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Discrete Financial Mathematics
Wim Schoutens
Leuven, 2004-2005
Lecture Notes to the Course
(G0Q20a) Discrete Financial Mathematics
Abstract
The aim of the course is to give a rigorous yet accessible introduction to the
modern theory of discrete financial mathematics. The student should already
be comfortable with calculus and probability theory. Prior knowledge of basic
notions of finance is useful.
We start with providing some background on the financial markets and the
instruments traded. We will look at different kinds of derivative securities, the
main group of underlying assets, the markets where derivative securities are
traded and the financial agents involved in these activities. The fundamen-
tal problem in the mathematics of financial derivatives is that of pricing and
hedging. The pricing is based on the no-arbitrage assumptions. We start by dis-
cussing option pricing in the simplest idealised case: the Single-Period Market.
Next, we turn to Binomial tree models. Under these models we price European
and American options and discuss pricing methods for the more involved exotic
options. Monte-Carlo issues come into play here.
Finally, we set up general discrete-time models and look in detail at the
mathematical counterpart of the economic principle of no-arbitrage: the exis-
tence of equivalent martingale measures. We look when the models are complete,
i.e. claims can be hedged perfectly. We discuss the Fundamental theorem of
asset pricing in a discrete setting.
To conclude the course, we make a bridge to continuous-time models. We
look at them as limiting cases of discrete models. The discrete models will guide
us in the analysis of continuous-time models in the Continuous Mathematical
Finance Course.
2
Contents
1 Derivative Background 1
1.1 Financial Markets and Instruments . . . . . . . . . . . . . . . . . 1
1.1.1 Basic Instruments . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 The Bank Account . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Derivative Instruments . . . . . . . . . . . . . . . . . . . . 9
1.1.4 Markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.1.5 Contract Specifications . . . . . . . . . . . . . . . . . . . 14
1.1.6 Types of Traders . . . . . . . . . . . . . . . . . . . . . . . 15
1.1.7 Modelling Assumptions . . . . . . . . . . . . . . . . . . . 17
1.2 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.3 Arbitrage Relationships . . . . . . . . . . . . . . . . . . . . . . . 23
1.3.1 The Put-Call Parity . . . . . . . . . . . . . . . . . . . . . 23
1.3.2 The Forward Contract . . . . . . . . . . . . . . . . . . . . 26
1.3.3 Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.4 Currencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1
1.3.5 Commodities . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.6 The Cost of Carry . . . . . . . . . . . . . . . . . . . . . . 32
2 Binomial Trees 34
2.1 Single Period Market Models . . . . . . . . . . . . . . . . . . . . 34
2.2 Two-Step Binomial Trees . . . . . . . . . . . . . . . . . . . . . . 43
2.2.1 European Call . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2.2 Matching Volatility with u and d . . . . . . . . . . . . . . 46
2.3 Binomial Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3.1 European Call and Put Options . . . . . . . . . . . . . . 49
2.3.2 American Options . . . . . . . . . . . . . . . . . . . . . . 53
2.4 Moving towards The Black-Scholes Model . . . . . . . . . . . . . 59
3 Mathematical Finance in Discrete Time 62
3.1 Information and Trading Strategies . . . . . . . . . . . . . . . . . 63
3.2 No-Arbitrage Condition . . . . . . . . . . . . . . . . . . . . . . . 67
3.3 Risk-Neutral Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Complete Markets . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5 The Fundamental Theorem of Asset Pricing . . . . . . . . . . . . 75
3.5.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4 Exotic Options 82
4.1 Monte Carlo Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2 Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.3 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2
4.4 Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 The Black-Scholes Option Price Model 100
5.1 Continuous-Time Stochastic Processes . . . . . . . . . . . . . . . 101
5.1.1 Information and Filtration . . . . . . . . . . . . . . . . . . 101
5.1.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.2 Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.3 Ito’s Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3.1 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . 107
5.3.2 Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . 110
5.5 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . 112
5.6 The Market Model . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.7 Equivalent Martingale Measures and Risk-Neutral Pricing . . . . 117
5.7.1 The Pricing of Options under the Black-Scholes Model . . 120
5.7.2 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.8 The Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.9 Drawbacks of the Black-Scholes Model . . . . . . . . . . . . . . . 129
6 Miscellaneous 132
6.1 Decomposing Options into Vanilla Position . . . . . . . . . . . . 132
6.2 Variance Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3
Chapter 1
Derivative Background
1.1 Financial Markets and Instruments
A market typically consist out of a riskfree Bank account and some other risky
assets. On these basic instruments other financial contracts are written; these
financial contracts are so-called derivative securities. This text is on the (risk-
neutral) pricing of derivative securities.
This section provides the institutional background on the main group of
underlying assets, the related derivative securities, the markets where derivatives
securities are traded and the financial agents involved in these activities.
1.1.1 Basic Instruments
Next, we highlight some of the most common underlying securities.
1
Stocks – Equity
The basis of modern economic life are companies owned by their shareholders;
the shares provide partial ownership of the company, pro rata with investment.
Shares have value, reflecting both the value of the company’s real assets and
the earning power of the company’s dividends. With publicly quoted companies,
shares are quoted and traded on the Stock Exchange. Stock is the generic term
for assets held in the form of shares. Stock markets date back to at least 1531,
when one was started in Antwerp, Belgium. Today there are over 150 stock
exchanges.
Interest Rates – Fixed-Income
The value of some financial assets depends solely on the level of interest rates,
e.g. Treasury notes, municipal and corporate bonds. These are fixed-income
securities by which national, state and local governments and large companies
partially finance their economic activity. Fixed-income securities require the
payment of interest in the form of a fixed amount of money at predetermined
points in time, as well as the repayment of the principal at maturity of the secu-
rity. Interest rates themselves are notional assets, which cannot be delivered. A
special fixed-income product is the bank account, which we typically assume to
be riskfree. We go into more detail on possible bank account models in Section
1.1.2.
2
Currencies – Foreign Exchange
A currency is the denomination of the national units of payment (money) and
as such is a financial asset. Companies may wish to hedge adverse movements
of foreign currencies and in doing so use derivative instruments.
A foreign currency has the property that the holder of the currency can earn
interest at the risk-free interest rate prevailing in the foreign country. We thus
have to kind of interest rates, the domestic and the foreign interest rate.
Commodities
Commodities are a kind of physical products like gold, oil, cattle, fruit juice.
Trade in these assets can be for different purposes: for using them in the pro-
duction process or for speculation. Derivative instruments on these asset can
be used for hedging and speculation. Special care has to be taken with com-
modities because of storage costs (see Section 1.3.5) In Figure 1.1, one sees the
some prices (of the market on the 16th of October 2004) of (futures on) energy,
metal, livestock and other commodities.
3
Figure 1.1: Commodities future prices on 16-10-2004
Miscellaneous
Indexes
An index tracks the value of a basket of stocks (FTSE100, S&P500, Dow
Jones Industrial, NASDAQ Composite, BEL20, EUROSTOXX50, ...), bonds,
and so on. Derivative instruments on indices may be used for hedging if no
4
derivative instruments on a particular asset in question are available and if the
correlation in movement between the index and the asset is significant. Further-
more, institutional funds (such as pension funds), which manage large diversi-
fied stock portfolios, try to mimic particular stock indices and use derivative on
stock indices as a portfolio management tool. On the other hand, a speculator
may wish to bet on a certain overall development in a market without exposing
him/herself to a particular asset.
In Figure 1.2, one sees the Belgian Bel-20 Index over a period of more than
4 years.
Figure 1.2: BEL-20
A new kind of index was generated with the Index of Catastrophe Losses
(CAT-index) by the Chicago Board of Trade (CBOT) lately. The growing num-
ber of huge natural disasters (such as hurricanes and earthquakes) has led the
insurance industry to try to find new ways of increasing its capacity to carry
risks. Currently investors are offered options on the CAT-index, thereby taking
5
in effect the position of traditional reinsurance.
Credit Risk Market
Credit risk captures the risk on a financial loss that an institution incurs
when it lends money to another institution or person. This financial loss real-
izes whenever the borrower does not meet all of its obligations under its borrow-
ing contract. Because credit risk is so important for financial institutions the
banking world has developed instruments that allows them to evacuate credit
risk rather easily. The most commonly known and used example is the credit
default swap. These instruments can best be considered as tradable insurance
contracts. This means that today I can buy protection on a bond and tomorrow
I can sell that same protection as easily as I bought it. Credit default swaps
work just as an insurance contract. The protection buyer (the insurance taker)
pays a fee and in exchange he gets reimbursed his losses if the company on which
he bought protection defaults.
In Figure 1.3, one sees credit default swap bid and offer rates for major
aerospace/transport and auto (parts) companies.
1.1.2 The Bank Account
In the next Chapter we will start building models for the stock or asset price
process. Here we focus on one instrument which we typically assume to be
available in all the later on encountered market models: the bank account. There
are two (quite related) regimes under which we will work: discrete or continuous
compounding. This has all to do with when and how frequently the interest
6
Figure 1.3: Credit Default Swap rates
gained on the invested money is paid out. Typically the discrete compounding
will be only used in discrete time models; the continuous compound can be used
in almost any situation.
Consider an amount A is invested for n years at an interest rate of R per
annum. If the rate is compounded once per annum, the terminal value of the
investment is
A(1 +R)n.
If it is compounded m times per annum, the terminal value of the investment is
A
(
1 +R
m
)mn
.
We note that there is a difference. Indeed, take A = 100 euro and n = 1, when
7
the interest rate is 10 percent a year. The first regime leads to 110 euros after
one year. However, with quarterly payments (m = 4), i.e. with payments every
three month, we have after one year 100× (1.0025)4 = 110.38 euros. In Figure
1.4 one sees the effect of increasing the compounding frequency from a yearly
compounding to a daily compounding.
0 50 100 150 200 250 300 350110
110.1
110.2
110.3
110.4
110.5
110.6discrete compound interest rates (A=100, n=1, r=0.10)
compounding frequency (m)
end
valu
e
Figure 1.4: Discrete compounding
The limit as m tends to infinity is known as continuous compounding. We
have
limm→∞
A
(
1 +R
m
)mn
= A exp(nR).
With such a continuous compounding the invested amountA grows toA exp(nR)
after n years. Note that it is because of the above discussion, important to state
the units/frequency in which the interest rate is measured/compounded. For
example, an interest rate of 10 percent continuous compounding is the same as
8
(1 − exp(0.10))/100 = 10.517 percent annual compounding.
Throughout the text we will make use of a bank account on which we can
put money and borrow money on a fixed continuously compounded interest rate
r. This means that 1 euro on the bank account at time 0 will give rise to ert
euro on time t > 0. Similarly, if we borrow 1 euro now, we have to pay back ert
euro at time t > 0. Or equivalently, if we borrow now e−rt euro we have to pay
back 1 euro at time t.
One euro on the bank account will grow over time; at some time t we denote
its value by B(t). Note thus that we set B(0) = 1. We call B = B(t), t ≥ 0
the bank account price process or bond price process.
Related to all this is the time value of money. An investor will prefer 100 euro
in his pocket today to 100 euro in his pocket one year from now. The interest
paid on the riskless bank account expresses this. Using continuous compounding
with a rate r = 0.10, 100 euro is equivalent with 110.517 euros in one year. If
we receive a cash-flow X at some future time T , the equivalent now is equal to
exp(−rT )X . 100 euro in one year is equivalent with 90.484 euros now. This
procedure is called discounting and exp(−rT ) is the discounting factor.
1.1.3 Derivative Instruments
In practitioner’s terms a ’derivative security’ is a security whose value depends
on the value of other more basic underlying securities. We adopt the more
precise definition:
A derivative security, or contingent claim, is a financial contract whose value
9
at expiration date T (more briefly, expiry) is determined exactly by the price
process of the underlying financial assets (or instruments) up to time T.
Derivative securities can be grouped under three general headings: Options,
Forwards and Futures, and Swaps. During this text we will mainly deal with
options although our pricing techniques may be readily applied to forwards,
futures and swaps as well.
Options
An option is a financial instrument giving one the right but not the oblig-
ation to make a specified transaction at (or by) a specified date at a specified
price.
A lot of different type of options exists. We give here the basic types. Call
options give one the right to buy. Put options give one the right to sell. European
options give one the right to buy/sell on the specified date, the expiry date,
on when the option expires or matures. American options give one the right
to buy/sell at any time prior to or at expiry. Asian options depend on the
average price over a period. Lookback options depend on the maximum or
minimum price over a period and barrier options, depend on some price level
being attained or not.
The price at which the transaction to buy/sell the underlying assets (or
simply the underlying), on/by the expiry date (if exercised), is made is called
the exercise price or strike price. We usually use K for strike price, time t = 0,
for the initial time (when the contract between the buyer and the seller of the
10
option is struck), time t = T for the expiry or final time.
Consider, say, an European call option, with strike price K; write St for the
value (or price) of the underlying at time t. If St > K, the option is in the
money, if St = K, the option is said to be at the money and if St < K, the
option is out the money. This terminology is of course motivated by the payoff,
the value of the option at maturity, from the option which is
ST −K if ST > K and 0 otherwise
(more briefly written as (ST − K)+). This payoff function for K = 100 is
visualized in Figure 1.5
80 85 90 95 100 105 110 115 1200
2
4
6
8
10
12
14
16
18
20Payoff of European Call (K=100)
stock price at maturity
Payo
ff
Figure 1.5: Payoff of Call Option (K=100)
There are two sides to every option contract. On one side there is the person
who has bought the option (the long position); on the other side you have the
11
person who has sold or written the option (the short poistion). The writer
receives cash up front but has potential liabilities later.
In Figure 1.6, one can see that by investing in an option one can make huge
gains, but also if markets goes the opposite direction as anticipate, it is possible
to loose all money one has invested.
Figure 1.6: Stock Prices and European Call Option at time t = 0 and t = T .
In 1973, the Chicago Board Options Exchange (CBOE) began trading in
options on some stocks. Since then, the growth of options has been explosive.
Risk Magazine (12/1997) estimated $35 trillion as the gross figure for worldwide
derivatives markets in 1996.
In Figure 1.7, one sees some of the prices of call options written on the SP500-
index. The main aim of this text is to give a basic introduction to models for
determining these kind of option prices.
12
Forwards, Futures
A forward contract is an agreement to buy or sell an asset at a certain future
date T for a certain price K. It is usually between two large and sophisticated
financial agents (banks, institutional investors, large corporations, and broker-
age firms) and not traded on an exchange. The agents who agrees to buy the
underlying asset is said to have a long position, the other agent assumes a short
position. The payoff from a long position in a forward contract on one unit of
an asset with price ST at the maturity time T of the contract is
ST −K.
Compared with a call option with the same maturity and strike price K we see
that the investor now faces a downside risk, too. He has the obligation to buy
the asset for price K.
A futures contract, like a forward contract, is an agreement to buy or sell an
asset at a certain future date for a certain price. The difference is that futures
are traded. As such, the default risk is removed from the parties to the contract
and borne by the clearing house.
Swaps
A swap is an agreement whereby two parties undertake to exchange, at known
dates in the future, various financial assets (or cash flows) according to a pre-
arranged formula that depends on the value of one or more underlying assets.
Examples are currency swaps (exchange currencies) and interest-rate swaps (ex-
13
change of fixed for floating set of interest payments) and the nowadays popular
credit default swaps as in Figure 1.3.
1.1.4 Markets
Financial derivatives are basically traded in two ways: on organized exchanges
and over-the-counter (OTC). Organized exchanges are subject to regulatory
rules, require a certain degree of standardization of the traded instruments
(strike price, maturity dates, size of contract, etc.). Examples are the Chicago
Board Options Exchange (CBOE), the London International Financial Futures
Exchange (LIFFE).
The exchange clearinghouse is an adjunct of the exchange and acts as an
intermediary in the transactions. It garantuess the performanjce of the parties
to each transaction. Its main task is to keep track of all the transactions that
take place during a day so it can calculate the net poistions of each of its
members.
OTC trading takes between various commercial and investments banks such
as Goldman Sachs, Citibank, Deutsche Bank.
1.1.5 Contract Specifications
It is very important that the financial contract specifies in detail the exact nature
of the agreement between the two parties. It must specify the contract size (how
much of the asset will be delivered under one contract), where delivery will be
made, when exactly the delivery is made, etc. When the contract is traded at
14
an exchange, it should be made clear how prices will be quoted, when trade is
allowed, etc.
Financial assets in derivatives are generally well defined and unambiguous,
e.g. it is clear what a Japanese Yen is. When the asset is a commodity, there
may be quite a variation in the quality and it is important that the exchange
stipulates the grade or grades of the commodity that are acceptable.
Most contracts are refered to by its delivery month and year. The contract
must specify in detail the period of that month when delivery can be made. For
some future the delivery period is the entire month. For other contract delivery
must be at a special day, hour, etc.
Some contracts are in terms of a so-called settlement price. For example
derivatives on indices (like the SP-500). The settlement price is calculated by
the exchange by a very detailed algorithm. It can be e.g. averages of the index
taken every five minutes during one hour, but also just the closing price of the
asset.
Other specification by the exchange deal with movement limits. Trade will be
halted if these limits are exceeded. The purpose of price limits is to prevent large
movements from occurring because of speculative excesses, extremal situation
(11th of September), ...
1.1.6 Types of Traders
We can classify the traders of derivatives securities in three different classes.
15
Hedgers
Successful companies concentrate on economic activities in which they to best.
They use the market to insure themselves against adverse movements of prices,
currencies, interest rates etc. Hedging is an attempt to reduce exposure to risk.
Hedgers prefer to forgo the chance to make exceptional profits when future un-
certainty works to their advantage by protecting themselves against exceptional
loss.
Speculators
Speculators want to take a position in the market – they take the opposite
position to hedgers. Indeed, speculation is needed to make hedging possible, in
that a hedger, wishing to lay off risk, cannot do so unless someone is willing to
take it on.
