIV t + Ω2∆GAINSt−1 + Ω3Dt + εt (5.2)
is estimated, where ∆GAINSt represents the difference in gains between the delta-hedged com-
bination forecast strategy and the delta-hedged naive straddle strategy of the options expiring
at day t, MIV t is the estimated average index implied volatility of the expiring options at the
moment of initiating, i.e. the lagged 30-days index implied volatility at day t, and Dt is an indi-
cator variable equal to one when there exists a strangle combination within the expiring options
of the combination forecast strategy. Hence testing whether trading the slope is not a lucrative
extension on the naive straddle strategy is equivalent to testing the null-hypothesis that Ω3 ≤ 0
in Eq. (5.2). Note that in this regression model a lagged value of GAINSt is added to correct for
autocorrelation in the residuals. Nevertheless the model is still estimated with the Newey-West
covariance matrix and the lags are based on the rule of thumb, T 0.25, which is approximately
equal to 5 for the sample size of our data set. Also, the 30-day lagged index implied volatility
is used to approximate the 30-day historical realised volatility prior day t because the latter
variable is highly correlated with its own lags by construction (overlapping data). Furthermore,
using the realised volatility in the regression complicates the analysis because most likely it is
an endogenous variable for the difference in gains.
Based on the DECS tracking portfolio, Ω3 = 0.23 with a t-statistic (p-value) of 0.23 (0.91)
and is thus not significant different from zero. The lagged value of GAINSt has an estimated
coefficient of Ω2 = 0.44 and is with a t-statistic (p-value) of 5.71 (0.00) highly significant at the
1% level. When the regression model is estimated with the PCA tracking portfolio, Ω3 = −0.54
with a t-statistic (p-value) of -0.21 (0.83), Ω2 = 0.45 and is again highly significant with a
t-statistic (p-value) of 5.95 (0.00). In both models, R2 = 0.21, and the coefficient of MIVt is
negative and insignificant at the 5% level.
The previous results imply that trading the slope offers no significant improvement over the
naive straddle strategy.
5.5 Mixing signals
In this section the two different market signal strategies are consolidated into one complete
strategy. This means that at the end of each day the combination as well as the positions on
the options are forecasted whereafter the corresponding dispersion trade is executed.
The results are shown in Tables 5.2 and 5.3. It can be observed that the DECS mixing proce-
dure using EGARCH forecasting achieves the highest annualised SRs of all strategies considered
in this study with a value of 1.791 and 1.821 for the delta-hedged and non-hedged strategy, re-
spectively, moreover the bootstrapped standard deviations decrease for both to approximately
55
0.57. The average daily returns are equal to 0.43% and 0.45% and the HAC standard devia-
tions are equal to 3.91% and 4.05%, for the same order of strategies. Also, both bootstrapped
t-statistics for the average daily returns are significantly different from zero at the 5% level and
the null-hypothesis of normally distributed returns is rejected by the Lilliefors test at the same
level. In accordance with the previous results, the beta in the CAPM model is significant at
the 5% level for the non-hedged strategy with a value of 0.3712 and a t-statistic equal to 3.04,
alpha is positive but insignificant. The CAPM beta for the delta-hedged strategy is insignifi-
cant with a value of 0.0992 and a t-statistic of 0.83, additionally alpha is significant positive at
the 5% level with a value of 0.0042 and a t-statistic of 2.28. Hence next to the fact that the
mixing strategy attains the highest SR in this study, the delta-hedged strategy is fully in accor-
dance with the market neutrality assumption. Moreover, despite the indefinite conclusions for
combination forecasting from the previous section, the EGARCH mixing strategy outperforms
the EGARCH straddle strategy and hence combination forecasting can be seen as a profitable
extension on the latter procedure.
The EGARCH mixing strategy based on the PCA tracking portfolio results in an average
daily return of 0.31% and a HAC standard deviation of 4.30% when daily delta-hedging is
applied to the portfolio, the SR is equal to 1.224. Furthermore, the daily returns are not
normally distributed by the Lilliefors test with a p-value of 0.00 and the beta in the CAPM
model is not significant different from zero at the 5% level with a value of 0.1391 and a t-statistic
of 0.94, alpha is positive but insignificant with a t-statistic of 1.35. The non-hedged strategy
performs roughly the same as the delta-hedged strategy, however it has a lower average daily
return and a higher standard deviation. Likewise the DECS method, it can be concluded that
the mixing strategy is an improvement on the EGARCH straddle strategy for the PCA tracking
portfolio.
