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Electronic copy available at: http://ssrn.com/abstract=1428555
July 2009 1
st revision: Sept 2009
1,2,3,4, The views expressed here are those of the authors, and do not necessarily represent those of their employers. 1Bloomberg, 2Credit-Suisse, 3Citi, 4Natixis *corresponding author contact email: [email protected]
General Auto-Regressive Asset Model
Jiaxin Wang1, Andrea Petrelli2, Ram Balachandran3, Olivia Siu4, Jun Zhang2, Rupak Chatterjee3, & Vivek Kapoor3,*
Abstract. Equity returns are addressed by a new General Auto-Regressive Asset Model
(GARAM). In this model, two stochastic processes are employed to represent the return
magnitude and return sign. Empirical auto-covariance and cross-covariance functions of the
return magnitude and return sign are key model inputs, and result in a realistic structure of the
clustering of volatility, dynamic asymmetry (leverage-effect), and the associated fat-tails. The
term-dependence of the asset return density, including the slow decay of kurtosis and the
buildup and slow decay of skewness are encompassed by GARAM. The resulting framework
for unconditional and conditional Monte-Carlo simulation of asset returns is illustrated.
Keywords: asymmetry, skewness, leverage-effect, kurtosis, filtering, conditional simulation, financial time-series JEL Classification: G11: Portfolio Choice; Investment Decisions; D81: Criteria for Decision-Making under Risk & Uncertainty
1. Introduction………………………………...... 2
• Motivation……………………………………….. 2
• Key Features of GARAM………………………. 2
• Overview of Financial Time Series Models…... 3 - Brownian Motion………………………………………... 3 - Jump-Diffusion…………………………………………... 4 - Heston…………………………………………………… 4 - Variance-Gamma………………………………………... 5 - GARCH…………………………………………………. 5 - Multi-Fractal Cascades………………………………….. 5
• Organization…………………………………….. 6
2. Empirical Features of an Equity Index..…. 7
• 2008……………………………………………….. 7
• Heteroskedasticity………………………………. 7
• Term-Structure of Return Skewness & Kurtosis…………………………….10
• Temporal Correlation of Squared Return and Return Sign………………………... 12
- Leverage Effect…………………………………………... 14
3. General Auto-Regressive Asset Model…. 15 • Specification…………………………………….. 15
• Monte-Carlo Simulation……………………….. 19 - Unconditional Simulation………………………………. 19 - Term Structure of GARAM Return Distribution……… 21 - Conditional Simulation…………………………………. 22
• Evolution of GARAM………………………….. 24
4. Discussion…………………………………… 25
• Risk Taking Cultures & Asset Return Descriptions…………………….. 25
• Volatility Trading………………………………. 25
• Future Work…………………………………….. 26
Appendix A GARAM Parameter Estimation……. 27 Appendix B Stationary Stochastic Processes: Simulation & Filtering………………. 33
References……………………………………... 39 Acknowledgements………………………….. 40
Electronic copy available at: http://ssrn.com/abstract=1428555
General Auto-Regressive Asset Model
2
1. Introduction
Motivation Equity investment strategies and derivative products continue to expand in size and breadth (e.g.,
index Variance swaps, options on VIX, systematic managed-futures, etc.) and challenge us to
understand the probabilistic structure of equity returns. The temporal dynamics of equity returns
are central to start analyzing the second generation products rationally, and analyze the first
generation products with aplomb. In pursuit of that we develop a stochastic model - for an equity
asset-type - that attempts to be suitably realistic to understand the risk-return of the panoply of
equity derivative products and investment strategies.
Investment strategies (e.g., managed futures, “CTAs”) obviously involve wagering bets on the asset
return distribution with all its real-world richness. Also, while trading derivatives, barring few
trivial situations (e.g., buy & sell identical contracts or sell a put & buy a call and short the
underlying – put-call parity) accounting for hedge slippage in the real-world and developing a
hedging strategy in search of profitability requires a probabilistic description of real-world returns.