In speculation, available funds are invested opportunistically in the hope of
making a profit: the underlying itself is irrelevant to the speculator, who is only
interested in the potential for possible profit that trade involving it may present.
Arbitrageurs
Arbitrageurs try to lock in riskless profit by simultaneously entering into trans-
actions in two or more markets. An arbitrage opportunity exists, for example, if
a security can be bought in New York at one price and sold at a slightly higher
price in London. The underlying concept of the here presented theory is the
absence of arbitrage opportunities.
16
1.1.7 Modelling Assumptions
We will discuss contingent claim pricing in an idealized case. We will not allow
market frictions; there is no default risk, agents are rational and there is no
arbitrage. More concrete this means
• no transaction costs
• no bid/ask spread
• no taxes
• no margin requirements
• no restrictions on short sales
• no transaction delays
• same interest for borrowing and lending
• market participants act as price takers
• market participants prefer more to less
We develop the theory of an ideal – frictionless – market so as to focus
irreducible essentials of the theory and as a first-order approximation to real-
ity. Understanding frictionless markets is also a necessary step to understand
markets with frictions.
The risk of failure of a company – bankruptcy – is inescapably present in
its economic activity: death is part of life. Moreover those risks also appear
at the national level: quite apart from war, recent decades have seen default
17
of interest payments of international debt, or the threat of it (see for example
the 1998 Russian crisis). We ignore default risk for simplicity while developing
understanding of the principal aspects.
We assume financial agents to be price takers, not price makers. This implies
that even large amounts of trading in a security by one agent does not influence
the security’s price. Hence agents can buy or sell as much of any security as
they wish without changing the security’s price.
To assume that market participants prefer more to less is a very weak as-
sumption on the preferences of market participants. Apart from this we will
develop a preference-free theory.
The relaxation of all these assumptions is subject to ongoing research.
We want to mention the special character of the no-arbitrage assumption.
It is the basis for the arbitrage pricing technique that we shall develop, and we
discuss it in more detail below.
1.2 Arbitrage
The essence of arbitrage is that it should not be possible to guarantee a profit
without exposure to risk. Were it possible to do so, arbitrageurs would do so,
in unlimited quantity, using the market as a money-pump to extract arbitrarily
large quantities of riskless profit. This would, for instance, make it impossible
for the market to be in equilibrium. We shall see that arbitrage arguments
suffice to determine prices - the arbitrage pricing technique.
18
To explain the fundamental arguments of the arbitrage pricing technique we
use the following:
Example: Consider an investor who acts in a market in which only three
financial assets are traded: (riskless) bonds B (bank account), stocks S and
European Call options C with strike K = 100 on the stock S. The investor may
invest today, time t = 0, in all three assets, leave his investment until time
t = T and gets his returns back then. We assume the option C expires at time
t = T . We assume the current prices (in euro, say) of the financial assets are
given by
B(0) = 1, S(0) = 100, C(0) = 20
and that at t = T there can be only two states of the world: an up-state with
euro prices
B(T, u) = 1.25, S(T, u) = 175, and therefore C(T, u) = 75,
and a down-state with euro prices
B(T, d) = 1.25, S(T, d) = 75, and therefore C(T, d) = 0.
Now our investor has a starting capital of 25000 euro from which he buy the
following portfolio,
Portfolio I:
Asset Number Total amount in euro
Bond 10000 10000
Stock 100 10000
Call option 250 5000
19
Depending of the state of the world at time t = T the value of his portfolio
will differ: In the up state the total value of his portfolio is 48750 euro:
Asset Number × Price Total amount in euro
Bond 10000 × 1.25 12500
Stock 100 × 175 17500
Call option 250 × 75 18750
TOTAL 48750
whether in the down-state his portfolio has a value of 20000 euro:
Asset Number × Price Total amount in euro
Bond 10000 × 1.25 12500
Stock 100 × 75 7500
Call option 250 × 0 0
TOTAL 20000
Can the investor do better ? Let us consider the restructured portfolio with
initial investment of 24600 euro:
Portfolio II:
Asset Number Total amount in euro
Bond 11800 11800
Stock 70 7000
Call option 290 5800
We compute its return in the different possible states. In the up-state the total
20
value of his portfolio is again 48750 euro:
Asset Number × Price Total amount in euro
Bond 11800 × 1.25 14750
Stock 70 × 175 12250
Call option 290 × 75 21750
TOTAL 48750
and in the down-state his portfolio has again a value of 20000 euro:
Asset Number × Price Total amount in euro
Bond 11800 × 1.25 14750
Stock 70 × 75 5250
Call option 290 × 0 0
TOTAL 20000
We see that this portfolio generates the same time t = T return while costing
only 24600 euro now, a saving of 400 euro against the first portfolio. So the
investor should use the second portfolio and have a free lunch today!
In the above example the investor was able to restructure his portfolio, re-
ducing the current (t = 0) expenses without changing the return at the future
date t = T in both possible states of the world. So there is an arbitrage pos-
sibility in the above market situation, and the prices quoted are not arbitrage
prices. If we regard (as we shall do) the prices of the bond and the stock
(our underlying) as given, the option must be mispriced. Let us have a closer
look between the differences between Portfolio II, consisting of 11800 bonds, 70
stocks and 29 call options, in short (11800, 70, 290) and Portfolio I, of the form
21
(10000, 100, 250) The difference is the portfolio, Portfolio III say, of the form
(11800, 70, 290)− (10000, 100, 250) = (1800,−30, 40).
Asset Number Total amount in euro
Bond 1800 1800
Stock -30 -3000
Call option 40 800
So if you sell short 30 stocks, you will receive 3000 euro from which you
buy 40 options, put 1800 euro in your bank account and have a gastronomic
lunch of 400 euro. But what is the effect of doing that ? Let us consider the
consequences in the possible states of the world. We see in both cases that
the effects of the different positions of Portfolio III offset themselves: In the
up-state:
Asset Number × Price Total amount in euro
Bond 1800 × 1.25 2250
Stock -30 × 175 -5250
Call option 40 × 75 3000
TOTAL 0
In the down state:
Asset Number × Price Total amount in euro
Bond 1800 × 1.25 2250
Stock -30 × 75 -2250
Call option 40 × 75 0
TOTAL 0
22
But clearly the portfolio generates an income at t = 0 of which you had a free
lunch, and a good one. Therefore it is itself an arbitrage opportunity.
If we only look at the position in bonds and stocks, we can say that this
position covers us against possible price movements of the option, i.e. having
1800 euro in your bank account and being 30 stocks short has the opposite time
t = T value as owning 40 call options. We say that the bond/stock position is
a hedge against the position in options.
Let us emphasize that the above arguments were independent of the prefer-
ences and plans of the investor.
1.3 Arbitrage Relationships
We will in this section use arbitrage-based arguments to develop general bounds
on the value of options. In our analysis here we use non-dividend paying stocks
as the underlying, with price process S = St, t ≥ 0. We assume we have a
risk-free bank account available which uses continuously compounding with a
fixed interest rate r.
1.3.1 The Put-Call Parity
Next, we will deduce a fundamental relation between put and call options, the
so-called put-call parity. Suppose there is a stock (with value St at time t), with
European call and put options on it, with value Ct and Pt respectively at time
t, with expiry time T and strike-price K. Consider a portfolio consisting of one
stock, one put and a short position in one call (the holder of the portfolio has
23
written the call); write Πt for the time t value of this portfolio. So
Πt = St + Pt − Ct.
Recall that the payoffs at expiry are
for the call : CT = maxST −K, 0 = (ST −K)+,
for the put : PT = maxK − ST , 0 = (K − ST )+.
For the above portfolio we hence get at time T the payoff
if ST ≥ K : ΠT = ST + 0 − (St −K) = K,
if ST ≤ K : ΠT = ST + (K − St) − 0 = K.
This portfolio thus guarantees a payoff K at time T . How much is it worth
at time t? The riskless way to guarantee a payoff K at time T is to deposit
Ke−r(T−t) in the bank at time t and do nothing (we assume continuously com-
pounded interest here). Under the assumption that the market is arbitrage-free
the value of the portfolio at time t must thus be Ke−r(T−t), for it acts as a
synthetic bank account and any other price will offer arbitrage opportunities.
Let us explore these arbitrage opportunities.
If the portfolio is offered for sale at time t too cheaply–at price Πt <
Ke−r(T−t) – we can buy it, borrow Ke−r(T−t) from the bank, and pocket a
positive profit Ke−r(T−t) −Πt > 0. At time T our portfolio yields K, while our
bank debt has grown to K. We clear our cash account – use the one to pay off
the other – thus locking in our earlier profit, which is riskless.
If on the other hand the portfolio is priced at time t at a too high price –
at price Πt > Ke−r(T−t) – we can do the exact opposite. We sell the portfolio
24
short – that is,we buy its negative: buy one call, write one put, sell short one
stock, for Πt and invest Ke−r(T−t) in the bank account, pocketing a positive
profit Πt −Ke−r(T−t) > 0. At time T , our bank deposit has grown to K, and
again we clear our cash account – use this to meet our obligation K on the
portfolio we sold short, again locking in our earlier riskless profit.
We illustrate the above with so-called arbitrage tables. In such a table we
simply enter the current value of a given portfolio and then compute its value
in all possible states of the world when the portfolio is cashed in. In the case
Πt < Ke−r(T−t):
Transactions Current cash flow Value at expiry
ST < K ST ≥ K
buy 1 stock −St ST ST
buy 1 put −Pt K − ST 0
write 1 call Ct 0 −ST +K
borrow Ke−r(T−t) −K −K
TOTALKe−r(T−t) − St
−Pt + Ct > 0
0 0
Thus the rational price for the portfolio at time t is exactly Ke−r(T−t). Any
other price presents arbitrageurs with an arbitrage opportunity (to make and
lock in a riskless profit) – which they will take ! Therefore
Proposition 1 We have the following put-call parity between the prices of the
underlying asset and its European call and put options with the same strike price
25
and maturity on stocks that pay no dividends:
St + Pt − Ct = Ke−r(T−t).
The value of the portfolio above is the discounted value of the riskless equiv-
alent. This is a first glimpse at the central principle, or insight, of the entire
subject of option pricing. Arbitrage arguments allow one to calculate precisely
the rational price – or arbitrage price – of a portfolio. The put-call parity
argument above is the simplest example of the arbitrage pricing technique.
1.3.2 The Forward Contract
Next, we will deduce a fair price (based on the no-arbitrage assumption) for the
following forward contract: The contract states that party A (the buyer) must
buy from party B (the seller) the (non-dividend paying) stock at time T at the
price K (the strike price).
We claim that F = S0 − exp(−rT )K is the correct initial price of this
derivative which party A will pay to party B at time t = 0. Indeed, suppose
you are party B, so you sold the forward contract and has received at time
t = 0, the amount F . At time zero, you do the following:
• buy 1 stock for the price S0;
• borrow exp(−rT )K.
To buy the stock you need S0. You already have from the forward F = S0 −
exp(−rT )K and receives from your loan exp(−rT )K. So you spent all the
available money and at time t = 0 you have the following portfolio:
26
• long 1 stock.
• short 1 forward;
• short exp(−rT )K bonds.
Look what happens at time T . You must deliver the stock to party A. You
give away your stock in your portfolio, for this you receive K. The forward
contract ends and you pay back your bank. You have to pay back the amount
exp(rT ) exp(−rT )K = K. This you can do exactly with the money you received
from party A. In the end everything is settled, you have no gain, no lost. Note
that your initial investment is also zero.
Note that any other price for the forward would have led to an arbitrage
situation. Indeed, suppose you received F > F . Then by following the above
strategy you pocket at time t = 0 the difference F −F > 0 which you can freely
spent. At time T you just close all the position as described above. Without
any initial investment and risk, you have then spent at time 0, F −F > 0. This
is clearly an arbitrage opportunity (for party B). In case F < F , party A con
set up a portfolio will always leads to an arbitrage opportunity (check this !).
Forward/future contracts in practice are almost always struck at the price
K, such that F = S0 − exp(−rT )K = 0, i.e.
K = exp(rT )S0.
By doing this entering (in both ways: long or short) a forward contract is at
zero cost. For this reason, exp(rT )S0 is called the time T forward price of the
stock.
27
Finally, note that the put-call parity can be simply rewritten in terms of
the call price, the put price and the forward contract price as: C − P = F (all
derivatives have the same strike and time to maturity). From this one can see
that at the forward price of the stock, i.e. in case K = exp(rT )S0 and hence
F = 0, call and puts have the same value: C = P .
1.3.3 Dividends
Up to now, we have assumed that the risky asset pays no dividends, however
in reality stocks can pay some dividend to the stock holders at some moments.
We assume that the amount and timing of the dividends during the live of an
option can be predicted with certainty. Moreover, we will assume that the stock
pays a continuous compound dividend yield at rate q per annum. Other ways
of dividend payments can be considered and techniques are described in the
literature to deal with this.
Continuous payment of a dividend yield at rate q means that our stock is
following a process of the form:
St = exp(−qt)St,
where S is describing the stock prices behavior not taking into account divi-
dends. A stock which pays continuously dividends and an identical stock that
not pays dividends should provide the same overall return, i.e. dividends plus
capital gains. The payment of dividends causes the growth of the stock price to
be less than it would otherwise be by an amount q. In other words, if, with a
continuous dividend yield of q, the stock price grows from S0 to ST at time T ,
28
then in absence of dividends it would grow from S0 to exp(qt)ST . Alternatively,
in absence of dividends it would grow from exp(−qt)S0 to ST . This argumen-
tation brings us to the fact that we get the same probability distribution for
the stock price at time T in the following cases: (1) The stock starts at S0
and pays a continuous dividend yield at rate q and (2) the stock starts at price
exp(−qt)S0 and pays no dividend yield.
The put-call parity for a stock with dividend yield q can be obtained from the
put-call parity for non-dividend-paying stocks. With no dividends we obtained
:
St + Pt − Ct = K exp(−r(T − t)).
If we now take into account dividends, the change comes down to replacing St
with St exp(−q(T − t)). We have:
exp(−q(T − t))St + Pt − Ct = K exp(−r(T − t)).
This relation can be proven by considering the portfolio consisting of exp(−q(T−
t)) number of stocks, one put option and minus one call option. We reinvest
the dividends on the shares instantaneously in additional shares, i.e. at some
future time point t ≤ s ≤ T , we have exp(−q(T − s)) number of stock; at the
expiry date of the option we own one stock, one put and minus one call. The
value of the portfolio at that time thus always equals K. By the no-arbitrage
argument the time T value of the portfolio must equal K exp(−r(T − t)), the
time t-value of a future payment (at time T ) of K.
If our asset is an index, the dividend yield is the (weighted) average of the
29
dividends yields on the stocks composing the index.In practice, the dividend
yield can be determined from the forward price of the asset. It is the agreement
to buy or sell an asset at a certain future time for a certain price, the delivery
price. At the time the contract is entered into, the delivery price is chosen so
that the value of the forward is zero. This means that it costs nothing to buy or
sell the contract. For an asset paying a continuous yield at rate q, the delivery
price of a forward contract expiring at time T , is given by (proof this yourself !)
F = S0 exp((r − q)T ). (1.1)
Assuming the short rate r and the delivery price of the forward as given, q can
easily be obtained.
1.3.4 Currencies
If the underlying is not a stock but a currency, we must take into account the
domestic as well as the foreign interest rate. Let us continuous compounding
and denote these interest rates by rd and rf , respectively.
We are in Europe so our domestic currency is the euro. Consider a forward
contract on the USD: you must buy N USD at some point in the future T for
the price of K EUR/USD. Assume the currence exchange rate is S0 EUR/USD.
What is the value of this future contract and for what value of K such that
the contract has a zero value (the forward price of the USD). It will turn out
now
K = exp((rd − rf )T )S0.
30
Indeed, suppose K > exp((rd − rf )T )S0. An investor can then do the following
(at time 0).
• Borrow N × S0 exp(−rfT ) EUR at rate rd.
• Use this cash to buy N × exp(−rfT ) USD and put this on an USD-
bankaccount at rate rf
• Short the forward contract.
Then the holding of the foreign currency grows to N because of the interest (rf )
earned. Under the terms of the contract this holding is exchanged for N ×K
at time T . An amount exp(rdT )N × S0 exp(−rfT ) is required to repay the
borrowing. Hence a net profit of N × (K −S0 exp((rd − rf )T )) > 0 is, therefor,
made at time T . In case K < exp((rd − rf )T )S0 you can the following
• Borrow N × exp(−rfT ) USD at rate rf .
• Use this cash to buy N × S0 exp(−rfT ) EURD and put this on an EUR-
bankaccount at rate rd
• Take a long position in the forward contract.
then the domestic currency grows to N × S0 exp((rd − rf )T )), you pay N ×K
to receive N USD and uses these dollars to pay the loan. In total you earned
N × (S0 exp((rd − rf )T )) −K) which in this case was assumed to be positive.
31
1.3.5 Commodities
We now consider the cas of commodities. Important here is the impact of storage
costs. If the storage costs incurred at any time are proportional to the price of
the commodity, they can be regarded as providing a negative dividend yield. In
this case from equation (1.1),
F = S0 exp((r + u)T ),
where u is the storage costs per annum as a proportion of the spot price.
1.3.6 The Cost of Carry
The relationship between all above future/forward prices and spot prices can be
summerized in terms of what is known as the cost of carry. This measures the
storage cost plus the interest that is paid to finance the asset less the income
earned on the asset. For a non-dividend paying stock, the cost of carry is r since
there are no storage costs and no income is earned; for a stock index, it is r− q
since income is earned at rate q on the asset; for a currency it is rd − rf ; for a
commodity with storage costs that are a proportion u of the price, it is r + u;
and so on.
Define the cost of carry as c. For an investment asset, the future price is
F = S0 exp(cT ).