When Bollinger Bands ares used instead of the EGARCH model, the performance becomes
worse than the BBs straddle strategy, both for the PCA as the DECS tracking portfolio and
independent from delta-hedging. The average daily returns decrease and the standard deviations
increase, as a consequence the SRs are reduced as well.
In the previous section it was found, by the estimation of Eq. (5.2), that the strangle indicator
variable has no significant and univocal relationship with the difference in gains between the
combination forecast strategy and the naive straddle strategy. From the present, by combining
position forecasting with combination forecasting, it becomes clear that the profitability of
trading the slope of the volatility smile is dependent on the forecast methodology of the implied
correlation. For this reason it must be stressed that based on these results no specific and
unambiguous verdict can be made about the ceteris paribus effect of position forecasting within
dispersion trading.
56
5.6 Robustness checks
5.6.1 Do the strategy returns have finite variance?
To have an idea of the potential extreme return events it is important to investigate the tail
behaviour of the different strategy return series. Because it was found that the naive strategies
have the most variability in the daily return series, the tail index is estimated with the Hill
estimator for these series. For different values of the parameter k, with a maximum of k = 26
such that b kT c = 0.05, it becomes clear that the tail index indeed strongly depends on the
choice of k. Also it appears that the shape parameter of the left tail, ξl, is larger than the
shape parameter of the right tail, ξr. Hence this indicates that the daily return series may have
a heavier left tail. At the one-sided 5% critical value, i.e. z = −1.645, the null-hypothesis of
finite variance is rejected for the right-tail index in the case of the naive strangle combination
strategies when k > 8. For the the left tails and the remainder of the naive strategies, the null-
hypothesis is not rejected for the evaluated values of k. However, due to the fact that the shape
parameter does not become very stable for the k ≤ 26, it is expected that the null-hypothesis
of finite variance is rejected for more return series and tails when k is large. Because of this
strong dependence on the parameter k, the validity of the tests is precarious and hence it can
not be concluded indefinitely whether the return series have finite variance.
5.6.2 Is dispersion trading profitable under a transaction costs scenario?
It was explained that the main component of the transaction costs is the bid-ask spread. How-
ever, it is essentially impossible to have an exact measure of these costs. Furthermore, due to
limitations of the available data it was not possible to get the bid-ask spreads of the option
prices over the sample period studied in this thesis. Because an option tends to be less liquid
when it becomes more in-the-money or more out-of-the-money, the bid-ask spread costs for
options have roughly a convex parabolic shape with a minimum on the ATM strike price and is
increasing in the strike prices in either direction. Moreover, because OTM options are naturally
cheaper than ATM and ITM options, the costs of the bid-ask spread cannot be modeled as
a fixed proportion of the option’s price, as this will underestimate (overestimate) the costs of
OTM (ITM) options. For this reason the costs involved in option trading are modeled by a fixed
proportion of the underlying security price, such that all options document the same transaction
costs and in addition, optimistically, the costs of (nearly) ATM options are overestimated. In
this way an upper bound for the bid-ask spread costs is constructed.
The costs are set equal to 5, 10 or 20 bps of the underlying security price for a single stock
option. Because CAC 40 index options are amongst the most actively traded index options in
the world (Blancard and Chaudhury, 2001), they are presumable more liquid than stock options.
Deng (2008) shows that bid-ask spreads for the S&P 500 index options is around two-thirds
of the single stock options as a percentage of the mid-price. Therefore, in this study the cost
for a single index option is set proportional to 2/3 of the 5, 10 or 20 bps of the underlying
57
index price. This corresponds (on average) to about e1.17, e2.33 and e4.67 as an one-way fee
per index option. In this study, a transaction costs scenario is only implemented for the delta-
hedged straddle combinations as it is believed that the results can be generalised to the strangle
strategies and furthermore to simplify the analysis, the transaction costs of delta-hedging, i.e.
equity trading, are not taken into account.