Indeed, such real-world probabilistic descriptions of the underlying do not pose any special
problem for the modern methodology for analyzing derivatives (Optimal Hedge Monte-Carlo
(OHMC), see Potters & Bouchaud [2001] & Bouchaud & Potters [2003]), as it is independent of the
dynamics of the underlying assets and informs the user about residual risks inherent in attempted
replication. The flexibility of such a modern derivative analysis approach encourages the
development of realistic models of the underlying asset evolution that can combine empirically
observed features as well as beliefs about the asset1.
The Optimal Hedge Monte-Carlo (OHMC) method is not only applicable to vanilla equity options -
it has been applied by Kapoor and co-workers to a wide range of derivative contracts including
structured products with equity underlying, including Cliquets & Multi-Asset Options (Kapoor et al
[2003], Petrelli et al [2008] & [2009]). OHMC has also been applied to derivatives with credit
underlyings, such as CDS swaptions, & CDOs (Petrelli et al [2006], Zhang et al [2007]). In all
these aforementioned works, hedge slippage en-route to attempted replication of derivative
contracts was assessed, in addition to the expected hedging costs, and real-world descriptions of the
underlying with fat-tails and jumps to default were considered. There is simply no need for
derivative analysis to limit the development of realistic stochastic models of the underlying
markets.
Key Features of General Auto-Regressive Model (GARAM) The new model is named General Auto-Regressive Asset Model (GARAM). This is because the
building blocks for the model are a pair of classical auto-regressive stationary stochastic processes.
One stochastic process directly controls the return magnitude (via the squared return), and the other
stochastic process directly controls the return sign. The auto-covariance and cross-covariance of
these processes is empirically specified. The non-Gaussian features of the asset return and its
variation over different time-scales is controlled by the shapes of these general covariance
functions. We specify the marginal density of the return magnitude over the base-time scale (i.e.,
1 In contrast, the risk-neutral approach is based on the contrivance of a perfect hedge that is based on a narrow and
unrealistic description of the underlying. Such risk-blind modeling is associated with lack of effective risk management
and the concomitant widespread heist of risk-capital by derivative trading.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
3
observation time-scale, which can vary), and the model controls the term structure of the
distribution of asset returns via covariance functions inferred from observations at the base time-
scale.
The slowly decaying auto-covariance of return magnitude results in GARAM providing a multi-
scale description of asset returns. The negative cross-covariance between asset return sign indicator
and future return magnitude captures they dynamic asymmetry and the long-term persistence of
asset return skewness (i.e., the leverage effect). GARAM results in a stochastic description of
return with a temporal dependence structure that is empirically driven and is realistically slow in
approaching Gaussianity under aggregation over time.
Overview of Financial Time Series Models
A modeling framework that can produce a realistic probabilistic description of asset returns over a
large range of time-scales is needed to analyze option trading and investment strategies and
associated risk-management. With that backdrop, our reasons for developing the GARAM
approach are introduced while providing an overview of other modeling approaches.
Brownian Motion If the asset underlying a derivative is driven by such a process, in the absence of transaction costs, a
theoretical perfect replication strategy can be effected via continuous dynamic hedging. It is
amazing that this model is used in practical finance – despite incontrovertible evidence of skewness
and kurtosis in returns for even the most benign asset, and the degree of hedge slippage commonly
experienced by any market agent attempting to replicate by dynamic delta hedging.
The commonly measured skewness and kurtosis of asset returns is incompatible with the rapid IID
(identically and independently distributed) motion that provided the physical motivation for
Brownian motion. The widely experienced hedge slippage is a testimonial that the scale-disparity
implicit in Einstein’s useful physical theory of Brownian motion (Einstein [1905]) is not applicable
to the time-scales relevant to a trader who has sold a put and is attempting to replicate the payoff.
While the interesting mathematics of Brownian motion (Bachelier [1900]) may be sufficient to build
confidence on the atomic/molecular nature of matter (Einstein [1905]), the continued disregard of
empirical features incompatible with Brownian motion, by the mathematical finance community, is
a mistake of historical proportions2.