32
Figure 1.7: Call options on S&P 500 Index
33
Chapter 2
Binomial Trees
2.1 Single Period Market Models
Our aim here is to show in the simplest possible non-trivial model how the
theory based on the principle of no-arbitrage works.
Example
Let our financial market consist of two financial assets, a riskless bank account
(or bond) B and a risky stock S, with today’s price S0 = 20 euro. We look at
a single-period model and assume that starting from today (t = 0) the world
can only be in one of two states at time t = T : the stock price will either be
ST = 22 euro or ST = 18 euro. We are interested in valuing a European call
option to buy the stock for 21 euro at time t = T . At time t = T , this option
can have only two possible values. It will have value 1 euro, if the stock price
34
Figure 2.1: One-period binomial tree example
is 22 euro; if the stock price turns out to be 18 euro at time t = T , the value of
the option will be zero. The situtation is illustrated in Figure 2.1.
It turns out that we can price the option by the assumption that no arbitrage
opportunities exist. We set up a portfolio of the stock and the option in such a
way that there is no uncertainty about the value of the portfolio at the time of
expiry, t = T . We then argue that, because the portfolio has no risk, the return
earned on it must equal the risk-free interest rate of the bank account. This
enables us to work out the cost of setting up the portfolio and, therefore, the
option’s price.
Consider a portfolio consisting of a long postion in ∆ shares of the stock and
a short position in one call option. We calculate the value of ∆ that makes the
portfolio riskless. If the stock price moves up from 20 to 22 euro, the value of
the shares is 22∆ and the value of the option is 1 euro, so that the total value
of the portfolio is 22∆ − 1 euro. If the stock price moves down from 20 to 18
euro, the value of the shares is 18∆ euro and the value of the option is zero, so
that the total value of the portfolio is 18∆ euro. The portfolio is riskless if the
35
value of ∆ is chosen so that the final value of the portfolio is the same for both
alternatives. This means
22∆− 1 = 18∆
or
∆ = 0.25
A riskless portfolio is, therefore,
• Long 0.25 shares.
• Short 1 option.
If the stock price moves up to 22 euro, the value of the portfolio is
22 × 0.25− 1 = 4.5.
If the stock price moves down to 18 euro, the value of the portfolio is
18× 0.25 = 4.5.
Regardless of whether the stock price moves up or down, the value of the port-
folio is always 4.5 euro at the end of the life of the option.
Riskless portfolios must, in the absence of arbitrage opportunities, earn the
risk free rate of interest. Suppose that in this case the risk-free rate is 12 percent
per annum and that T = 0.5, i.e. six months. It follows that the value of the
portfolio today must be the present value of 4.5 euro, or
4.5e−0.12×0.5 = 4.238
36
The value of the stock today is known to be 20 euro. Suppose the option price
is denoted by f . The value of the portfolio today is
20× 0.25− f = 5− f
It follows that
5 − f = 4.238
or
f = 0.762.
This shows that, in the absence of arbitrage opportunities, the current value of
the option must be 0.762. If the value of the option were more than 0.762 euro,
the portfolio would cost less than 4.238 euro to set up and would earn more than
the risk-free rate. If the value of the option were less than 0.762 euro, shorting
the portfolio would provide a way of borrowing money at less than the risk-free
rate.
In other words, if the value of the option were more than 0.762 euro, for
example 1 euro, you can borrow for example 42380 euro and buy 10000 times
the above portfolio at a cost of
10000(0.25× 20− 1) = 40000euro.
You pocket 2380 euro and after 6 months, you sell 10000 portfolio and cashes in
45000, because the value of one portfolio is always 4.5 euro. With this money
you pay back the bank for the money you borrowed plus the interests on it, i.e.
you pay the bank an amount of 42380 × e0.12×0.5 = 45000 euro. At the end
37
Figure 2.2: General one-period binomial tree
of all this you earned 2380 euro without taking any risk and without an initial
capital. If the value of the option were less than 0.762 euro, you do the opposite.
Generalization
We can generalize the argument just presented by considering a stock whose
price is initially S0 and an option on the stock whose current price is f . We
suppose that the option lasts for time T and that during the life of the option
the stock can move up from S0 to a new level, S0u or down from S0 to a new
level, S0d (u > 1; 0 < d < 1). If the stock price moves up to S0u, we suppose
that the payoff from the option is fu; if the stock price moves down to S0d, we
suppose the payoff from the option is fd. The situation is illustrated in Figure
2.2.
As before, we imagine a portfolio constisting of a long position in ∆ shares
and a short position in one option. We calculate the value of ∆ that makes the
portfolio riskless. If there is an up movement in the stock price, the value of the
38
portfolio at the end of the life op the option is
S0u∆ − fu.
If there is a down movement in the stock price, the value becomes
S0d∆ − fd.
The two are equal when
S0u∆ − fu = S0d∆ − fd,
or
∆ =fu − fdS0u− S0d
. (2.1)
In this case, the portfolio is riskless and must earn the riskless interest
rate. If we denote the risk-free interest rate by r, the present value of the
portfolio is
(S0u∆ − fu)e−rT = (S0d∆ − fd)e
−rT .
The cost of setting up the portfolio is
S0∆ − f.
It follows that
(S0u∆ − fu)e−rT = S0∆ − f,
or
f = S0∆ − (S0u∆ − fu)e−rT .
39
Substituting from equation (2.1) for ∆ and simplifying, this equation reduces
to
f = e−rT [pfu + (1 − p)fd] (2.2)
where
p =erT − d
u− d(2.3)
Remark 2 If we assume that u > erT , together with u > 1 and 0 < d < 1,
one can easily show that the value of p given in (2.3) satisfies 0 < p < 1. Note
that it is natural to assume that u > erT , because it means that after a time T ,
you can gain more (a factor u) by investing in the risky stocks, than you can
earn with a riskless investment in bond (a factor erT ). If this was not the case
no one would invest in stocks. Ofcourse, you can also lose money (d factior by
investing in stocks.
Remark 3 Equation (2.1) shows that ∆ is the ratio of the change in the option
price to the change in the stock price.
Remark 4 The option pricing formula in (2.2) does not involve the probabili-
ties of the stock moving up or down. This is suprising and seems counterintu-
itive. The key reason is that the probabilities of future up or down movements
are already incorporated into the price of the stock.
Risk-Neutral Valuation
Although we do not need to make any assumptions about the probabilities of
an up and down movement in order to derive Equation (2.2), it is natural to
40
interpret the variable p in Equation (2.2) as the probability of an up movement
in the stock price. The variable 1−p is then the probability of a down movement,
and the expression
pfu + (1 − p)fd
is the expected payoff from the option. With this interpretation of p, Equation
(2.2) then states that the value of the option today is its expected future value
discounted at the risk-free rate.
We now investigate the expected return from the stock when the probability
of an up movement is assumed to be p. The expected stock price at time T ,
Ep[ST ], is given by
Ep[ST ] = pS0u+ (1 − p)S0d
= pS0(u− d) + S0d.
Substituting from (2.3) for p, this reduces to
Ep[ST ] = S0erT (2.4)
showing that the stock price grows, on average, at the risk-free rate. Setting the
probability of an up movement equal to p is therefore, equivalent to assuming
that the return on the stock equals the rsik-free rate. In a risk-neutral world
the expected return on all securities is the risk-free interest rate. Equation (2.4)
shows that we are assuming a risk-neutral world when we set the proability of
an up movement to p. Equation (2.2) shows that the value of the option is its
expected payoff in a risk-neutral world discounted at the risk-free rate.
41
This result is an example of an important genereal principle in option pricing
known as risk-neutral valuation. The principle states that it is valid to assume
the world is risk neutral when pricing options. The resulting option prices
are correct not just in a risk-neutral world, but in the real world as
well.
The Single-Period Example Revisited
We now turn back to the numerical example in Figure 2.1 to illustrate that risk-
neutral valuation gives the same answers as no-arbitrage arguments. In Figure
2.1, the stock price is currently 20 euro and will move either up to 22 euro or
down to 18 euro at the end of six months. The option considered is a European
call option with strike price of 21 euro and an expiration date in six months.
The risk-free interest rate is 12 percent per annum.
We define p as the probability of an upward movement in the stock price in
a risk-neutral world. (We know from the analysis given earlier in this section
that p is given by Equation (2.3). However, for the purpose of this illustration
we suppose that we do not know this.) In a risk-neutral world the expected
return on the stock must be the risk-free rate of 12 percent. This means that p
must satisfy
22p+ 18(1− p) = 20e0.12×0.5
or
p =20e0.12×0.5 − 18
4= 0.8092
At the end of the six months, the call option has a 0.8092 probability of being
42
worth 1 euro and a 0.1908 probability of being worth zero. Its expected value
is, therefore,
0.8092× 1 + 0.1908× 0 = 0.8092
In a risk-neutral world, this should be discounted at the risk-free rate. The
value of the option today is, therefore,
0.8092e−0.12×0.5 = 0.7620
This is the same value as the value obtained earlier, illustrating that no-arbitrage
arguments and risk-neutral valuation give the same answer.
2.2 Two-Step Binomial Trees
We can extend the analysis to a two-step binomial tree. The objective of the
analysis is to calculate the option price at the initial node of the tree. This can
be done by repeatedly applying the principles established earlier in the chapter.
2.2.1 European Call
We can first apply the analysis to a two-step binomial tree. Here the stock price
starts at 20 euro and in each of the two time steps may go up by 10 percent or
down by 10 percent. We suppose that each time step is six months long and
the risk-free interest rate is 12 percent per annum.
We consider a European call option with a strike price of 21 euro. Figure
2.3 shows the tree with both the stock price and the option price at each node.
(The stock price is the upper number and the option price is the lower number.)
43
Figure 2.3: Two-period binomial tree example
The option prices at the final nodes of the tree are easily calculated. They are
the payoffs from the option. At node D, the stock price is 24.2 euro and the
option price is 24.2− 21 = 3.2 euro; at nodes E and F, the option is out of the
money and its value is zero.
At node C, the option price is zero, because node C leads to either node E
or node F and at both nodes the option price is zero. Next, we calculate the
option price at node B.
Using the notation introduced earlier in the chapter, u = 1.1, d = 0.9,
r = 0.12, and T = 0.5 so that p = 0.8092. Equation (2.2) gives the value of the
option at node B as
e−0.12×0.5[0.8092× 3.2 + 0.1908× 0] = 2.4386
It remains for us to calculate the option at the initial node, A. We do so by
focusing on the first step of the tree. We know that the value of the option at
node B is 2.4386 and that at node C it is zero. Equation (2.2), therefore, gives
44
Figure 2.4: General two-period binomial tree
the value at node A as
e−0.12×0.5[0.8092× 2.4386 + 0.1908× 0] = 1.8583
The value of the option is 1.8583 euro.
We can generalize the case of two time steps by considering the situation in
Figure 2.4.
The stock price is initially S0. During each step, it either moves up to u
times its value or moves down to d times its value. The notation for the value
of the option is shown on the tree. For example, after two up movements, the
value of the option is fuu. We suppose that the risk-free interest rate is r and
the length of the time step is ∆t years.
Repeated application of Equation (2.2) gives
fu = e−r∆t[pfuu + (1 − p)fud] (2.5)
fd = e−r∆t[pfud + (1 − p)fdd] (2.6)
f = e−r∆t[pfu + (1 − p)fd] (2.7)
45
Substituting the first two equations in the last one, we get
f = e−2r∆t[p2fuu + 2p(1− p)fud + (1 − p)2fdd]. (2.8)
This is constistent with the principle of risk-neutral valuation mentioned earlier.
The variable p2, 2p(1 − p), and (1 − p)2 are the probabilities that the upper,
middle, and lower final nodes will be reached. The option price is equal to
its expected payoff in a risk-neutral world discounted at the risk-free
interest rate.
As we add more steps to a binomial tree, the risk-neutral valuation principle
continues to hold. The option price is always equal to the present value (dis-
counting at the risk-free interest rate) of its expected payoff in a risk-neutral
world.
2.2.2 Matching Volatility with u and d
In practice, when constructing a binomial tree to represent the movements in
a stock price, we choose the parameters u and d to match the volatility of the
stock price. To see how this is done, suppose that the expected return on a stock
in the real world is µ: The expected stock price at the end of the first time
step is S0(1+µ∆t). The volatility of a stock price, σ, is defined so that σ2∆t is
the variance of the return in a short period of time of length ∆t. Suppose from
empirical data we estimated that the probability of an up movement in the real
world is equal to q. In order to match the expected return on the stock, we
46
must therefore, have
qS0u+ (1 − q)S0d = S0(1 + µ∆t),
or
q =(1 + µ∆t) − d
u− d(2.9)
The variance of the stock price return is
qu2 + (1 − q)d2 − [qu+ (1 − q)d]2.
In order to match the real world stock price volatility we must therefore have
qu2 + (1 − q)d2 − [qu+ (1 − q)d]2 = σ2∆t.
or equivalently
q(1 − q)(u− d)2 = σ2∆t. (2.10)
Substituting from Equation (2.9)into Equation (2.10) we get
((1 + µ∆t) − d) (u− (1 + µ∆t)) = σ2∆t
When terms in (∆t)2 and higher powers of ∆t are ignored (remember ∆t is
supposed to be small), one solution to this equation is
u = (1 + σ√
∆t) (2.11)
d = (1 − σ√
∆t) (2.12)
47
Indeed,
((1 + µ∆t) − d) (u− (1 + µ∆t))
= −(1 + µ∆t)2 + (1 + µ∆t)(u+ d) − ud
= −1− 2µ∆t− (µ∆t)2 + 2(1 + µ∆t) − (1 + σ√
∆t)(1 − σ√
∆t)
= −(µ∆t)2 + σ2∆t
Another setting is
u = eσ√
∆t (2.13)
d = e−σ√
∆t, (2.14)
which is, because ∆t is supposed to be small, approximatelly the same as (2.11).
These are the values proposed by Cox, Ross and Rubinstein. Note that in both
cases the values of u and d are independent of µ, which implies that if we move
from the real world to the risk-neutral world the volatility on the stock remains
the same (at least in the limit as ∆t tends to zero). This is an illustration
of an important general result known as Girsanov’s theroem. When we move
from a world with one set of risk preferences to a world with another set of risk
preferences, the expected growth rates change, but their volatilities remain the
same. Moving from one set of risk preferences to another is sometimes referred
to as changing the measure.
48
2.3 Binomial Trees
The above one- and two-steps binomial trees are very imprecise models of reality
and are used only for illustrative purposes. Clearly an analyst can expect to
obtain only a very rough approximation to an option price by assuming that
the stock movements during the life of the option consist of one or two binomial
steps. When binomial trees are used in pratice, the life of the option is typically
divided into 30 or more time steps of length ∆t. In each time step there is
a binomial stock movement. With 30 time steps this means that 31 terminal
stock prices and 230 possible stock price paths are considered.
2.3.1 European Call and Put Options
Consider the evaluation of an option on a non-dividend-paying stock. We start
by dividing the life of the option into a large number of small intervals of length
∆t. We assume that in each time interval the stock price moves from its initial
value S to one of two new values Su and Sd. In general, u > 1 and 0 < d < 1.
The movement from S to Su is, therefore, an ”up” movement and the movement
from S to Sd is a ”down” movement. In the above sections we introduced what
is known as the risk-neutral valuation principle. This states that any security
which is dependent on a stock can be valued on the assumption that the world
is risk neutral. It means that for the purposes of valuing an option, we can
assume:
• The expected return from all traded securities is the risk-free interest rate.
49
• Future cash flows can be valued by discounting their expected values at
the risk-free interest rate.
We make use of this when using a binomial tree. The tree is designed to represent
the behavior of a stock price in a risk-neutral world. In this risk-neutral world
the probability of an up movement will be denoted by p. The probability of a
down movement is 1 − p; as seen above in (2.3):
p =er∆t − d
u− d.
As mentioned above, a popular way of chosing the parameters u and d is
u = eσ√
∆t
d = e−σ√
∆t
Figure 2.5 illustrates the tree of stock prices over 5 time periods that is
considered when the binomial model is used.
At time zero, the stock price S0 is known. At time ∆t there are two possible
stock prices, S0u and S0d; at time 2∆t, there are three possible stock prices,
S0u2, S0ud, and S0d
2; and so on. In general, at time i∆t, i+ 1 stock prices are
considered. These are
S0ujdi−j , j = 0, . . . , i.
European call and put options are evaluated by starting at the end of the tree
(time T ) and working backward. The value of the option is known at time T .
For example, a European put option is worth maxK − ST , 0 and a European
call option is worth maxST −X, 0, where ST is the stock price at time T and
50
Figure 2.5: General binomial tree for stock price
K is the strike price. Because a risk-neutral world is being assumed, the value
at each node at time T −∆t can be calculated as the expected value at time T
discounted at rate r for a time period ∆t. Similarly, the value at each node at
time T −2∆t can be calculated as the expected value at time T −∆t discounted
for a time period ∆t at rate r, and so on. Eventually, by working back through
all the nodes, the value of the option at time zero is obtained. This procedure
is illustrated in Figure 2.6.
Another way of calculating the option prices is by directly taking the dis-
counted value of the expected payoff of the option in the risk-neutral world. For
example the European put, with strike price K and maturity T has a value:
e−rTN∑
j=0
N
j
maxK − S0ujdN−j , 0pj(1 − p)N−j
For more complex options, but where the payoff only depends on the final stock
price, i.e. the payoff is a function of ST , g(ST ) say, a similar expression can be
51
Figure 2.6: General binomial tree for stock price
derived; the current value of the option is then given by:
e−rTEp[g(ST )] = e−rTN∑
j=0
N
j
g(S0ujdN−j)pj(1 − p)N−j ,
where Ep denotes the expectation in the risk-neutral world, i.e. with a probabil-
ity p given by (2.3) of an up-move of size u , and a probability of a down-move
of (1 − p), or equivalently, with a probability
N
j
pj(1 − p)N−j (2.15)
of ending with a time T stock price of S0ujdN−j . The distribution (2.15) is
called the Binomial distribution.