The strategies perform well under a transaction costs scenario of 5 bps of the underlying
asset price (Table 5.5). For EGARCH forecasting, the DECS tracking portfolio obtains a SR
of 1.309 and the PCA tracking portfolio a SR of 0.752, bootstrapped standard deviations are
equal to 0.57 and 0.52 for the same order of portfolios. For the naive and combination forecast
strategies, both tracking portfolios have (as expected) a lower but positive SR as well. When
the transaction costs are changed to 10 bps, the naive and combination forecast strategies based
on the PCA tracking portfolio become unprofitable. The profitability for the DECS portfolio
is sealed for all strategies based the 10 bps scheme. When the transaction costs are raised to
20 bps of the underlying asset’s price per option, the PCA tracking portfolio is unprofitable
for each strategy. The DECS tracking portfolio obtains a slightly positive SR with EGARCH
forecasting, SR is 0.045, yet the other strategies are fruitless.
5.6.3 Remarks
In this study it was not assumed that the options on the short leg of the strategy could be
exercised prior maturity when a short position was entered on the tracking portfolio, i.e. the
American single stock options. The P&L of the short positions in the EGARCH-, Bollinger
Bands- and mixing strategies are therefore likely to overestimate the gains and hence the daily
returns of dispersion trading. The total effect of early exercising on the P&L of these strategies
is unknown and left as an interesting subject for future research.
58
Tab
le5.
5:D
elta
-hed
ged
stra
tegi
esu
nd
ertr
ansa
ctio
nco
sts.
Th
ista
ble
rep
orts
the
aver
age
dai
lyre
turn
sof
the
dai
lyd
elta
-hed
ged
naiv
est
rad
dle
stra
tegy,
the
dail
yd
elta
-hed
ged
stra
dd
leco
mb
inati
on
-an
dp
osi
tion
fore
cast
stra
tegi
esan
dth
ed
aily
del
ta-h
edge
dst
rad
dle
mix
ing
stra
tegie
s,in
com
bin
ati
on
wit
hH
AC
stan
dard
dev
iati
on
s,t-
stats
from
ast
ati
on
ary
boots
trap
pro
ced
ure
,th
eL
illi
efor
ste
stst
atis
tics
andp-v
alu
es,
Bre
usc
h-G
od
frey
test
stati
stic
san
dp-v
alu
es,
an
nu
ali
sed
SR
sw
ith
stan
dard
dev
iati
on
sfr
om
ast
ati
on
ary
boot
stra
p,
and
the
CA
PM
mod
elco
effici
ent
esti
mat
esan
dt-
stati
stic
s.T
he
pro
bab
ilit
yof
the
sequen
ceev
ent,L
i=m,m
=1,2,...
,in
the
afo
rem
enti
on
ed
stat
ion
ary
boot
stra
pis
equ
alto
(1−p)m
−1p,
wh
erep
=T
−1/3,
an
dT
isth
enu
mb
erof
ob
serv
ati
on
sin
the
retu
rnse
ries
.
HA
CB
oots
trap
Lil
lief
ors
test
Bre
usc
h-G
od
frey
test
CA
PM
CA
PM
Str
ateg
yC
osts
Typ
eM
ean
Std
.D
ev.
t-st
at
LF
(p-v
alu
e)χ2 2
(p-v
alu
e)S
R(S
td.
Dev
.)α
(t-s
tat)
β(t
-sta
t)
Nai
ve5
bp
sD
EC
S0.
21%
4.61
%1.0
90.0
9b
(0.0
0)
1.8
5(0
.40)
0.8
54(0
.61)
0.0
021
(1.0
5)
0.1
899
(1.4
8)
5b
ps
PC
A0.
03%
5.77
%0.1
10.
10b
(0.0
0)
0.0
2(0
.99)
0.1
37(0
.61)
0.0
003
(0.1
3)
0.3
068
(1.8
8)
10b
ps
DE
CS
0.12
%4.