52
2.3.2 American Options
If the option is American, the procedure only changes slightly. It is necessary
to check at each node to see whether early exercise is preferable to holding the
option for a further time period ∆t. Eventually, again by working back through
all the nodes the value of the option at time zero is obtained.
American put option
Consider a five-month American put option on a non-dividend-paying stock
when the current stock price is 50 euro, the strike price is also 50 euro, the risk-
free interest rate is 10 percent per annum, and the volatility is 40 percent per
annum. With our usual notation, this means that S0 = 50, K = 50, r = 0.10,
σ = 0.40, and T = 152/365 = 0.416. Suppose that we divide the life of the
option into five intervals of length one month (= 0.0833 year) for the purposes
of constructing a binomial tree. Then
∆t = 0.0833
u = eσ√
∆t = 1.1224
d = e−σ√
∆t = 0.8909
p = (er∆t − d)/(u− d) = 0.5073
Figure 2.7 shows the related binomial tree.
At each node there are two numbers. The top one shows the stock price
at the node; the lower one shows the value of the option at the node. The
53
Figure 2.7: Binomial tree for American put option
probability of an up movement is always 0.5073; the probability of a down
movement is always 0.4927. The stock price at the jth node (j = 0, 1, . . . , i) at
time i∆t (i = 0, 1, 2, 3, 4, 5) is calculated as S0ujdi−j .
The option prices at the final nodes are calculated as maxK −ST , 0. The
option prices at the penultimate nodes are calculated from the option prices at
the final nodes. First, we assume no exercise of the option at the nodes. This
means that the option price is calculated as the present value of the expected
option price one step later. For example at node C, the option price is calculated
as
(0.5073× 0 + 0.4927× 5.45)e−0.10×0.0833 = 2.66
54
whereas at node A it is calculated as
(0.5073× 5.45 + 0.4927× 14.64)e−0.10×0.0833 = 9.90
We then check to see if early exercise is preferable to waiting. At node C,
early exercise would give a value for the option of zero because both the stock
price and the strick price are 50 euro. Clearly it is best to wait. The correct
value for the option at node C is, therefore, 2.66 euro. At node A, it is a different
story. If the option is exercised, it is worth 50 − 39.69 = 10.31 euro. This is
more than 9.90. If node A is reached, the option should therefore, be exercised
and the correct value for the option at node A is 10.31 euro.
Option prices at earlier nodes are calculated in a similar way. Note that it
is not always best to exercise an option early when it is in the money. Consider
node B. If the option is exercised, it is worth 50− 39.69 = 10.31 euro. However,
if it is held, it is worth
(0.5073× 6.38 + 0.4927× 14.64)e−0.10×0.0833 = 10.36
The option should, therefore, not be exercised at this node, and the correct
option value at the node is 10.36 euro.
Working back through the tree, we find the value of the option at the initial
node to be 4.49 euro. This is our numerical estimate for the option’s current
value. In practice, a smaller value of ∆t, and many more nodes, would be used.
It can be shown that with 30, 50, and 100 time steps we get values for the option
of 4.263, 4.272, and 4.278.
In general suppose that the life of an American put option on a non-dividend-
55
paying stock is divided into N subintervals of length ∆t. We will refer to the
jth node at time i∆t as the (i, j) node. Define fi,j as the value of the option
at the (i, j) node. The stock price at the (i, j) node is S0ujdi−j . Because the
value of an American put at its expiration date is maxK − ST , 0, we know
that
fN,j = maxK − S0ujdN−j , 0, j = 0, 1, . . . , N
There is a probability, p, of moving from the (i, j) node at time i∆t to the
(i+1, j+ 1) node at time (i+1)∆t, and a probability 1− p of moving from the
(i, j) node at time i∆t to the (i + 1, j) node at time (i + 1)∆t. Assuming no
early exercise, risk-neutral valuation gives
fi,j = e−r∆t(pfi+1,j+1 + (1 − p)fi+1,j)
for 0 ≤ i ≤ N −1 and 0 ≤ j ≤ i. When early exercise is taken into account, this
value for fi,j must be compared with the option’s intrinsic value, and we obtain
fi,j = max
K − S0ujdi−j , e−r∆t(pfi+1,j+1 + (1 − p)fi+1,j)
Note that, because the calculations start at time T and work backward, the
value at time i∆t captures not only the effect of early exercise possibilities at
time i∆t, but also the effect of early exercise at subsequent times. In the limit as
∆t tends to zero, an exact value for the American put is obtained. In practice,
N = 30 usually gives reasonable results.
56
It is never optimal to exercise an American call option
We are now going to proof that for a non-dividend paying stock the price of a
European call and an American call are the same. This means that an early
exercise of an American call is never optimal. To prove this striking result we
first proof
Proposition 5 The current price C of a European (and American) call option,
with strike price K and time to expiry T , on a non-dividend paying stock with
current price S satisfies :
C ≥ maxS − e−rTK, 0.
Proof: That C ≥ 0 is obvious, otherwise ’buying’ the call would give a riskless
profit now and no obligations later.
To prove the remaining lower bound, we setup an arbitrage table (Table 2.1)
to examine the cash flows of the following portfolio:
sell 1 stock short, buy 1 call, invest in bank account e−rTK.
Assuming the condition C ≥ S− e−rTK is violated, i.e. C < S− e−rTK we
get the arbitrage Table 2.1.
So in all possible states of the world at expiry we have a non-negative return
for a portfolio, which has a positive current cash flow. This is clearly an arbitrage
opportunity and hence our assumption was wrong. •
Suppose now that the American call is exercised at some time t strictly less
than expiry T , i.e. t < T . The financial agent thereby realises a cash-flow
57
Portfolio Current cash flow Value at expiry
ST ≤ K ST > K
Short 1 stock S −ST −ST
Buy 1 call −C 0 ST −K
Bank account −e−rTK K K
Balance S − C − e−rTK ≥ 0 K − ST ≥ 0 0
Table 2.1: Arbitrage table for bounds on calls
St−K. From the above proposition we know that the value of the call must be
greater or equal to St− e−r(T−t)K, which is greater than St−K. Hence selling
the call would have realised a higher cash-flow and the early exercise of the call
was suboptimal. In conclusion:
CA = CE
There are two reasons why an American call should not be exercised early.
• Insurance: An investor which holds the call option does not care if the
share price falls far below the strike price - he just discards the option -
but if he held the stock, he would. Thus the option insures the investor
against such a fall in stock price, and if he exercises early, he loses this
insurance.
• Interest on the strike price: When the holder exercises the option, he buys
the stock and pays the strike price, K. Early exercise at time t < T
58
deprives the holder of the interest on K between times t and T : the later
he pays out K, the better.
Notice how this changes when we consider American puts in place of calls:
The insurance aspect above still holds, but the interest aspect above is reversed
(the holder receives cash K at the exercise time, rather than paying it out).
2.4 Moving towards The Black-Scholes Model
By creating a tree with more and more time steps, that is by taking smaller and
smaller time-steps, we can get finer and finer graduations at the final stage and
thus hopefully a more accurate price. However, we have to be a little careful
about how we do this in order to get the prices to converge to a meaningful
value. Which limiting price we obtain will depend on how we make the tree
finer - this essentially comes down to assumptions we make about the random
process the asset follows.
Let us try to price an option with payoff function f(ST ) and we will refine
the Cox-Ross-Rubenstein model with choices
u = eσ√
∆t (2.16)
d = e−σ√
∆t. (2.17)
Taking N time steps we have that risk-neutral probaility of moving upwards
equals:
pN =exp(rT ) − exp(−σ
√
T/N)
exp(σ√
T/N − exp(−σ√
T/N))
59
Let us now investigate the risk-neutral limiting distribution of ST :
ST = S0
N∏
j=1
eZjσ√T/N = S0 exp
σ√
∆tN∑
j=1
Zj
,
where Zj are independent random variables taking the values −1 and 1, with
probabilities pN and 1 − pN respectively, for j = 1, . . . , N .
In other words:
logST = logS0 + σ√
T/NN∑
j=1
Zj .
Now we can apply the Central Limiting Theorem (CLT).
Theorem 6 (CLT) Assume X1, X2, . . . is a series of independent random varoables,
all with the same distribution as X of which the second moment is finite. Then
∑Nj=1 −NE[X ]√
NVar[X ]→D N ,
with N a standard Normal distributed variable (with mean zero and variance
equal to one).
We note that
E[Zj ] = 2pN − 1;
Var[Zj ] = 4pN(1 − pN ).
Hence, a simple calculation, using
NE[Zj ]√
T/Nσ → (r − (1/2)σ2)T ;
√
Var[Zj ]N√
T/Nσ → σ√T .
60
leads to
logST →D logS0 + σ√TN +
(
r − 1
2σ2
)
T
whenN → +∞. The distribution of the logarithm of the stock price thus follows
a Normal distribution with mean(
r − 12σ
2)
T and variance σ2T ; the stock price
itself is thus lognormally distributed.
The price of the derivative in the limit will be given by
limN→∞
exp(−rT )EpN[g(ST )] = exp(−rT )E
[
g
(
S0 exp(σ√TN +
(
r − 1
2σ2
)
T
)]
.
In case of the European call option with strike K and time to maturity T , one
can with a little effort show that its initial price is given by:
C(K,T ) = S0N(d1) −K exp(−rT )N(d2),
where
d1 =ln(S0/K) + (r + σ2
2 )T
σ√T
(2.18)
d2 =ln(S0/K) + (r − σ2
2 )T
σ√T
= d1 − σ√T (2.19)
and N(x) is the cumulative probability distribution function for a variable that
is standard normal distributed. This is the famous Black-Scholes formula. This
lognormal model (the Black-Scholes model), will be studied in detail in the
course ”Continuous Financial Mathematics”.
61
Chapter 3
Mathematical Finance in
Discrete Time
Any variable whose value changes over time in an uncertain way is said to
follow a stochastic p.rocess. Stochastic processes can be classified as discrete-
time or continuous-time. A discrete-time stochastic process is one where the
value of the variable can change only at certain fixed points in time, whereas
a continuous-time stochastic process is one where changes can take place at
any time. Stochastic processes can also be classified as continuous-variables or
discrete-variables. In a continuous-variable process, the underlying variable can
take any value within a certain range, whereas in a discrete-variable process,
only certain discrete values are possible. Binomial tree models belong to the
discrete-time, discrete-variable stochastic processes.
62
In this chapter we study so-called finite markets, i.e. discrete-time models
of financial markets in which all relevant quantities take a finite number of val-
ues. We specify a time horizon T , which is the terminal date for all economic
activities considered. For a simple option pricing model the time horizon typi-
cally corresponds to the expiry date of the option. We thus work with a finite
probability space (Ω, P ), with a finite number |Ω| of possible outcomes ω, each
with a positive probability: P (ω) > 0.
3.1 Information and Trading Strategies
Access to full, accurate, up-to-date information is clearly essential to anyone
actively engaged in financial activity or trading. Indeed, information is arguably
the most important determinant of success in financial life. We shall confine
ourselves to the situation where agents take decisions on the basis of information
in the public domain, available to all. We shall further assume that information
once known remains known and can be accessed in real time.
Our financial market contains two financial assets. A risk-free asset (the
bond) with a deterministic price process Bi, and a risky assets with a stochastic
price process Si. We assume B0 = 1 (we reckon in units of the initial value of
the bond) and Bi > 0; we say it is a numeraire. 1/Bi is called the discounting
factor at time i.
As time passes, new information becomes available to all agents. There
exists a mathematical object to model this information flow, unfolding with
63
time: filtrations. The concept filtration is not that easy to understand. The full
theory will lead us too far. In order to clear this out a bit, we explain the idea
of filtration in a very idealized situation. We will consider a stochastic process
X which starts at some value, zero say. It will remain there until time t = 1,
at which it can jump with positive probability to the value a or to a different
value b. The process will stay at that value until time t = 2 at which it will
jump again with positive probability to two different values: c and d say if is
was at time t = 1 at a and f and g say if the process was at time t = 1 at state
b. From then on the process will stay in the same value. The universum of the
probability space consists of all possible paths the process can follow, i.e. all
possible outcomes of the experiment. We will denote the path 0 → a → c by
ω1, similarly the paths 0 → a → d, 0 → b → f and 0 → b → g are denoted by
ω2, ω3 and ω4 respectively. So we have Ω = ω1, ω2, ω3, ω4.
In this situation we will take the following flow of information, i.e. filtrations:
Ft = ∅,Ω 0 ≤ t < 1;
Ft = ∅,Ω, ω1, ω2, ω3, ω4 1 ≤ t < 2;
Ft = D(Ω) = F 2 ≤ t.
We set here F = D(Ω), the set of all subsets of Ω.
To each of the filtrations given above, we associate resp. the following par-
64
titions (i.e. the finest possible one) of Ω:
P0 = Ω 0 ≤ t < 1;
P1 = ω1, ω2, ω3, ω4 1 ≤ t < 2;
P2 = ω1, ω2, ω3, ω4 2 ≤ t.
At time t = 0 we only know that some event ω ∈ Ω will happen, at time
t = 2 we will know which event ω∗ ∈ Ω has happened. So at times 0 ≤ t < 1 we
only know that some event ω∗ ∈ Ω. At time point after t = 1 and strictly before
t = 2, i.e. 1 ≤ t < 2, we know to which state the process has jumped at time
t = 1: a or b. So at that time we will know to which set of P1, ω∗ will belong:
it will belong to ω1, ω2 if we jumped at time t = 1 to a and to ω3, ω4 if we
jumped to b. Finally, at time t = 2, we will know to which set of P2, ω∗ will
belong, in other words we will know then the complete path of the process.
During the flow of time we thus learn about the partitions. Having the
information Ft revealed is equivalent to knowing in which set of the partition
of that time, the event ω∗ is. The partitions become finer in each step and thus
information on ω∗ becomes more detailed.
We thus keep in mind that a filtration F = (Fi, i = 0, 1, . . . , T ) exists of a
sequence of mathematical objects (σ-algebras), F0 ⊂ F1 ⊂ · · · ⊂ FT , describing
the information available. At time i we have access to information in Fi. It is
clear that the price of the stock Si at time i (and i− 1, i− 2, ..., 0) is contained
in the information Fi.
If a random variable X is known with respect to the information G we say
65
it is G-measurable. So we have that Si is Fi-measurable. A stochastic process
Xi, i = 0, 1, . . . , T is called adapted to the filtration G = (Gi, i = 0, 1, . . . , T )
(or just G−adapted) if at every time point i = 0, 1, . . . , T the random variable
Xi is Gi-measurable. So we have that S = Si, i = 0, 1, . . . , T is F-adapted.
A trading strategy ϕ = ϕi = (βi, ζi), i = 1, . . . , T is a real vector stochastic
process such that each ϕi is Fi−1-adapted. Here βi, ζi denotes the numbers of
bonds ands stocks resp. held at time i and to be determined on the basis of
information available strictly before time i: Fi−1; i.e. the investor selects his
time i portfolio after observing the prices Si−1. The components βi, ζi may
assume negative values as well as positive values, reflecting the fact that we
allow short sales and assume that the assets are perfectly divisible.
The value of the portfolio ϕ at time i, V ϕi = Vi, is called the wealth or value
process of the trading strategy:
V ϕi = Vi = βiBi + ζiSi, i = 1, 2, . . . , T
We will denote by V0 the initial investment or endowment of the investor.
Now βiBi−1 + ζiSi−1 reflects the market value of the portfolio just after it
has been established at time i − 1, whereas βiBi + ζiSi is the value just after
time i prices are observed, but before changes are made in the portfolio. Hence
βi(Bi −Bi−1) + ζi(Si − Si−1)
is the change in the market value due to changes in security prices which occur
between time i− 1 and i. We call Gϕ = G = Gi, i = 1, . . . , T, where
Gϕi = Gi =
i∑
j=1
(βj(Bj −Bj−1) + ζj(Sj − Sj−1))
66
the gains process.
After the new prices (Bi, Si) are quoted at time i, the investor adjusts his
portfolio from ϕi = (βi, ζi) to ϕi+1 = (βi+1, ζi+1). We do not allow him bringing
in or consuming any wealth, so we must have
V0 = β1B0 + ζ1S0, Vi = βi+1Bi + ζi+1Si, i = 1, . . . , T
We say our trading strategy is self-financing and denote this by ϕ ∈ Φ.
To avoid negative wealth and unbounded short sales we also introduce the
concept of admissible strategies. A self-financing trading strategie ϕ ∈ Φ is
called admissible if V ϕi ≥ 0 for each i = 0, 1, . . . , T . We write Φa for the class
of admissible trading strategies. Clearly Φa ⊂ Φ.
3.2 No-Arbitrage Condition
The central principle in the Binomial tree models was the absence of arbitrage
opportunities, i.e. the absence of risk-free plans for making profits without any
investment. As mentioned there this principle is central for any market model,
and we now define the mathematical counterpart of this economic principle in
our setting.
We call a self-financing trading strategy ϕ an arbitrage opportunity if P (V ϕ0 =
0) = 1 and the terminal wealth of ϕ satisfies
P (V ϕT ≥ 0) = 1 and P (V ϕT > 0) > 0
So an arbitrage opportunity is a self-financing strategy with zero initial value,
67
which produces a non-negative final value with probability one and has a postive
probability of a positive value.
We say that our market is arbitrage-free if there are no self-financing trading
strategies which are arbitrage opportunities.
We will link the economic principle of an arbitrage free market to a mathe-
matical one: the existence of an equivalent martingale.
We say a probability measure P ∗ on Ω is equivalent to P , if it has the same
null sets. Here it means P ∗(ω) > 0. We say a probability measure Q on
Ω is a martingale measure for a process X = Xi, i = 0, 1, . . . , T, if Xi is a
Q-martingale with respect to the filtration F, i.e.
• X is F-adapted
• EQ[Xi|Fi−1] = Xi−1, i = 1, . . . , T
Note that in a more general context a third condition is required: EQ[|Xi|] <∞.
Because we work in a finite probability space this condition is in our setting
automatically satisfied.
One can show that the secound condition is equivalent to
EQ[Xi|Fj ] = Xj , 0 ≤ j ≤ i ≤ T.
We denote by P(X) the class of equivalent martingale measures for X and
will use the notation X for the discounted version of the process X : Xi =
B−1i Xi. For eaxmple, we will denote by S the discounted stock price process :
Si = B−1i Si.
68
As a kind of example of the above concepts, we show the following proposit-
tion which we will later on need to prove one direction of the No-Arbitrage
Theorem.
Proposition 7 Let P ∗ ∈ P(S) and ϕ ∈ Φ, then V is a P ∗-martingale.
Proof: First note that
V ϕi = B−1i (βiBi + ζiSi)
and since Bi, Si, βi, ζi ∈ Fi, we also have that V ϕi ∈ Fi. Hence V ϕ is F-adapted.
Next, we will prove EP∗ [Vi|Fi−1] = Vi−1, i = 1, . . . , T . We have
EP∗ [Vi|Fi−1] = EP∗ [B−1i (βiBi + ζiSi)|Fi−1];
= βi + ζiEP∗ [B−1i Si)|Fi−1];
= βi + ζiB−1i−1Si−1;
= B−1i−1(βiBi−1 + ζiSi−1);
= B−1i−1(βi−1Bi−1 + ζi−1Si−1);
= Vi−1,
where the third line is because S is a P ∗-martingale and the fifth line is because
of the self-financing property of ϕ. ♦
The next result is the key-result in discrete mathematical finance.
Theorem 8 (No-Arbitrage Theorem) The market is arbitrage-free if and
only if there exists an equivalent martingale measure for the discounted price
process of the stock Si = B−1i Si, i.e. P(S) 6= ∅.
69
Proof : We only prove that P(S) 6= ∅ implies that the market is arbitrage-
free; the other direction can be proven using the Hahn-Banach theorem from
Functional Analysis.
Assume P(S) 6= ∅ and let P ∗ ∈ P(S). For any self-financing strategy ϕ ∈ Φ,
we have from the above proposition that V ϕ is a P ∗-martingale. So
EP∗ [V ϕT ] = V ϕ0 .
Suppose ϕ is an arbitrage opportunity. Then P (V ϕ0 = 0) = 1, so P ∗(V ϕ0 = 0) =
1 and thus EP∗ [V ϕT ] = 0. We must have
P ∗(V ϕT ≥ 0) = 1 and P ∗(V ϕT > 0) > 0.
Together with P ∗(ω) > 0, this leads to a contradiction. ♦
One can show that a security market which has no arbitrage opportunities
in Φa, is also arbitrage-free with respect to Φ.
3.3 Risk-Neutral Pricing
We now turn to the main underlying question of this text, namely the pricing
of contingent claims (i.e. financial derivatives). First we have to model these
financial instruments in our current framework. This is done in the following
fashion.
Definition 9 A contingent claim X with maturity date T is an arbitrary non-
negative FT -measurable random variable. We denote the class of all contingent
claims by X .
70
We say that the claim is attainable if there exists an (admissible) self-
financing strategy ϕ ∈ Φ such that
V ϕT = X.
The self-financing strategy ϕ ∈ Φ is said to be a replicating strategy. It generates
the same time T cash-flow as X does.
We now return to the main question of the section: given a contingent claim
X , i.e. a cash-flow at time T , how can we determine its value (price) at time
i < T ? For attainable contingent claims this value should be given by the
value of any replicating strategy (perfect hedge) at time i, i.e. there should be
a unique value process (say V Xi ) representing the time i value of the claim X .
The following proposition ensures that the value process of replicating strategies
coincide, thus proving uniqueness of the value process.
Proposition 10 Suppose the market is arbitrage-free. Then any attainable con-
tingent claim X is uniquely replicated: for all ϕ, ψ ∈ Φ such that
V ϕT = V ψT = X
we have that for all 0 ≤ i ≤ T
V ϕi = V ψi
This uniqueness property allows us now to define the important concept of
an arbitrage price process.
Definition 11 Suppose the market is arbitrage free. Let X be any attainable
contingent claim with time T to maturity. Then the arbitrage price process πXi ,
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0 ≤ i ≤ T or simply the arbitrage price of X is given by the value process of
any replicating strategy ϕ for X.
The construction of hedging strategies that replicate the outcome of a con-
tingent claim is an important problem in both practical and theoretical ap-
plications. Hedging is central to the theory of option pricing. The classical
arbitrage valuation models, such as the Binomial tree models and the Black-
Scholes Model (see the next Chapters), depend on the idea that an option can
be perfectly hedged using the underlying risky asset and a risk-free asset.
Analysing the arbitrage-pricing approach we observe that the derivation of
the price of a contingent claim doesn’t require any specific preferences of the
agents other than that they prefer more to less, which rules out arbitrage. So,
the pricing formula for any attainable contingent claim must be independent
of all preferences that do not admit arbitrage. In particular, an economy of
risk-neutral investors must price a contigent claim in the same manner. This
fundamental insight simplifies the pricing formula enormously. In its general
form the price of an attainable contingent claim is just the expected value of
the discounted payoff with respect to an equivalent martingale measure.
Proposition 12 The arbitrage price process of any attainable contingent claim
X is given by the risk-neutral valuation formula
πXi =BiBT
EP∗ [X |Fi], i = 0, 1, . . . , T
where EP∗ is the expectation operator with respect to an equivalent martingale
measure P ∗.
72
Proof: Since we assume that the market is arbitrage-free there exists (at
least) an equivalent martingale measure P ∗ for the discounted price process
Si. Furthermore because the claim is attainable there exists (at least) one
self-financing replicating strategy ϕ. First we prove that the discounted value
process V ϕi = B−1i V ϕi is a P ∗-martingale: Indeed, by the self-financing property
of ϕ = (βi, ζi)
EP∗ [V ϕi |Fi−1] − V ϕi−1
= EP∗ [V ϕi − V ϕi−1|Fi−1]
= EP∗ [B−1i V ϕi −B−1
i−1Vϕi−1|Fi−1]
= EP∗ [B−1i (βiBi + ζiSi) −B−1
i−1(βi−1Bi−1 + ζi−1Si−1)|Fi−1]
= EP∗ [B−1i (βiBi + ζiSi) −B−1
i−1(βiBi−1 + ζiSi−1)|Fi−1]
= EP∗ [ζi(B−1i Si −B−1
i−1Si−1)|Fi−1]
= ζiEP∗ [Si − Si−1|Fi−1]
= 0.
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The last equality follows because Si = B−1i Si is P ∗-martingale. So we have for
each i = 0, 1, . . . , T
πXi = V ϕi
= BiVϕi
= BiEP∗ [V ϕT |Fi]
= BiEP∗ [B−1T V ϕT |Fi]
= (Bi/BT )EP∗ [V ϕT |Fi]
= (Bi/BT )EP∗ [X |Fi]
3.4 Complete Markets
The last section made clear that attainable contingent claims can be priced using
an equivalent martingale measure. In this section we will discuss the question
of the circumstances under which all contingent claims are attainable. This
would be a very desirable property of the market, because we would then have
solved the pricing question (at least for contingent claims) completely under
the assumption that the market is arbitrage free. Since contingent claims are
merely non-negative FT -measurable random variables in our setting, it should
be no suprise that we can give a criterion in terms of probability measures. We
start with:
Definition 13 A market is complete if every contingent claim is attainable, i.e.
for every non-negative FT -measurable random variables X ∈ X there exists a
74
replicating strategy ϕ ∈ Φ such that V ϕT = X.
In the case of an arbitrage-free discrete market, one can insist on replicating
contingent claims by an admissible strategy ϕ ∈ Φa.
Based on the no-arbitrage assumption one can prove:
Theorem 14 (Completeness Theorem) An arbitrage free market is com-
plete if and only if there exists a unique probability measure P ∗ equivalent to P
under which the discounted price process of the stock Si = B−1i Si is a martin-
gale, i.e. P(S) = P ∗.
3.5 The Fundamental Theorem of Asset Pricing
We summarise what we have achieved so far. We call a measure P ∗ under
which the discounted price S is a P ∗-martingale a martingale measure. Such
a P ∗ equivalent to the actual probability measure P is called an equivalent
martingale measure. Then:
• No-Arbitrage Theorem: A market is arbitrage free if and only if at least
one equivalent martingale measure exists.
• Completeness Theorem: An arbitrage-free market is complete (all contin-
gent claims can be replicated) if and only if there exists a unique equivalent
martingale measure.
So
75
Theorem 15 (Fundamental Theorem of Asset Pricing) In an arbitrage-
free complete market, there exists a unique equivalent martingale measure P ∗.
The above theorem establishes the equivalence of an economic modelling condi-
tion such as no-arbitrage and completeness to the existence of the mathematical
modelling condition, viz. the existence and uniqueness of equivalent martingale
measures.
Assume now that the market is arbitrage-free and complete and let X ∈ X
be any contingent claim, ϕ a replicting strategy (which exists by completeness),
then:
V ϕT = X
Furthemore, we have seen that
πXi = V ϕi =BiBT
EP∗ [X |Fi], i = 0, 1, . . . , T
and call πXi = V ϕi the the arbitrage price of the contingent claim X at time i.
For, if an investor sells the claim X at time i for πXi , he can follow strategy
ϕ to replicate X at time T and clear the claim; an investor selling this value
is perfectly hedged. To sell the claim for any other amount would provide an
arbitrage opportunity. We note that, to calculate prices as above, we need to
know only:
• Ω the set of all possible states,
• the filtration F,
• P ∗.
76
We do not need to know the underlying probability measure P (only its null
sets, to know what ’equivalent to P ’ means and actually in our finite model
there are no non-empty null-sets, so we do not need to know even this).
Now pricing of contingent claims is our central task, and for pricing purposes
P ∗ is vital and P itself irrelevant. We thus may – and shall – focus attention
on P ∗, which is called the risk-neutral probability measure.
To summarize, we have:
Theorem 16 (Risk-Neutral Pricing Formula) In an arbitrage-free complete
market, arbitrage prices of contingent claims are their discounted expected values
under the risk neutral (unique equivalent martingale measure) P ∗.
3.5.1 Examples
The One-step Binomial Model
We return to model given in Figure 2.2. There exists only two possible outcomes.
There is an upperstate u if price after one time step equals S1 = uS0 and a
down-state d if the stock price changes to S1 = dS0, Ω = u, d. In both
cases the riskfree asset goes from 1 to a price b say (b is typically equal to
er or 1 + r′). A probability measure on Ω is completely determined by the
number 0 < P (u) < 1; we then have P (d) = 1 − P (u). In order that
the discounted price process is a martingale with respect to a (P -equivalent)
probability measure P ∗, with say 0 < P ∗(u) = p∗ < 1, on Ω, it has to satisfy
only one equation:
EP∗ [b−1S1|F0] = S0
77
or equivalently
b−1uS0p∗ + b−1dS0(1 − p∗) = S0. (3.1)
Rewriting (3.1) gives
p∗ =b− d
u− d(3.2)
In order that this gives rise to a probability measure, we should have 0 < p∗ < 1,
which is equivalently with
u > b > d ≥ 0. (3.3)
In conclusion a martingale measure P ∗ ∈ P for the discounted stock price exists
if and only if (3.3) is satisfied. If (3.3) holds true, then there is a unique such
measure in P characterised by (3.2). So in conclusion, if (3.3) is satisfied the
one-step binomial model is arbitrage free and complete.
Note that (3.3) means that by investing in a stock one can have a bigger
return than the risk-free return (u > b), but also can have a greater loss (b > d).
Note also that one can easily show that the multi-period model of Section
2.3 is complete if and only if the underlying single-period model is complete.
If we now have a contingent claim with payoff fu in the upstate and fd in
the down state, the initial price of this claim is equal to
f = b−1p∗fu + b−1(1 − p∗)fd
78
Figure 3.1: The One-step Trinomial Model
In order to hedge or replicated this claim one has to solve the equations
ξuS0 + ηb = fu
ξdS0 + ηb = fd
Note that this system of equations has a unique solution if and only if
det
uS0 b
dS0 b
6= 0,
which is equivalent with S0 6= 0, b 6= 0, and u 6= d (all which are ruled out).
The One-step Trinomial Model
Suppose now the following one-step trinomial model: In one time step there
exists three possible outcomes as shown in picture 3.1. There is an upperstate u
if the stock price changes to S1 = uS0, a middle state m if the stock price after
one step is S1 = mS0, and a down-state d if the stock price changes to S1 = dS0,
0 ≤ d < m < u: Ω = u,m, d. Again, in all cases the riskfree asset changes in
a deterministic way from 1 to a price b say. A probability measure on Ω is now
79
completely determined by two numbers 0 < P (u) < 1 and 0 < P (m) < 1;
we then have P (d) = 1 − P (u) − P (m). In order that the discounted
price process is a martingale with respect to a probability measure P ∗, with say
0 < P ∗(u) = p∗ < 1 and 0 < P ∗(m) = q∗ < 1, on Ω, it has to satisfy again
only one equation:
EP∗ [b−1S1|F0] = S0
or equivalently
b−1uS0p∗ + b−1mS0q
∗ + b−1dS0(1 − p∗ − q∗) = S0.
Unfortunately this equation has more than one solution as can be easily been
seen after a simple rewriting:
p∗ =(b− d) − (m− d)q∗
u− d
For every 0 < q∗ < 1 there is a corresponding p∗. If we then take also into
account that the values of p∗ and q∗ must give rise to a probability distribution,
i.e. 0 < p∗, q∗ < 1 and p∗ + q∗ < 1, there still are infinitely many solutions.
In conclusion there exist more then one martingale measure for the discounted
stock price. So the one-step trinomial model is arbitrage free, but is not com-
plete.
If we have a contingent claim with payoff fu in the upstate, fm in the middle
state and fd in the down state it can only be replicated if there exists a solution
80
to the equations
ξuS0 + ηb = fu
ξmS0 + ηb = fm
ξdS0 + ηb = fd
This is only the case if
det
uS0 b fu
mS0 b fm
dS0 b fd
= 0.
Because we assume that S0 6= 0 and b 6= 0, this is equivalent with
det
u 1 fu
m 1 fm
d 1 fd
= 0.
So only contingent claims which payoff function satisfies the above condition
are attainable and can be replicated and priced in an arbitrage-free way.
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Chapter 4
Exotic Options
Derivatives with more complicated payoffs than the standard European or Amer-
ican calls and puts are referred to as exotics options. Most exotics options are
traded in the OTC market and have been designed to meet particular needs of
investors.
In this chapter we describe different types of exotic options and discuss their
valuation. Option of an European nature can typically be price by Monte-Carlo
simulation. The main problem with using Monte-Carlo simulation to value
path-dependent derivatives is that the computation time necessary to achieve
the required level of accuracy can be unaccaptable high. Moreover American-
type option can not be handled.
82
4.1 Monte Carlo Pricing
When the payoff depends on the path followed by the underlying variable S
in theory one has to consider every possible path. When using 30 time steps
in the Binomial tree model, there are about a billion different paths and one
has to relay on (Monte Carlo) simulations, which are computationally very time
consuming. The expected payoff in a risk-neutral world is calculated using a
sampling procedure. It is then discounted at the risk-free interest rate:
1. Sample a random path for S in a risk-neutral world.
2. Calculate the payoff from the derivative.
3. Repeat steps one and two to get many sample values of the payoff from
the derivative in a risk neutral world.
4. Calculate the mean of the sample payoff to get an estimate of the expected
payoff in a risk-neutral world.
5. Discount the estimated expected payoff at the risk-free rate to get an
estimate of the value of the derivative.
4.2 Lookback Options
In everything we have encountered so far, uncertainty has unfolded with time,
and our task has been to make optimal use of the information available to date.
For options, at expiry T the investor is in posssesion of the history of the price
evolution over time interval [0, T ] of the option’s life, and it may well be that
83
one could have been doing better. It is only natural to look back with regret.
If only one could buy at the low, and sell at the high ...
In order to provide some investor the right to do that lookback options were
created.
We write S for the stock price process and consider a time interval [0, T ].
Let us denote the maximum and minimum process, resp., of a process X =
Xt, 0 ≤ t ≤ T as
MXt = supXu; 0 ≤ u ≤ t and mX
t = infXu; 0 ≤ u ≤ t, 0 ≤ t ≤ T.
The two basic types of continuously montiored lookback options are the
lookback call, with payoff
LCcont(T ) = ST −mST ,
giving one the right to buy at the low over [0, T ], and the lookback put with
payoff
LP cont(T ) = MST − ST ,
giving one the right to sell at the high over [0, T ].
One can approximate their value by their discretely monitored counterparts.
Consider an a partition of [0, T ] into n equal time intervals with size ∆t = T/n.
Write Si for the stock price value at time i∆t and MSi , mS
i for its maximum
and minimum over [0, i∆t]. The discretely monitored versions payout
LCdiscr(T ) = Sn −mSn ,
and
LP discr(T ) = MSn − Sn,
84
respectively.
We illustrate first how such a discrtely monitored European-type lookback
put can be priced using binomial trees. Later, we will comment on the American
version. When exercised, this provides a payoff equal to the excess of the current
maximum stock price over the current stock price.
Set
Yi =MSi
Si
and produce a binomial tree for the stock price (using the Cox-Ross-Rubenstein
setting). See the left tree in Figure 4.1. From this tree produce a corresponding
tree for Y . Initially Y0 = 1, becauseMS0 = S0. If there is an up-move in S during
the first step, both the maximum and the stock price increase by a proportional
amount u and remains Y = 1. If there is a down movement in S during the
first step, the maximum stays at S0, so that Y = 1/d = 1/ exp(−σ√
∆t) = u.