61%
0.6
10.0
9b
(0.0
0)
1.8
5(0
.40)
0.5
01
(0.5
9)
0.0
012
(0.5
8)
0.1
905
(1.4
8)
10b
ps
PC
A-0
.07%
5.77
%-0
.26
0.1
0b
(0.0
0)
0.0
2(0
.99)
-0.1
46
(0.6
0)
-0.0
006
(-0.2
4)
0.3
075
(1.8
8)
20b
ps
DE
CS
-0.0
7%4.
65%
-0.3
60.
09b
(0.0
0)
1.8
6(0
.39)
-0.2
00
(0.6
0)
-0.0
007
(-0.3
5)
0.1
917
(1.4
9)
20b
ps
PC
A-0
.25%
5.79
%-1
.00
0.1
0b
(0.0
0)
0.0
1(1
.00)
-0.7
08
(0.5
8)
-0.0
025
(-0.9
7)
0.3
087
(1.8
8)
Com
bin
atio
n5
bp
sD
EC
S0.
22%
5.06
%1.0
60.
10b
(0.0
0)
1.5
6(0
.46)
0.8
24(0
.62)
0.0
022
(1.0
3)
0.2
363
(1.7
0)
5b
ps
PC
A0.
03%
6.21
%0.1
10.
11b
(0.0
0)
0.1
4(0
.93)
0.1
32(0
.62)
0.0
003
(0.1
2)
0.3566a
(2.0
4)
10b
ps
DE
CS
0.13
%5.
06%
0.6
10.1
0b
(0.0
0)
1.5
8(0
.45)
0.5
01
(0.6
0)
0.0
013
(0.6
0)
0.2
370
(1.7
0)
10b
ps
PC
A-0
.06%
6.21
%-0
.24
0.1
1b
(0.0
0)
0.1
5(0
.93)
-0.1
31
(0.6
0)
-0.0
006
(-0.2
2)
0.3572a
(2.0
5)
20b
ps
DE
CS
-0.0
6%5.
09%
-0.2
70.
10b
(0.0
0)
1.6
9(0
.43)
-0.1
42(0
.58)
-0.0
006
(-0.2
6)
0.2
382
(1.7
0)
20b
ps
PC
A-0
.25%
6.23
%-0
.94
0.1
0b
(0.0
0)
0.1
7(0
.92)
-0.6
55
(0.5
7)
-0.0
025
(-0.9
1)
0.3585a
(2.0
5)
EG
AR
CH
5b
ps
DE
CS
0.28
%3.
59%
1.8
90.0
9b
(0.0
0)
1.0
2(0
.60)
1.3
09(0
.57)
0.0
027
(1.5
9)
0.1
164
(1.0
6)
5b
ps
PC
A0.
17%
4.04
%1.0
20.
09b
(0.0
0)
10.2
5b(0
.01)
0.7
52
(0.5
2)
0.0
017
(0.7
8)
0.1
580
(1.1
4)
10b
ps
DE
CS
0.18
%3.
61%
1.2
30.0
9b
(0.0
0)
0.9
4(0
.62)
0.8
84
(0.5
6)
0.0
018
(1.0
5)
0.1
170
(1.0
6)
10b
ps
PC
A0.
08%
4.06
%0.4
50.
09b
(0.0
0)
10.0
8b(0
.01)
0.3
72
(0.5
0)
0.0
008
(0.3
5)
0.1
586
(1.1
4)
20b
ps
DE
CS
-0.0
1%3.
66%
-0.0
40.
09b
(0.0
0)
0.6
7(0
.71)
0.0
45
(0.5
7)
-0.0
001
(-0.0
4)
0.1
183
(1.0
7)
20b
ps
PC
A-0
.11%
4.11
%-0
.67
0.0
9b
(0.0
0)
9.4
9b
(0.0
1)
-0.3
78
(0.4
9)
-0.0
011
(-0.5
1)
0.1
598
(1.1
5)
a=
sign
ifica
nt
atth
e5%
leve
lb
=si
gnifi
cant
atth
e1%
leve
l
59
Chapter 6
Conclusion
In this thesis the intriguing relationship between the implied volatilities derived from CAC
40 index options and portfolios of single stock options was studied. In the current body of
literature it is generally observed that index options trade against a premium compared to their
theoretical prices derived from the Black-Scholes model by reason of a dissimilarity between
the implied volatility of an index and its components. Dispersion trading is a trading strategy
based on monetising these market discrepancies.
Various dispersion trading strategies were discussed. Herein, the first step was to create
a tracking portfolio with minimal correlation risk with the index. Two optimisation methods
were discussed: the PCA method and the DECS method. It was of primary interest whether
search heuristics to encounter simultaneously the combinatorial and the continuous numerical
problem, implemented with market impact constraints, was able to produce significantly better
weighting schemes and therefore trading results than a linear dimension reduction method.
A naive trading strategy was tested and evaluated on the PCA and DECS tracking portfolio
weighting schemes, wherein combinations of options were used to monetise the hypothesized
discrepancies in the implied volatility of the CAC 40 compared to the implied volatilities of these
tracking portfolios. The gains from the naive trading strategy based on DECS indeed surpassed
the gains of using PCA. Moreover, the daily return volatility of using strangles (corrected for
autocorrelation) was more than twice the size of using straddles. Almost all naive strategies
showed positive performance and low market correlation when a daily delta-hedged procedure
was enforced. Notwithstanding, large daily losses were not uncommon.
The aim was to improve the results from the naive strategies by forecasting the implied
correlation between the index and the tracking portfolios, using either an EGARCH model or
Bollinger Bands. This study was able to show that both approaches improve the profitability
of dispersion trading significantly, while reducing the daily risk. Trading the volatility smile
had no univocal effect on the daily earnings compared to the naive straddle strategies, hence its
appositeness is dubious. However, a mixture between EGARCH implied correlation forecasting
and trading the volatility smile is found to be the most lucrative addition on a naive dispersion
trade.
60
In all situations, the trading strategies based on the DECS tracking portfolio had less correla-
tion risk with the CAC 40 index compared to the strategies based on the simple PCA tracking
portfolio, especially because the latter method was unable to capture unusual market events.
Introduction of a transaction costs model reduced the SRs significantly. Yet, the DECS com-
bination forecast strategy was able to gain little under the most intensive upper bound for
transaction costs.
Several simplifications were made in this study, it is interesting for future studies to inves-
tigate some of them. In the first place the transaction costs and also if possible the market
impact costs on an actively traded dispersion strategy must be elaborated. Furthermore, it was
neglected that in a short dispersion trade the buyer of the American single stock options had the
opportunity to exercise prior maturity, reversing the hedged dispersion position and augmenting
the correlation risk with the benchmark. A related issue is whether exit rules are remunera-
tive. Another field of interest is whether trading the volatility smile fosters the profitability of
dispersion trading.
61
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64
Appendix A
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
Date
Implie
d C
orr
ela
tion D
EC
S S
trangle
Implied Correlation
MA
MA plus 2 std
MA minus 2 std
Figure A.1: Implied correlation DECS tracking portfolio (strangle).
65
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12
0.7
0.8
0.9
1
1.1
1.2
1.3
Date
Implie
d C
orr
ela
tion P
CA
Str
addle
Implied Correlation
MA
MA plus 2 std
MA minus 2 std
(a) PCA straddle implied correlation
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr12
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Date
Implie
d C
orr
ela
tion P
CA
Str
angle
Implied Correlation
MA
MA plus 2 std
MA minus 2 std
(b) PCA strangle implied correlation
Figure A.2: Implied correlations PCA tracking portfolio.
66
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr120.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Date
Implie
d v
ola
tilit
y
CAC 40 impl. vol.
DECS Straddle impl. vol.
(a) DECS implied volatility based on 126-days historical returns
Jul10 Oct10 Jan11 Apr11 Jul11 Oct11 Jan12 Apr120.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Date
Implie
d v
ola
tilit
y
CAC 40 impl. vol.
PCA Straddle impl. vol.
(b) PCA implied volatility based on 126-days historical returns
Figure A.3: Implied volatilities for the straddle combination.