Continuing with these types of arguments, we produce the tree shown in Figure
4.1 (σ = 0.40, r = 0.10). The rules defining the geometry of the tree are
1 When Yi = 1, then Yi+1 is either u or 1.
2 When Yi = um, then Yi+1 is either um+1 or um−1.
An up-movement in Y corresponds to a down-movement in S and vice versa.
The probability of an up movement in Y is, therefore, always 1 − p, with p
the probability of a up-movement in the stock. Note thus that p is also the
probability of a down-movement of Y .
We will use the Y -tree to value the lookback option in units of the stock
85
price (rather then in euros). In euros, the payoff from the option is
SnYn − Sn.
In stock price units, the payoff from the option is
Yn − 1.
We roll back through the tree in the usual way, valuing a derivative that provides
this payoff except that we adjust for the differences in the stock prices (i.e. the
units of measurement) at the nodes. If fi,j is the value of the lookback at the
jth node at time iδt, and Yi,j is the value of Y at this node: Yi,j = uj , the
rollback procedure gives
feuri,j = exp(−r∆t)((1 − p)fi+1,j+1d+ pfi+1,j−1u),
when j ≥ 1. Note that fi+1,j+1 is multiplied by d and fi+1,j−1 is multiplied by u
in this equation. This takes into account that the stock price at node (i, j) is the
unit of measurement. The stock price at node (i+ 1, j + 1), which was the unit
of measurement for fi+1,j+1 is d times the stock price at node (i, j). Similarly,
the stock price at node (i + 1, j − 1), which was the unit of measurement for
fi+1,j−1 is u times the stock price at node (i, j). When j = 0, the roll back
procedure gives
feuri,0 = exp(−r∆t)((1 − p)fi+1,1d+ pfi+1,0u),
The tree is initialized at the final nodes with the boundary conditions
feurn,j = Yn,j − 1
feurn,0 = 0.
86
Figure 4.1: Lookback tree example
The tree (with 5 time steps) in the Figure 4.1 estimates the value of the
option at time zero (in stock price units) as 0.230 for the European version.
This means that the value of the option is 0.230× S0 = 11.50 euros.
In case of an American type option, these two equations can be adjusted by
comparing the european price with the early exercise price (Yi,j −1) and taking
the maximum of both:
fameri,j = max Yi,j − 1, exp(−r∆t)((1 − p)fi+1,j+1d+ pfi+1,j−1u) , j ≥ 1
fameri,0 = exp(−r∆t)((1 − p)fi+1,1d+ pfi+1,0u),
87
the boundary conditions remain the same:
famern,j = Yn,j − 1
famern,0 = 0.
Increasing the number of time-steps n, will give a more precise estimate of a
continuously montinored lookback options. It is quite well known that the tree-
values converge slowly to this value. This is due because, one is actually missing
all the situations where a maximum/minimum has been attained in between two
discrete montoring points and where the stock price has fallen/risen back before
the end of that interval.
4.3 Barrier Options
The payoff of a barrier option depends on whether the price of the underlying
asset crosses a given threshold (the barrier) before maturity. The simplest bar-
rier options are “knock in” options which come into existence when the price of
the underlying asset touches the barrier and “knock-out” options which come
out of existence in that case. For example, an up-and-out call has the same
payoff as a regular plain vanilla call if the price of the underlying asset remains
below the barrier over the life of the option but becomes worthless as soon as
the price of the underlying asset crosses the barrier.
Let us denote with 1(A) the indicator function, which has a value 1 if A is
true and zero otherwise.
For single barrier options, we will focus on the following types of call options:
88
• The down-and-out barrier call is worthless unless its minimum remains
above some low barrier H , in which case it retains the structure of a
European call with strike K. Its initial price is given by:
DOBC = exp(−rT )EQ[(ST −K)+1(mST > H)].
• The down-and-in barrier call is a standard European call with strike K,
if its minimum went below some low barrier H . If this barrier was never
reached during the life-time of the option, the option is worthless. Its
initial price is given by:
DIBC = exp(−rT )EQ[(ST −K)+1(mST ≤ H)].
• The up-and-in barrier call is worthless unless its maximum crossed some
high barrier H , in which case it retains the structure of a European call
with strike K. Its price is given by:
UIBC = exp(−rT )EQ[(ST −K)+1(MST ≥ H)].
• The up-and-out barrier call is worthless unless its maximum remains below
some high barrier H , in which case it retains the structure of a European
call with strike K. Its price is given by:
UOBC = exp(−rT )EQ[(ST −K)+1(MST < H)].
The put-counterparts, replacing (ST − K)+ with (K − ST )+, can be defined
along the same lines.
89
We note that the value, DIBC, of the down-and-in barrier call option with
barrier H and strike K plus the value, DOBC, of the down-and-out barrier
option with same barrier H and same strike K, is equal to the value C of the
vanilla call with strike K. The same is true for the up-and-out together with
the up-and-in:
DIBC +DOBC = C = UIBC + UOBC. (4.1)
The above options are so-called continuously monitored. Their value can
be approximated by the discretely monitored counterparts, like in the lookback
case.
These discretely monitored barrier options (of european and american type)
can again be priced using the binomial tree setup. For example an American
down-and-out put can be valued as in the same way as an regular American
option except that, when we encounter a node below the barrier, we set the
value at that note equal to zero.
With the usual notation we have for 0 ≤ i < n
fi,j = max
K − S0ujdi−j , exp(−r∆t)((1 − p)fi+1,j+1 + pfi+1,j)
if S0ujdi−j ≥ H
fi,j = 0 if S0ujdi−j < H
and for i = n
fn,j = K − S0ujdn−j if Sn ≥ H
fn,j = 0 if Sn < H
Similar schemes can be easily deduced for the other combinations. Unfortu-
nately, convergence of the price of the discretely monitored option to the price
90
of the continuouss is also here very slow when this approach is used. A large
number of time steps is required to obtain a reasonably accurate result. The
reason for this is that the barrier being assumed by the tree is different fom the
true barrier. Define the inner barrier as the barrier formed by nodes on the side
of the true barrier (i.e., closer to the center of the tree) and the outer barrier
as the barrier formed by nodes just outside the true barrier (i.e., farther away
from the center of the tree). Figure 4.2 shows the inner and outer barrier for a
trinomial tree on the assumption the true barrier is horizontal. Figure 4.3 does
the same for a binomial tree. The usual tree calculations implicitly assume that
the outer barrier is the true barrier because the barrier conditions are first used
at nodes on this barrier.
For coping with this barrier-problem, one alternative is to calculate when
rolling back through the tree, two values of the derivative (for the nodes on
the inner barrier). The first one is obtained by assuming the inner barrier is
correct; the second one is obtained by assuming the outer barrier is correct. A
final estimate for the value of the derivative for the true barrier is then obtained
by interpolating between these two values.
For example, suppose that at time i∆t, the true value barrier is 0.2 above the
inner barrier and 0.6 below the outer barrier and suppose further that the value
of the derivative on the inner barrier is zero if the inner barrier is assumed to be
correct and 1.6 if the outer barrier is assumed to be correct. The interpolated
value (for the inner barrier node) is then 0.4. Once we have adjusted the value
at the inner barrier node, we can roll back through the tree to obtain the initial
91
Figure 4.2: Trinomial tree: inner and outer barrier
92
Figure 4.3: Trinomial tree: inner and outer barrier
93
value of the derivative in the usual way.
4.4 Asian Options
In this section we consider the pricing of a European-style arithmetic average
call option with strike price K, maturity T and n averaging days 0 ≤ t1 < . . . <
tn ≤ T .
Its payoff is given by
AAC =
(∑nk=1 Stkn
−K
)+
.
The american versions allows early exercise and in that case pays out the surplus
over the strike price K of the running average. For the put version just switch
the sum and the strike price :
AAP =
(
K −∑nk=1 Stkn
)+
.
Average price options are typically less expensive than regular options and are
arguably more appropriate than regular options for meeting some of investors
needs. Asian options are widely used in pratice - for instance, for oil and foreign
currencies. The averaging complicates the mathematics, but e.g., protects the
holder against speculative attemps to manipulate the asset price near expiry.
Assume for simplicity that t = 0 and that the averaging has not yet started.
First note, that for any K1, . . . ,Kn ≥ 0 with K =∑nk=1 Kk, we have
(
n∑
k=1
Stk − nK
)+
=(
(St1 −nK1) + · · ·+ (Stn −nKn))+
≤n∑
k=1
(Stk − nKk)+.
94
Hence the intial price AAC0(K,T )
AAC0(K,T ) =exp(−rT )
nEP∗
(
n∑
k=1
Stk − nK
)+∣
∣
∣F0
≤ exp(−rT )
n
n∑
k=1
EP∗
[
(Stk − nKk)+∣
∣
∣F0
]
=exp(−rT )
n
n∑
k=1
exp(rtk)EC0(κk, tk), (4.2)
where EC0(κk, tk) denotes the price of a European call option at time 0 with
strike κk = nKk and maturity tk.
In terms of hedging, this means that we have the following static super-
hedging strategy: for each averaging day tk, buy exp(−r(T − tk))/n European
call options at time t = 0 with strike κk and maturity tk and hold these until
their expiry. Then put their payoff on the bank account.
Since the upper bound (4.2) holds for all combinations of κk ≥ 0 that satisfy
∑nk=1 κk = nK, one still has the freedom to choose strike values.
Note that, if 0 ≤ r, the choice κk = K (k = 1, . . . , n) immediately implies,
since EC0(K, t) ≤ EC0(K, s), for t ≤ s, that
AAC0(K,T ) ≤ EC0(K,T ).
However, one naturally look for that combination of κk’s which minimizes
the right-hand side of (4.2). This can be done using comonotonic theory, but
will lead us to far in this course.
Next, we discuss the pricing of the European and American AAC using
binomial tree models. However, the procedure is not as simple as in the barrier
95
case and this because at each node we do not know the running average when
we reached that node. Typically, all different paths to reach the node lead to
different average prices, and the number of paths grow exponentially. Luckily,
the tree-approach can be extended to cope with this under certain circumstance.
We illustrate the nature of the calculation by condidering the case of an
European Asian arthimetic option. The payoff of this options depends on a
single function of the path followed, namely the average stock price. We call
this average function is the determining path function.
The trick is tp carry out, at each node, the calculations for a small number
of representative values of the path function. When the value of the derivative
is required for other values of the path function, we calculate it from the known
values using interpolation.
Suppose the initial stock price is 50, the strike price is 50, r = 0.10 and
the volatility is 0.40, and the time to maturity is one year. We use a tree with
20 time steps. The parameters describing the binomial tree parameters are
∆t = 0.05, u = 1.0936, d = 0.9144, p = 0.5056.
Figure 4.4 shows the calculations that are carried out in one small part of
the tree. Node X is the central node at time 0.2 year (at the end of the fourth
time step). Nodes Y and Z are the two nodes at time 0.25 years that are
reachable from node X . The stock price is X is 50. Forward induction shows
that the maximum average stock price achievable in reaching node X is 53.83.
The minimum is 46.65. (We include both the initial and final stock prices when
calculating the average, i.e. t1 = 0 and tn = T .) From node X , we branch to
96
Figure 4.4: Part of tree for Asian option
one of the two nodes Y and Z. At node Y , the stock price is 54.68 and the
bounds for the average are 47.99 and 57.39. At node Z, the stock price is 45.72
and the bounds for the average stock price are 43.88 and 52.48.
Suppose that we have chosen the representative values of the average to
be four equally spaced values at each node. This means that at node X , we
consider the averages 46.65, 49.04, 51.44, and 57.83. At node Y , we consider the
averages 47.99, 51.12, 54.26, and 57.39. At node Z, we consider the averages
43.88, 46.75, 49.61, and 52.48. We assume backward induction has already been
used to calculate the value of the option for each of the alternative valuesof the
average at node Y and Z. The values are shown in Figure 4.4. For exemple, at
97
node Y when the average is 51.12, the value of the option is 8.101.
Consider the calculation at node X for the case where the average is 51.44. If
the stock price moves up to node Y , the new average will be 5×51.44+54.68)/6 =
51.98. The value of the derivative at node Y for this average can be found by
interpolating between the values when the average is 51.12 and when it is 54.26.
It is
(51.98− 51.12)× 8.635 + (54.26− 51.98)× 8.101
54.26− 51.12= 8.247.
Similarly, if the stock price moves down to node Z, the new average will be
5 × 51.44 + 45.72)/6 = 50.49 and by interpolation the value of the derivative
is 4.182. The value of the derivative at node X when the average is 51.44 is,
therefore,
exp(−0.1× 0.05)(0.5056× 8.247 + (1 − 0.5056)× 4.182) = 6.206.
The other values at node X are calculated similarly. Once the values at all
nodes at time 0.2 year have been calculated, we can move on to the nodes at
time 0.15 year.
The value given by the full tree for the option at time zero is 7.17. As the
number of time steps and the number of averages considered at each node is
increased, the value of the option converges to the correct answer. With 60 time
steps and 100 averages at each node, the value of the option is 5.58. The true
value of the option is around 5.62.
A key advantage of the method here is that it can handle American options.
The calculations are as we have described them except that we test for early
98
exercise at each node for each of the alternative values of the path function at
the node.
The approach just described can be used in a wide range of different situa-
tions if the following conditions are satisfied:
• the payoff from the derivative must depend on a single function, the path
function, of the path followed by the underlying asset;
• it must be possible to calculate the value of the path function at time t+∆t
from the value of this function at time t and the value of underlying asset
at time t+ ∆t.
Efficiency is improved somewhat if quadratic rather than linear interpolation is
used at each node.
99
Chapter 5
The Black-Scholes Option
Price Model
In the early 1970s, Fischer Black, Myron Scholes, and Robert Merton made
a major breakthrough in the pricing of stock options by developing what has
become known as the Black-Scholes model. The model has had huge influence
on the way that traders price and hedge options. In 1997, the importance of the
model was recognized when Myron Scholes and Robert Merton were awarded
the Nobel prize for economics. Sadly, Fischer Black died in 1995, otherwise he
also would undoubtedly have been one of the recipients of this prize.
This chapter shows how the Black-Scholes model for valuing European call
and put options on a non-dividend-paying stock is derived.
100
5.1 Continuous-Time Stochastic Processes
This section develops a continuous-time, continuous-variable stochastic process
for stock prices. An understanding of this process is the first step to understand-
ing the pricing of options and other more complicated derivatives. It should
be noted that, in practice, we do not observe stock prices following continuous-
variable, continuous-time processes. Stock prices are restricted to discrete values
(often multiples of 0.01 euro) and changes can be observed only when the ex-
change is open. Nevertheless, the continuous-variable, continuous-time process
proves to be a useful model for many purposes.
5.1.1 Information and Filtration
The underlying set-up is as in the discrete time case. We assume a fixed finite
planning horizon T . We need a complete probability space (Ω,FT , P ), equipped
with a filtration, i.e. a nondecreasing family F = (Ft)0≤t≤T of sub-σ-fields of
FT : Fs ⊂ Ft ⊂ FT for 0 < s < t ≤ T ; here Ft represents the information
available at time t, and the filtration F = (Ft) represents the information flow
evolving with time.
We assume that the filtered probability space (Ω,FT , P,F) satisfies the ’usual
conditions’: a) F0 contains all P -null sets of F . This means intuitively that we
know which events are possible and which not, and b) (Ft) is right-continuous,
i.e. Ft = ∩s>tFs; a technical condition.
A stochastic process X = (Xt)0≤t≤T is a family of random variables defined
on (Ω,FT , P,F). We say X is F-adapted if Xt ∈ Ft (i.e. Xt is Ft-measurable)
101
for each t: thus Xt is known at time t.
5.1.2 Martingales
A stochastic process X = (Xt)t≥0 is a martingale relative to (P,F) if
• X is F-adapted
• E[|Xt|] <∞ for all t ≥ 0
• E[Xt|Fs] = Xs, P -a.s., (0 ≤ s ≤ t),
A martingale is ’constant on average’, and models a fair game. This can be seen
from the third condition: the best forecast of the unobserved future value Xt
based on information at time s, Fs, is the at time s known value Xs.
5.2 Brownian Motion
The Scottish botanist Robert Brown observed pollen particles in suspension
under a microscope in 1828 and 1829, and observed that they were in constant
irregular motion. In 1900 L. Bachelier considered Brownian motion as a possible
model for stock-market prices. In 1905 Albert Einstein considered Brownian
motion as a model of particles in suspension and used it to estimate Avogadro’s
number. In 1923 Norbert Wiener defined and constructed Brownian motion
rigorously for the first time. The resulting stochastic process is often called the
Wiener process in his honour.
102
Definition 17 A stochastic process X = Xt, t ≥ 0 is a standard Brownian
motion on some probability space (Ω,F , P ), if
1. X0 = 0 a.s.
2. X has independent increments.
3. X has stationary increments.
4. Xt+s−Xt is normally distributed with mean 0 and variance s: Xt+s−Xt ∼
N(0, s).
5. X has continuous sample paths.
We shall henceforth denote standard Brownian motion by W = Wt, t ≥ 0
(W for Wiener).
Construction
No construction of Brownian motion is easy: one needs both some work and
some knowledge of measure theory. We take the existence of Brownian motion
for granted. To gain some intuition on its behaviour, it is good to compare
Brownian motion with a simple symmetric random walk on the integers. More
precisely, let X = Xi, i = 1, 2, . . . be a series of independent and identically
distributed random variables with Pr(Xi = 1) = Pr(Xi = −1) = 1/2. Define
the simple symmetric random walk Z = Zn, n = 0, 1, 2, . . . as Z0 = 0 and
Zn =∑n
i=1Xi, n = 1, 2, . . . . Rescale this random walk as
Yk(t) = Zbktc/√k,
103
where bxc is the integer part of x. Then from the Central Limit Theorem,
Yk(t) →Wt as k → ∞,
with convergence in distribution (or weak convergence).
In Figure 5.1, one sees a realization of the standard Brownian motion. In
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1Standard Brownian Motion
Figure 5.1: A sample path of a standard Brownian motion
Figure 5.2, one sees the random-walk approximation of the standard Brownian
motion. The process Yk = Yk(t), t ≥ 0 is shown for k = 1 (i.e. the symmetric
random walk), k = 3, k = 10 and k = 50. Clearly, one sees the Yk(t) →Wt.
The universal nature of Brownian motion as a stochastic process is simply
the dynamic counterpart – where we work with evolution in time – of the uni-
versal nature of its static counterpart, the normal distribution – in probability,
statistics, science, economy etc. Both arise from the same source, the central
limit theorem. This says that when we average large numbers of independent
and comparable objects, we obtain the normal distribution in a static context,
104
0 10 20 30 40 50−10
−5
0
5
10k=1
0 10 20 30 40 50−10
−5
0
5
10k=3
0 10 20 30 40 50−10
−5
0
5
10k=10
0 10 20 30 40 50−10
−5
0
5
10k=50
Figure 5.2: Random walk approxiamtion for standard Brownian motion
or Brownian motion in a dynamic context. What the central limit theory really
says is that, when what we observe is the result of a very large number of indi-
vidually very small influences, the normal distribution or Brownian motion will
inevitably and automatically emerge.
Next, we look at some of the classical properties of Brownian motion.
Martingale Property
Brownian motion is one of the most simple examples of a martingale. We have
for all 0 ≤ s ≤ t,
E[Wt|Fs] = E[Wt|Ws] = Ws.
105
We also mention that one has:
E[WtWs] = mint, s.
Path Properties
One can proof that Brownian motion has continuous paths, i.e. Wt is a continu-
ous function of t. However the paths of Brownian motion are very erratic. They
are for example nowhere differentiable. Moreover, one can prove also that the
paths of Brownian motion are of infinite variation, i.e. their variation is infinite
on every interval.
Another property is that for a Brownian motion W = Wt, t ≥ 0, we have
that
Pr(supt≥0
Wt = +∞ and inft≥0
Wt = −∞) = 1.
This result tells us that the Brownian path will keep oscillating between positive
and negative values.
Scaling Property
There are a well-known set of transformations of Brownian motion which pro-
duce another Brownian motion. One of this is the scaling property which says
that if W = Wt, t ≥ 0 is a Brownian motion, then also for every c 6= 0,
W = Wt = cWt/c2 , t ≥ 0 (5.1)
is a Brownian motion.
106
5.3 Ito’s Calculus
5.3.1 Stochastic Integrals
Stochastic integration was introduced by K. Ito in 1941, hence its name Ito
calculus. It gives meaning to∫ t
0
XudYu
for suitable stochastic processes X = Xu, u ≥ 0 and Y = Yu, u ≥ 0, the
integrand and the integrator. We shall confine our attention here to the basic
case with integrator Brownian motion: Y = W .
Because Brownian motion is of infinite (unbounded) variation on every inter-
val, the first thing to note is that stochastic integrals with respect to Brownian
motion, if they exist, must be quite different from the classical deterministic in-
tegrals. We take for granted Ito’s fundamental insight that stochastic integrals
can be defined for a suitable class of integrands.
We only show how these integrals can be defined for some simple integrands
X .
Indicators
If Xt = 1[a,b](t), i.e. it equals 1 between a and b and is zero elsewhere, we define
∫
XdW :
It(X) =
∫ t
0
XsdWs =
0 if t ≤ a
Wt −Wa if a ≤ t ≤ b
Wb −Wa if t ≥ b
107
Simple deterministic functions
We can extend the above definition by linearity: if X is a linear combination of
indicators, Xt =∑n
i=1 ci1[ai,bi](t), we define∫
XdW :
It(X) =
∫ t
0
XsdWs =
n∑
i=1
ci
∫ t
0
1[ai,bi](s)dWs
Simple stochastic processes
X is called a simple stochastic process if there is a partition 0 = t0 < t1 <
· · · < tn = T <∞ and uniformly bounded Ftk -measurable random variables ξk
(|ξk| ≤ C for all k = 0, . . . , n for some C) and if Xt can be written in the form
Xt = ξ010(t) +
n−1∑
i=0
ξi1(ti,ti+1](t), 0 ≤ t ≤ T.
Then if tk ≤ t ≤ tk+1, k = 0, . . . , n− 1,
It(X) =
∫ t
0
XsdWs =
k−1∑
i=0
ξi(Wti+1−Wti) + ξk(Wt −Wtk )
It is not so hard to prove some simple properties of the stochastic integrals
defined so far:
• It(aX + bY ) = aIt(X) + bIt(Y ).
• It(X) is a martingale.
• Ito isometry: E[(It(X))2] =∫ t
0 E[(Xu)2]du.
The Ito isometry above suggests that∫ t
0XdW should be defined only for processes
with∫ t
0 E[(Xu)2]du <∞ for all t and this is indeed the case. Each such X may
be approximated by a sequence of simple stochastic processes and the stochas-
tic integral may be defined as the limit of this approximation. Furthermore
108
the three above properties remain true. We will not include the technical and
detailed proofs of this procedure in this book. Note that one also can construct
a closely analogous theory for stochastic integrals with the Brownian integrator
W above replaced by a (semi-)martingale integrator M .
5.3.2 Ito’s Lemma
The price of a stock option is a function of the underlying stock’s price and
time. More generally, we can say that the price of any derivative is a function of
the stochastic variables underlying the derivative and time. Therefore, we must
acquire some understanding of the behavior of functions of stochastic variables.
An important result in this area was discovered by K. Ito, in 1951. It is known
as Ito’s lemma.
Suppose that F : R2 → R is a function, which is continuously differentiable
once in its first argument (which will denote time), and twice in its second
argument: F ∈ C1,2. Denote the partial derivatives
Ft(t, x) =∂F
∂t(t, x)
Fx(t, x) =∂F
∂x(t, x)
Fxx(t, x) =∂2F
∂x2(t, x)
Theorem 18 (Ito’s lemma) Let W = Wt, t ≥ 0 be Standard Brownian
motion and let F (t, x) ∈ C1,2, then
F (t,Wt)−F (s,Ws) =
∫ t
s
Fx(u,Wu)dWu+
∫ t
s
Ft(u,Wu)du+1
2
∫ t
s
Fxx(u,Wu)du.
109
or
dF = FxdWt + Ftdt+1
2Fxxdt.
for short.
As an application of Ito’s lemma we compute∫ t
0 WudWu by using F (t, x) =
x2. Then
W 2t = W 2
0 +
∫ t
0
2WudWu +1
2
∫ t
0
2du = 2
∫ t
0
WudWu + t.
So that∫ t
0
WudWu =W 2t
2− t
2
Note the contrast with ordinary calculus ! Ito calculus requires the second term
on the right – the Ito correction term.
5.4 Stochastic Differential Equations
Like with any ordinary and partial differential equations in a deterministic set-
ting (ODEs and PDEs), the two most basic questions are those of existence and
uniqueness of solutions. To obtain existence and uniqueness results, one has to
impose reasonable regularity conditions on the coefficients occuring in the dif-
ferential equation. Naturally, stochastic differential equations (SDEs) contain
all the complications of their non-stochastic counterparts, and more besides.
Consider the stochastic differential equation
dXt = b(t,Xt)dt+ σ(t,Xt)dWt, Xs = x, (5.2)
110
where the coefficients b and σ satisfy the following Lipschitz and growth condi-
tions
|b(t, x) − b(t, y)| + |σ(t, x) − σ(t, y)| ≤ K|x− y|
|b(t, x)|2 + |σ(t, x)|2 ≤ K2(1 + |x|2)
for all t ≥ 0, x, y ∈ R, for some constant K > 0.
To see that the SDE (5.2) has a solution, we first define recursively
X(0)t = x,X
(n+1)t = x+
∫ t
s
b(u,X(n)u )du+
∫ t
s
σ(u,X(n)u )dWu.
One can then prove that X(n)t converges (in some sense), to Xt say; Xt is the
unique (strong) solution to (5.2), i.e.
X0 = x,Xt = x+
∫ t
s
b(u,Xu)du+
∫ t
s
σ(u,Xu)dWu.
The next result, which is an example for the rich interplay between probabil-
ity theory and analysis, links SDEs with PDEs. Suppose we consider a stochastic
differential equation (satisfying the above Lipschitz and growth conditions),
dXu = µ(u,Xu)du+ σ(u,Xu)dWu, Xs = x, s ≤ u ≤ T
Consider a function F (t, x) ∈ C1,2 of it. Then we have the following extension
of Ito’s lemma:
Theorem 19 (Ito’s lemma for SDE’s) Let F (t, x) ∈ C1,2, then
F (t,Xt) − F (s,Xs) =
∫ t
s
σ(u,Xu)Fx(u,Xu)dWu+ (5.3)
∫ t
s
(
Ft(u,Xu) + µ(u,Xu)Fx(u,Xu) +σ(u,Xu)
2
2Fxx(u,Xu)
)
du.
111
Now suppose that F satisfies the PDE
Ft(t, x) + µ(t, x)Fx(t, x) +(σ(t, x))2
2Fxx(t, x) = 0,
with boundary condition
F (T, x) = h(x).
Then the above expression for (5.3) gives
F (s,Xs) = F (t,Xt) −∫ t
s
σ(u,Xu)Fx(u,Xu)dWu
The stochastic integral on the right is a martingale, so has constant expectation,
which must be 0 as it starts at 0. So
F (s, x) = E[F (t,Xt)|Xs = x]
which leads for t = T to the Feynman-Kac Formula
F (s, x) = E[h(XT )|Xs = x].
The Feynman-Kac formula gives a stochastic representation to solutions of
PDEs. We shall return to the Feynman-Kac formula below in connection with
the Black-Scholes partial differential equation.
5.5 Geometric Brownian Motion
Now that we have both Brownian motion W and Ito’s Lemma to hand, we can
introduce the most important stochastic process for us, a relative of Brownian
motion – geometric Brownian motion.
112
Suppose we wish to model the time evolution of a stock price St. Consider
how S will change in some small time interval from the present time t to a time
t+∆t in the near future. Writing ∆St for the change St+∆t−St, the return on
S in this interval is ∆St/St. It is economically reasonable to expect this return
to decompose into two components, a systematic part and a random part. The
systematic part could plausibly be modeled by µ∆t, where µ is some parameter
representing the mean rate of the return of the stock. The random part could
plausibly be modeled by σ∆Wt, where ∆Wt represent the noise term driving the
stock price dynamics, and σ is a second parameter describing how much effect
the noise has – how much the stock price fluctuates. Thus σ governs how volatile
the price is, and is called the volatility of the stock. The role of the driving noise
term is to represent the random buffeting effect of the multiplicity of factors at
work in the economic environment in which the stock price is determined by
supply and demand.
Putting this together, we have the following SDEs
∆St = St(µ∆t+ σ∆Wt), S0 > 0.
In the limit as ∆t→ 0, we have the stochastic differential equation
dSt = St(µdt+ σdWt), S0 > 0.
The differential equation above has the unique solution
St = S0 exp
((
µ− σ2
2
)
t+ σWt
)
.
For, writing
f(t, x) = exp
((
µ− σ2
2
)
t+ σx
)
113
Ito’s lemma gives
df(t,Wt) =δf
δtdt+
δf
δxdWt +
1
2
δ2f
δx2dt
=
(
µ− σ2
2
)
fdt+ σfdWt +1
2σ2fdt
= f(µdt+ σdWt)
so f(t,Wt) is a solution of the stochastic differential equation. This means that
logSt = logS0 +
(
µ− σ2
2
)
t+ σWt
has a normal distribution. Thus St itself has a lognormal distribution. This
geometric Brownian motion model, and the log-normal distribution which it
entails, are the basis for the Black-Scholes model for stock-price dynamics in
continuous time.
In Figure 5.3 one sees the realization of the geometric Brownian motion
based on the sample path of the standard Brownian motion of Figure ??.
5.6 The Market Model
We consider a frictionless security market in which two assets are traded con-
tinously. Investors are allowed to trade continuously up to some fixed finite
planning horizon T , where all economic activity stops.
The first asset is one without risk (the bank account). Its price process is
given by Bt = ert, 0 ≤ t ≤ T . The second asset is a risky asset, usually refered
to as stock. The price process of this stock, St, 0 ≤ t ≤ T , is modelled by the
114
Figure 5.3: Sample path of a geometric Brownian motion (S0 = 100, µ =
0.05, σ = 0.40)
linear stochastic differential equation
dSt = St(µdt+ σdWt), S0 = x > 0,
where Wt is standard Brownian motion, defined on a filtered probability space
(Ω,F , P,F). This means that under P , Wt has a Normal(0, t) distribution.
Furthermore, in the previous chapter we derived that St follows a geometric
Brownian motion:
St = S0 exp
((
µ− σ2
2
)
t+ σWt
)
.
µ is reflecting the drift and σ models the volatility and are assumed to be
constant. We assume as underlying filtration, the Brownian filtration F =
(Ft), basically Ft = σ(Ws, 0 ≤ s ≤ t), slightly enlarged to satisfy the usual
conditions. Consequently, the stock price process St follows a strictly positive
adapted process. We call this market model the Black-Scholes model.
115
Our principle task will be the pricing and hedging of contingent claims, which
we model as non-negative FT -measurable random variables. This implies that
the contingent claims specify a stochastic cash-flow at time T and that they
may depend on the whole path of the underlying in [0, T ] – because FT contains
all information.
We will often have to impose further (integrability) conditions on the con-
tingent claims under consideration. As before, the fundamental concept in (ar-
bitrage) pricing and hedging contingent claims is the interplay of self-financing
replicating portfolios and risk-neutral probabilities. Although the current (time-
continuous) setting is on a much higher level of sophistication, the key ideas
remain the same.
We call a two-dimensional adapted (predictable), locally bounded process
ϕ = ϕt = (βt, ξt), t ∈ [0, T ]
a trading strategy or dynamic portfolio process. The conditions ensure that the
stochastic integral∫ t
0ξtdWt exists. Here βt denotes the money invested in the
riskless asset and ξt denotes the number of stocks held in the portfolio at time
t.
Remark: In a more general setting the trading strategy has to be pre-
dictable in stead of adapted. Predictability of these processes imply that (βt, ξt)
has to be determined on the basis of information available strictly before time
t, Ft−: the investor selects his time t portfolio just before the observation of the
price St. Because our Brownian filtration is continuous we have Ft− = Ft and
predictablity and adaptedness are the same. •
116
The components of ϕt may assume negative as well as positive values, reflect-
ing the fact that we allow short sales and assume that the assets are perfectly
divisible.
Definition 20 (i) The value of the portfolio ϕ at time t is given by
Vt = V ϕt = βtBt + ξtSt = βtert + ξtSt
The process V ϕt is called the value process, or wealth process, of the trading
strategy ϕ.
(ii) The gains process Gϕt is defined by
Gt = Gϕt =
∫ t
0
βudBu +
∫ t
0
ξudSu
(iii) A trading strategy ϕ is called self-financing if the wealth process V ϕt satis-
fies
V ϕt = V ϕ0 +Gϕt for all t ∈ [0, T ].
The financial implications of the above equations are that all changes in the
wealth of the portfolio are due to market changes, as opposed to withdrawals of
cash or injections of new funds.
5.7 Equivalent Martingale Measures and Risk-
Neutral Pricing
Next, we develop a pricing theory for contingent claims. Again the underlying
concept is the link between the no-arbitrage condition and certain probability
117
measures. We begin with
Definition 21 A trading strategy ϕ is called tame, if the associated wealth
process is always positive:
V ϕt ≥ 0, for all t ∈ [0, T ].
Similarly as in the discrete case (admissible strategies), tame strategies prevent
the broker from unbounded short sales. Using tame strategies the investor’s
wealth may never go negative at a time, even if he is able to cover his debt at
the final date. If we would later on allow non-tame strategies, one can show
that it is possible to construct doubling strategies that can attain arbitrarily
large values of wealth starting with zero initial capital. Such strategies are
examples of arbitrage opportunities, which we define in general as:
Definition 22 A self-financing trading strategy ϕ is called an arbitrage oppor-
tunity if the wealth process V ϕ satisfies the following set of conditions:
V ϕ0 = 0, P (V ϕT ≥ 0) = 1 and P (V ϕT > 0) > 0.
Arbitrage opportunities represent the limitless creation of wealth through risk-
free profit and thus should not be present in a well-functioning market.
We say that our market is arbitrage-free if there are no tame self-financing
arbitrage opportunities.
The main tool in investigating arbitrage opportunities is the concept of
equivalent martingale measures:
Definition 23 We say that a probability measure P ∗ defined on (Ω,FT ) is an
equivalent martingale measure if:
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(i) P ∗ is equivalent to P
(ii) the discounted price process St = e−rtSt is a P ∗-martingale.
We denote the set of equivalent martingale measures by P.
As in the discrete case, one can prove that one can preclude arbitrage oppor-
tunities if an equivalent martingale measure exists. Furthermore, in the more
general continuous-time setting, we have the following partial analogue of the
completeness theorem in the discrete setting: If P = P ∗, then the market
is complete, in the restricted sense that for every contingent claim X satisfy-
ing EP∗ [X2] < ∞ there exists at least an admissible self-financing trading ϕ
strategy such that V ϕT = X .
Remark: Having seen the above results, a natural question is to ask whether
converse statements are also true. One has to put some further requirements
on portfolios to establish such converse results. These requirements should of
course be economically meaningful. A lot of effort has been put into solving this
question, and several alternatives have been proposed, but the details will lead
us to far. •
By the risk-neutral valuation principle the price Vt at time t, of a contingent
claim with payoff function G(Su, 0 ≤ u ≤ T) is given by
Vt = exp(−(T − t)r)EP∗ [G(Su, 0 ≤ u ≤ T)|Ft], t ∈ [0, T ], (5.4)
where P ∗ is an equivalent martingale measure. In a general setting their is not a
unique martingale measure ( incomplete market models). Roughly speaking in-
completeness means that a general contingent claim can not be perfectly hedged.