67
Jul1
0O
ct1
0Jan11
Apr1
1Jul1
1O
ct1
1Jan12
Apr1
22600
2800
3000
3200
3400
3600
3800
4000
4200
Date
Index
Jul1
0O
ct1
0Jan11
Apr1
1Jul1
1O
ct1
1Jan12
Apr1
2−
2000
02000
4000
6000
8000
10000
12000
14000
Cumulative Return
CA
C 4
0
DE
CS
Com
bin
ation
DE
CS
EG
AR
CH
Str
addle
DE
CS
BB
s S
traddle
DE
CS
EG
AR
CH
Str
angle
DE
CS
BB
s S
trangle
DE
CS
EG
AR
CH
Mix
DE
CS
BB
s M
ix
Fig
ure
A.4
:C
um
ula
tive
retu
rnD
EC
Ssi
gnal
trad
ing
(del
ta-h
edge
d).
68
Jul1
0O
ct1
0Jan11
Apr1
1Jul1
1O
ct1
1Jan12
Apr1
22000
3000
4000
5000
Date
Index
Jul1
0O
ct1
0Jan11
Apr1
1Jul1
1O
ct1
1Jan12
Apr1
2−
5000
05000
10000
Cumulative Return
CA
C 4
0
PC
A C
om
bin
ation
PC
A E
GA
RC
H S
traddle
PC
A B
Bs S
traddle
PC
A E
GA
RC
H S
trangle
PC
A B
Bs S
trangle
PC
A E
GA
RC
H M
ix
PC
A B
Bs M
ix
Fig
ure
A.5
:C
um
ula
tive
retu
rnP
CA
sign
altr
adin
g(d
elta
-hed
ged
).
69
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
20
40
60
80
100
120
140
160
Daily returns
Fre
que
ncy
Naive DECS Straddle
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
20
40
60
80
100
120
140
Daily returnsF
req
ue
ncy
Naive PCA Straddle
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
50
100
150
200
250
Daily returns
Fre
qu
en
cy
Naive DECS Strangle
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
20
40
60
80
100
120
140
160
180
200
Daily returns
Fre
qu
en
cy
Naive PCA Strangle
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
20
40
60
80
100
120
140
160
Daily returns
Fre
que
ncy
Combination DECS
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
20
40
60
80
100
120
140
Daily returns
Fre
que
ncy
Combination PCA
Figure A.6: Daily returns of the delta-hedged naive and combination strategies.
70
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20
50
100
150
Daily returns
Fre
que
ncy
EGARCH DECS Straddle
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
50
100
150
Daily returnsF
req
ue
ncy
EGARCH PCA Straddle
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
50
100
150
200
250
Daily returns
Fre
qu
en
cy
EGARCH DECS Strangle
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
20
40
60
80
100
120
140
160
180
200
Daily returns
Fre
qu
en
cy
EGARCH PCA Strangle
−0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.20
20
40
60
80
100
120
140
160
180
Daily returns
Fre
que
ncy
Mix EGARCH DECS
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
50
100
150
Daily returns
Fre
que
ncy
Mix EGARCH PCA
Figure A.7: Daily returns of the delta-hedged EGARCH strategies.
71
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
20
40
60
80
100
120
140
160
180
Daily returns
Fre
que
ncy
BBs DECS Straddle
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
50
100
150
Daily returnsF
req
ue
ncy
BBs PCA Straddle
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
50
100
150
200
250
Daily returns
Fre
qu
en
cy
BBs DECS Strangle
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.60
20
40
60
80
100
120
140
160
180
200
Daily returns
Fre
qu
en
cy
BBs PCA Strangle
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
20
40
60
80
100
120
140
160
180
Daily returns
Fre
que
ncy
Mix BBs DECS
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.30
50
100
150
Daily returns
Fre
que
ncy
Mix BBs PCA
Figure A.8: Daily returns of the delta-hedged Bollinger Bands strategies.
72
−0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 0.04 0.050
50
100
150
200
250
300
Daily returns
Fre
qu
en
cy
CAC 40
−0.06 −0.04 −0.02 0 0.02 0.04 0.060
50
100
150
200
250
300
Daily returns
Fre
qu
ency
Sanofi
−0.06 −0.04 −0.02 0 0.02 0.04 0.060
50
100
150
200
250
Daily returns
Fre
qu
ency
Total
−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.080
50
100
150
200
250
300
Daily returns
Fre
que
ncy
Vinci
−0.06 −0.04 −0.02 0 0.02 0.04 0.06 0.080
50
100
150
200
250
300
Daily returns
Fre
que
ncy
EDF
Figure A.9: Daily historical returns of the CAC 40 index and some constituents.
73
0 20 40 60 80 100 1200.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
Days to maturity (dtm)
Imp
l. v
ol
K=2200
K=3275
K=3525
K=3775
K=4025
K=4500
K=5500
Figure A.10: Implied volatility CAC 40 for different strike prices over the period 01-01-2010 to
31-05-2010.
74
Tab
leA
.1:
Au
gmen
ted
Dic
key-F
ull
erte
stre
turn
seri
es.
Th
eA
ugm
ente
dD
icke
y-F
ull
ert-
stati
stic
san
dp-v
alu
esfo
rth
eva
riou
sre
turn
seri
es.
Ret
urn
seri
esE
xce
ssre
turn
seri
es
Non
del
ta-h
edged
Del
ta-h
edged
Non
del
ta-h
edged
Del
ta-h
edged
Str
ateg
yC
omb
inat
ion
Typ
et-
stat
(p-v
alu
e)t-
stat
(p-v
alu
e)t-
stat
(p-v
alu
e)t-
stat
(p-v
alu
e)
Nai
veS
trad
dle
DE
CS
-23.0
5(0
.00)
-23.5
1(0
.00)
-23.1
4(0
.00)
-26.2
0(0
.00)
Str
add
leP
CA
-22.2
0(0
.00)
-22.8
3(0
.00)
-22.3
1(0
.00)
-25.1
8(0
.00)
Str
angl
eD
EC
S-2
3.2
0(0
.00)
-13.9
3(0
.00)
-23.3
0(0
.00)
-14.4
9(0
.00)
Str
angl
eP
CA
-22.6
9(0
.00)
-21.8
6(0
.00)
-22.7
6(0
.00)
-23.3
2(0
.00)
Com
bin
atio
n-
DE
CS
-22.2
1(0
.00)
-22.5
2(0
.00)
-22.3
9(0
.00)
-25.2
0(0
.00)
-P
CA
-21.9
0(0
.00)
-22.3
5(0
.00)
-22.0
7(0
.00)
-24.7
1(0
.00)
EG
AR
CH
Str
add
leD
EC
S-2
3.5
6(0
.00)
-23.4
6(0
.00)
-23.4
6(0
.00)
-24.8
6(0
.00)
Str
add
leP
CA
-20.1
1(0
.00)
-18.7
1(0
.00)
-20.1
8(0
.00)
-19.2
6(0
.00)
Str
angl
eD
EC
S-2
3.7
3(0
.00)
-24.1
3(0
.00)
-23.6
4(0
.00)
-26.5
4(0
.00)
Str
angl
eP
CA
-18.6
4(0
.00)
-23.5
5(0
.00)
-19.1
2(0
.00)
-25.8
3(0
.00)
BB
sS
trad
dle
DE
CS
-18.8
0(0
.00)
-22.0
3(0
.00)
-14.8
7(0
.00)
-14.8
4(0
.00)
Str
add
leP
CA
-20.0
7(0
.00)
-18.8
1(0
.00)
-14.9
7(0
.00)
-15.1
8(0
.00)
Str
angl
eD
EC
S-2
4.1
1(0
.00)
-22.2
4(0
.00)
-24.2
9(0
.00)
-24.1
4(0
.00)
Str
angl
eP
CA
-23.4
1(0
.00)
-22.1
5(0
.00)
-23.5
4(0
.00)
-23.7
4(0
.00)
Mix
-D
EC
S-2
2.4
6(0
.00)
-22.2
1(0
.00)
-22.3
9(0
.00)
-23.1
5(0
.00)
(EG
AR
CH
)-
PC
A-1
9.9
1(0
.00)
-18.4
0(0
.00)
-19.9
0(0
.00)
-18.7
9(0
.00)
Mix
-D
EC
S-2
2.5
5(0
.00)
-22.8
3(0
.00)
-22.6
1(0
.00)
-25.3
9(0
.00)
(BB
s)-
PC
A-1
8.2
1(0
.00)
-22.8
0(0
.00)
-18.6
8(0
.00)
-25.1
9(0
.00)
75
Table A.2: Characteristics of the implied volatilities.