119
Most models are not complete, and most practitioners believe the actual market
is not complete. we have to choose an equivalent martingale measure in some
way and this is not always clear. Actually, the market is choosing the martingale
measure for us.
In the Black-Scholes world however, one can prove (Girsanov Theorem) that
there is a unique equivalent martingale measure and we do not have to deal
with coosing an appropriate one. It is not hard to see that under P ∗, the stock
price is following a Geometric Brownian motion again. This risk-neutral stock
price process has the same volatility parameter σ, but the drift parameter µ is
changed to the continuously compounded risk-free rate r:
St = S0 exp
((
r − σ2
2
)
t+ σWt
)
.
Equivalent, we can say that under P ∗ our stock price process S = St, 0 ≤ t ≤
T is satisfying the SDE:
dSt = St(rdt+ σdWt), S0 > 0.
This SDE tells us that in a risk-neutral world the total return from the stock
must be r.
Next, we will calculate European call option prices under this model.
5.7.1 The Pricing of Options under the Black-Scholes Model
If the payoff function is only depending on the time T value of the stock, i.e.
G(Su, 0 ≤ u ≤ T) = G(ST ), then the above formula can be rewritten as (we
120
set for simplicity t = 0):
V0 = exp(−Tr)EP∗ [G(ST )]
= exp(−Tr)EP∗ [G(S0 exp((r − q − σ2/2)T + σWT ))]
= exp(−Tr)∫ +∞
−∞G(S0 exp((r − q − σ2/2)T + σx))fNormal(x; 0, T )dx.
Explicit Formula for European Call and Put Options
In some cases it is possible to evaluate explicitly the above expected value in
the risk-neutral pricing formula (5.4).
Take for example an European call on the stock (with price process S) with
strike K and maturity T (so G(ST ) = (ST −K)+). The Black-Scholes formulas
for the price C(K,T ) at time zero of this European call option on the stock
(with dividend yield q) is given by
C(K,T ) = C = S0N(d1) −K exp(−rT )N(d2),
where
d1 =log(S0/K) + (r + σ2
2 )T
σ√T
, (5.5)
d2 =log(S0/K) + (r − σ2
2 )T
σ√T
= d1 − σ√T , (5.6)
and N(x) is the cumulative probability distribution function for a variable that
is standard normally distributed (Normal(0, 1)).
From this, one can also easily (via the put-call parity) obtain the price
P (K,T ) of the European put option on the same stock with same strike K and
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same maturity T :
P (K,T ) = −S0N(−d1) +K exp(−rT )N(−d2).
For the call, the probability (under P ∗) of finishing in the money corresponds
with N(d2). Similarly, the delta (i.e. the change in the value of the option
compared with the change in the value of the underlying asset) of the option
corresponds with N(d1).
Black-Scholes PDE
If moreover G(ST ) is a sufficiently integrable function, then the price at time
t is only a function of t and St: Vt = F (t, St). We show that F solves the
Black-Scholes partial differential equation
∂
∂tF (t, s) + (r − q)s
∂
∂sF (t, s) +
1
2σ2s2
∂2
∂s2F (t, s) − rF (t, s) = 0, (5.7)
F (T, s) = G(s)
This will basically follow from the Feynman-Kac representation for Brownian
motion.
Indeed, let H(t, s) be a solution of
Ht(t, s) + rsHs(t, s) +1
2σ2s2Hss(t, s) = 0,
H(T, s) = e−rTG(s).
Then we know from the Feynman-Kac representation that H has the represen-
tation
H(t, St) = e−rTEP∗ [G(ST )|Ft];
122
Note that by the risk-neutral valuation principle
Vt = F (t, St) = exp(−r(T − t))EP∗ [G(ST )|Ft] = exp(rt)H(t, St).
By computing the partial derivatives of F we obtain the PDE (5.7).
5.7.2 Hedging
For a European call option on a non-dividend-paying stock, it can be shown
from the Black-Scholes formulas that
ξt = ∆call = N(d1) = N
(
ln(S0/K) + (r + σ2
2 )T
σ√T
)
For a European put option on a non-dividend-paying stock, it can be shown
from the Black-Scholes formulas that delta is given by
ξt = ∆put = N(d1) − 1 = N
(
ln(S0/K) + (r + σ2
2 )T
σ√T
)
− 1 = ∆call − 1
∆ is the rate of change of the option price with respect to the price of the
underlying asset. Suppose that the delta of a call option on as stock is 0.6. This
means that when the stock price changes by a small amount, the option price
changes by about 60 percent of that amount. Suppose further that the stock
price is 100 euro and the option price is 10 euro. Imagine an investor who has
sold 2000 option contracts – that is, options to buy 2000 shares. The investor’s
position could be hedged by buying 0.6 × 2000 = 1200 shares. The gain (loss)
on the option position would then tend to be offset by the loss (gain) on the
stock position. For example, if the stock goes up by 1 euro (producing a gain
of 1200 euro on the shares purchased), the option price will tend to go up by
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0.6 × 1 = 0.60 euro (producing a loss of 2000× 0.6 = 1200 euro on the options
written); if the stock price goes down by 1 euro (producing a loss of 1200 euro
on the stock position), the option price will tend to go down by 0.60 (producing
a gain of 1200 euro on the option position).
It is important to realize that, because delta changes (with time and stock
price movements), the investor’s position remains delta-hedged (or delta neu-
tral) for only a relatively short period of time. In order to have a perfect hedge,
the positions have to be adjusted continuously. In practice however one can only
adjust periodically. This is known as rebalancing. For example, suppose that
an increase in the stock leads to an increase in delta, say from 0.60 to 0.65. An
extra of 0.05× 2000 = 100 shares would then have to be purchased to maintain
the hedge.
Tables 5.1 and 5.2 provide two simulations of the operation of periodical
delta-hedging. The hedge is assumed to be rebalanced weekly. Assume we have
to hedge a position of 100000 written call options on a non-dividend paying
stock with strike price K, with
S0 = 49,K = 50, r = 0.05 (compound interest rate per year),
σ = 0.2 (per year) and T = 20 weeks = 0.3846 years
From this we can easily compute the initial value of the call: C = 2.40047;
and the delta which equals ∆ = 0.52160. This means that as soon as the option
is written we have to buy 0.52160×100000 = 52160 shares at a price of 49 euro,
for the total amount of 49 × 52160 = 2555840 euro. So we must borrow this
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amount of 2555840 euro to buy 52160 shares. Because the interest rate is 0.05,
the interest cost totaling 2555840(exp(0.05/52) − 1) = 2459 euro are incurred
in the first week.
In Table 5.1, the stock price falls to 48.125 euro by the end of the first week.
∆ is recomputed at the end of the first week using
S0 = 48.125,K = 50, r = 0.05 (compound interest rate per year),
σ = 0.2 (per year) and T = 19 weeks = 0.3654 years
and is equal to ∆ = 0.45835. A total of 52160 − (0.45835 × 100000) = 6325
shares must be sold to maintain the hedge. This realizes 6325×48.125 = 304391
cash and the cumulative borrowings at the end of week one are reduced to
2555840 − 304391 + 2459 = 2253908 euro. During the second week the stock
price reduces to 47.375 euro and the delta declines again; and so on. Towards the
end of the life of the option it becomes apparent that the option will be exercised
and the delta approaches 1. By week 20, therefore, the hedger has a fully covered
position. The hedger receives 5000000 euro for the stock held, so that the total
cost of writing the option and hedging it is 5000000− 5263157 = 263157 euro.
Table 5.2 illustrates an alternative sequence of events such that the option closes
out of the money. As it becomes clearer that the option will not be exercised,
delta approaches zero. By week 20, the hedger has a naked position and has
incurred costs totaling 256558 euro.
In Table 5.1 and 5.2, the costs of hedging the option, when discounted to
the beginning of the period, i.e. 258145 and 251672 are close to but not exactly
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Table 5.1: Hedging simulation; call option closes in the money
Table 5.2: Hedging simulation; call option closes out of the money
126
the same as the Black-Scholes price of 240047. If the hedging scheme worked
perfectly, the cost of hedging would, after discounting, be exactly equal to the
theoretical price of the option on every simulation. The reason that there is a
variation in the cost of delta hedging is that the hedge is rebalanced only once a
week. As rebalancing takes place more frequently, the uncertainty in the cost of
hedging is reduced. Of course the simulations above are idealized in that they
assume that the volatility and interest rate are constant and that there are no
transaction costs.
In Figure 5.4, one can see the underlying Standard Brownian Motion, the
related Geometric Brownian Motion, the option prices of a European call option
and the associated hedge over the one year life-time of the option (S0 = 100,
K = 105, r = 0.03, µ = 0.09, σ = 0.4). Note how fast ∆ is near maturuity
going to 1 (the option ends in the money).
5.8 The Greeks
The Black-Scholes option values depend on the (current) stock price S, the
volatility σ, the time to maturity T , the interest rate r, and the strike price K.
The sensitivities of the option price with respect to the first four parameters are
called the Greeks and are widely used for hedging purposes.
Recall the Black-Scholes formula for a European call:
C = C(S, T,K, r, σ) = SN
(
ln(S0/K) + (r + σ2
2 )T
σ√T
)
−Ke−rTN(
ln(S0/K) + (r − σ2
2 )T
σ√T
)
.
127
Figure 5.4: Wt, St, Ct and ∆t, t ∈ [0, 1] (S0 = 100, K = 105, r = 0.03, µ = 0.09,
σ = 0.4)
128
We therefore get
∆ =δC
δS= N
(
ln(S0/K) + (r + σ2
2 )T
σ√T
)
> 0
V =δC
δσ= S
√Tn
(
ln(S0/K) + (r + σ2
2 )T
σ√T
)
> 0
Θ =δC
δT=
Sσ
2√Tn
(
ln(S0/K) + (r + σ2
2 )T
σ√T
)
+
Kre−rTN
(
ln(S0/K) + (r − σ2
2 )T
σ√T
)
> 0
ρ =δC
δr= TKe−rTN
(
ln(S0/K) + (r − σ2
2 )T
σ√T
)
> 0
Γ =δ2C
δS2=
n
(
ln(S0/K)+(r+ σ2
2)T
σ√T
)
Sσ√T
> 0,
where as usual N is the cumulative normal distribution function and n is its
density. As discussed before ∆ measures the change in the value of the option
compared with the change in the value of the underlying asset. Furthermore, ∆
gives the number of shares in the replication portfolio for a call option.
Vega, V , measures the change of the option price compared with the change
in the volatility of the underlying, and similar statements hold for theta Θ, rho
ρ. Gamma Γ measures the sensitivity of our replicating portfolio to the change
in the stock price.
5.9 Drawbacks of the Black-Scholes Model
Over the last decades the Black-Scholes model turned out to be very popular.
One should bear in mind however, that this elegant theory hinges on several
129
Figure 5.5: Simulated Normally and Nasdaq Composite log-returns
crucial assumptions. We assumed that there were no market frictions, like taxes
and transaction costs or constraints on the stockholding, etc.
Moreover, most empirical evidence suggests that the classical Black-Scholes
model does not describe the statistical properties of financial time series very
well. Real markets exhibit from time to time very large discontinuous price
movements. Moreover, according to the Black-Scholes model, the log-returns,
i.e. differences of the form logSt+h−logSt, are independent and identically nor-
mally distributed. Figure 5.5 shows daily log-returns of the American Nasdaq-
Composite Index over the period 1-1-1990 until 31-12-2000 and simulated i.i.d.
normal variates with variance equal to the sample variance of the Nasdaq-
Composite log-returns.
130
This picture makes two stylized facts immediately apparent, which are typ-
ical for most financial time series.
• We see that large asset prize movements occur more frequently than in a
model with normal distributed increments. This feature is often refered
to as excess kurtosis or fat tails; it is the main reason for considering asset
price processes with jumps.
• There is evidence for volatility clusters, i.e. there seems to be a succession
of periods with high return variance and with low return variance. This
observation motivates the introduction of models for asset process where
volatility is itself stochastic.
Typically we enter the realm of incomplete markets whenever we want to
use models for asset price dynamics which are more ’realistic’ than the Black-
Scholes model. For example markets are incomplete if we consider asset price
processes with random volatility or with jumps of varying size.
131
Chapter 6
Miscellaneous
6.1 Decomposing Options into Vanilla Position
Consider an option with a payoff function that only depends on the terminal
stock price value, i.e. assume that the payoff function is of the form f(ST ).
Assume furthermore (for technical reasons) that the function f is twice differ-
entiable.
The fundamental theorem of calculus implies that for any fixed κ:
f(x) = f(κ) + 1(x > κ)
∫ x
κ
f ′(L)dL− 1(x < κ)
∫ κ
x
f ′(L)dL
= f(κ) + 1(x > κ)
∫ x
κ
[
f ′(κ) +
∫ L
κ
f ′′(K)dK
]
dL
−1(x < κ)
∫ κ
x
[
f ′(κ) −∫ κ
L
f ′′(K)dK
]
dL.
Noting that f ′(κ) does not depend on L and interchanging the order of integra-
132
tion (Fubini’s theorem) yields:
f(x) = f(κ) + f ′(κ)(x− κ) + 1(x > κ)
∫ x
κ
∫ x
K
f ′′(K)dLdK
+1(x < κ)
∫ κ
x
∫ K
x
f ′′(K)dLdK.
Performing the integral over L yields:
f(x) = f(κ) + f ′(κ)(x − κ) + 1(x > κ)
∫ x
κ
f ′′(K)(x−K)dK
+1(x < κ)
∫ κ
x
f ′′(K)(K − x)dK
= f(κ) + f ′(κ)(x − κ) +
∫ ∞
κ
f ′′(K)(x−K)+dK +
∫ κ−
0
f ′′(K)(K − x)+dK.
Thus, the payoff decomposes into bonds, forward contracts with delivery price
κ, calls struck above κ, and puts struck below κ. Letting V f0 denote the initial
value of the contract with payoff f(ST ) at T , then the absence of arbitrage
implies:
V f0 = B−1T EQ[f(ST )]
= f(κ)B−1T + f ′(κ)(S0 − κB−1
T ) +
∫ ∞
κ
f ′′(K)EC0(K,T )dK
+
∫ κ
0
f ′′(K)EP0(K,T )dK,
where EC0(K,T ) and EP0(K,T ) denotes the initial value of resp. an European
call and put option with strike K and time to maturity T . Note that if we
choose κ = BTS0, i.e. the forward price of the stock, the second term cancels
out.
133
6.2 Variance Swap
Consider a finite set of discrete times t0 = 0, t1, . . . , tn = T at which the
path of the underlying is monitored. We denote the price of the underlying at
these points, i.e. Sti , by Si for simplicity. Typically, t0 = 0, t1, . . . , tn = T
corresponds to daily closing times and Si is the closing price at day i. Note that
then
log(Si) − log(Si−1), i = 1, . . . , n,
correspond to the daily log-returns.
The so-called realized variance (or better, 2nd moment) is then calculated
using the estimator given by
1
n
(
n∑
i=1
(log(Si) − log(Si−1))2
)
.
A contract with as payoff
V S = N ×[
1
n
(
n∑
i=1
(log(Si) − log(Si−1))2
)
−K
]
,
where N denotes the notational amount, is called a variance swap. The value of
K is typically chosen such that the contract has a zero value when it is initiated
(just like in the case of futures and forwards). Basically this contract swaps
fixed (annualized) second moment, K, (variance) by the realized second moment
(variance) and as such provides protection against unexpected or unfavorable
changes in second moment (variance) or the related volatility.
Next, we will show how in a general setting this contract can be hedged
using more basic contracts.
134
We start with the following (Taylor-like) expansion of the 2nd power of the
logarithmic function
(log(x))2
= 2(
x− 1 − log(x) + O((x − 1)3))
.
Substituting x by Si/Si−1 leads to
(log(Si/Si−1))2
= 2
(
∆SiSi−1
− log(Si/Si−1) + O((∆Si/Si−1)3)
)
,
where ∆Si = Si − Si−1.
Summing over i gives the following decomposition:
n∑
i=1
(log(Si/Si−1))2
(6.1)
= 2n∑
i=1
(
∆SiSi−1
− log(Si/Si−1) + O((∆Si/Si−1)3)
)
= −2(log(ST ) − log(S0)) + 2
n∑
i=1
∆SiSi−1
+ O(
n∑
i=1
(∆Si/Si−1)3) (6.2)
due to telescoping. Thus up to 3rd-order terms the sum of the squared log-
returns decomposes into the payout from a log-contract (−2(log(ST )− log(S0)))
and a dynamic strategy (2∑ni=1
∆Si
Si−1).
The log-contract can be hedge by a dynamic trading strategy in combination
with a static position in bonds, European vanilla call and put options maturing
at time T. More precisely, first note that for any L > 0
log(ST ) − log(S0) =1
L(ST − S0) − u(ST ) + u(S0), (6.3)
for
u(x) =
(
x− L
L− log(x) + log(L)
)
.
135
Moreover with the technique explained in Section 6.1, one can show that
u(ST ) =
∫ L
0
1
K2(K − ST )+dK +
∫ +∞
L
1
K2(ST −K)+dK. (6.4)
Since ST − S0 =∑n
i=1 ∆Si, substituting (6.4) in (6.3) and (6.2) implies
n∑
i=1
(log(Si/Si−1))2 ≈
n∑
i=1
(
1
Si−1− 2
L
)
∆Si − 2u(S0)
+
∫ L
0
2
K2(K − ST )+dK +
∫ +∞
L
2
K2(ST −K)+dK
136