This table reports the statistics of the implied volatility, the correlation between the CAC 40
index implied volatility and the tracking portfolio implied volatility, and the number of times
that the tracking portfolio implied volatility exceeded the CAC 40 implied volatility.
Panel 1A: Straddle Implied Volatility
DECS tracking portfolio PCA tracking portfolio Index
30 Days 126 Days 252 Days 30 Days 126 Days 252 Days -
Mean 0.235 0.239 0.239 0.214 0.218 0.217 0.238
Median 0.212 0.222 0.220 0.194 0.203 0.205 0.220
St. dev. 0.078 0.070 0.063 0.063 0.055 0.047 0.063
Min 0.132 0.129 0.134 0.122 0.127 0.133 0.153
Max 0.480 0.462 0.446 0.413 0.400 0.384 0.442
Correlation 0.959 0.959 0.957 0.926 0.944 0.940 -
TIV > MIV 237 289 278 83 80 112 -
Panel 1B: Strangle Implied Volatility
DECS tracking portfolio PCA tracking portfolio Index
30 Days 126 Days 252 Days 30 Days 126 Days 252 Days -
Mean 0.240 0.244 0.244 0.219 0.223 0.223 0.242
Median 0.218 0.228 0.225 0.200 0.210 0.211 0.224
St. dev. 0.076 0.069 0.061 0.062 0.054 0.046 0.062
Min 0.142 0.138 0.144 0.126 0.134 0.140 0.163
Max 0.482 0.463 0.448 0.415 0.402 0.386 0.443
Correlation 0.958 0.959 0.959 0.930 0.950 0.946 -
TIV > MIV 238 284 278 79 74 107 -
76
Table A.3: Characteristics of the implied volatility spread.
This table reports the statistics of the implied volatility spread (i.e. TIV - MIV),
in combination with t-stats from the stationary bootstrap, and p-values from the
Augmented Dickey-Fuller test.
Panel 2A: Straddle Implied Volatility Spread
DECS tracking portfolio PCA tracking portfolio
30 Days 126 Days 252 Days 30 Days 126 Days 252 Days
Mean 0.003 -0.001 -0.001 0.024 0.020 0.020
Median 0.004 -0.002 -0.001 0.021 0.018 0.018
St. dev. 0.025 0.020 0.019 0.024 0.021 0.025
Min -0.069 -0.055 -0.047 -0.031 -0.026 -0.041
Max 0.057 0.047 0.043 0.099 0.091 0.085
t-stat 0.75 -0.41 -0.47 7.35b 6.72b 5.52b
ADF p-value 0.015 0.012 0.001 0.003 0.004 0.011
Panel 2B: Strangle Implied Volatility Spread
DECS tracking portfolio PCA tracking portfolio
30 Days 126 Days 252 Days 30 Days 126 Days 252 Days
Mean 0.002 -0.002 -0.002 0.023 0.019 0.019
Median 0.003 -0.002 -0.002 0.021 0.018 0.018
St. dev. 0.025 0.020 0.018 0.023 0.020 0.023
Min -0.070 -0.058 -0.050 -0.031 -0.028 -0.041
Max 0.057 0.048 0.043 0.087 0.080 0.085
t-stat 0.62 -0.63 -0.74 7.52b 6.92b 5.48b
ADF p-value 0.016 0.018 0.002 0.003 0.020 0.011
b = significant at the 1% level
77
Appendix B
Table B.1: Parameters of the DECS optimisation.
Parameter Value Description
P 100 Population
L 1 Lower bound for total assets in tracking portfolio
K 8 Upper bound for total assets in tracking portfolio
cf 0.7 Crossover factor
f 0.3 Scaling factor
ε 0.01 Lower bound for asset weight
ξ 1 Upper bound for asset weight
N 2500 Number of iterations
M 0.2 Maximum deviation from previous weight allocation
78