The continued usage of such a model brings into question the ingenuity and integrity of the
quantitative finance profession. The continued use of Brownian motion driven processes and the
associated fictitious perfect hedge also reflects the chokehold of interested parties on valuation
model development. These interested parties have gained by the ensuing unrealistic model that
hides the minimum risk-capital associated with any attempted option replication scheme, whether
they are vanilla options or exotics. The motivation for these tendentious and myopically self-
2 Financial contracts span time scales pertinent to the human participants, and the news transmission and human
reaction of greed and fear that feeds into the news digestion also occur over that time-scale. There is little merit to
imagining some external sterile higher frequency process bombarding the market – for human reaction is an integral
part of the market. The Brownian motion analogy to financial markets has no qualitative foundation. For a discussion
and illustration of the importance of psychology, sociology, and anthropology in economics, see Akerlof and Shiller
[2009] & Akerlof [1984].
General Auto-Regressive Asset Model
4
interested parties arises from the prevailing practice, where in the absence of any admission of
hedge slippage by the valuation model, any deviation from model price becomes arbitrage and
therefore recognizable day-1 P&L. Such model driven day-1 P&L is prevalent in customized
derivatives where there is no visible two way market. A valuation model that purports to address
“hedging costs”, but in reality ignores residual risk by invoking Brownian motion driven reference
assets is a recipe for repeat mischief and poor risk management.
Jump-Diffusion To deflect the criticisms of Brownian motion type process, jumps were added to the return process
(Merton [1976]). While that adds some elements of realism insofar as return can exhibit kurtosis
and skewness, the jump-diffusion description does not capture the temporal dynamics of returns that
suggest dependence between return sign and return volatility. Also, real assets do not diffuse in
between jumps! Rather, jumpiness is weaved into the real return process – and therefore it is a
fool’s errand to try to separate jumps from diffusion in any effort to empirically represent jump-
diffusion models.
The cardinal flaw in the popular usage of such a “jump-diffusion” model within the “risk-neutral”
regime is the ignoring of hedging errors associated with jumps. As a result, instead of helping build
a healthy risk-tolerance and awareness of jumps, the jump-diffusion model became a part of the
propaganda machine of risk-neutralistan – repeatedly falsely asserting unique price despite the
patent impossibility of hedging jumps!3 For interested parties, the jump-diffusion model provides
another fitting parameter and creates the ruse that somehow risks are better accounted for – while
that is simply not the case unless hedge slippage is explicitly quantified – but then the whole risk-
neutral “model” falls apart and the operational paradigm for model driven day-1 P&L recognition
on exotic options is threatened. Therefore the jump-diffusion model survives and quants continue
to please their masters by finding evermore rapid ways of finding risk-neutral expectations of
option payoffs to facilitate day-1 P&L on exotic options using a model that is silent about hedge
slippage and associated risk-capital needs.
Heston This is viewed as an advanced model in “risk-neutral” circles. The main reason for its usage is the
availability of analytical results in risk neutral option pricing (Heston [1993]). Risk neutral quants
simply fit the model to observed vanilla option prices as opposed to empirical return observations.
In keeping with the risk-neutral dictates, these published works are completely silent about hedge-
performance. These fitted parameters are then typically used to recognize day-1 P&L on exotics,
without any reference to irreducible hedging errors.
While the Heston model may be fit to empirical return distributions over a specific time-scale, it is
not possible to independently specify the slow decay of auto-covariance of return magnitude and
capture the temporal aggregation of the marginal return statistics. Furthermore, in this model the
past does not influence the future. So the Heston model ignores the empirical display of the
leverage effect - where the sign of the return of the past influences the realized volatility of the
3 A blatant assertion of diversification of jump risk is often made to further the mindless use of risk-neutral expectations
as the only way to value derivatives. Certainly we find significant asymmetry and kurtosis in equity index returns that
did not diversify away, despite there being hundreds of underlyings. It is incredulous that that disingenuous argument
has been with us for a few decades now and the risk-neutral jump-diffusion chicanery is still with us!
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
5
future. Whereas Heston is more complex than Brownian motion, it does not provide a satisfactory
basis to describe real asset returns over multiple-time-scales.
Variance-Gamma The Variance-Gamma process (Madan & Senata [1990]) is motivated by the non-uniformity of
arrival of news. Like the Heston model, this process lends itself to representing some realistic
features about return distribution at a specific observation time-scale. The ensuing fat-tails are a
welcome improvement over Brownian motion, and the non-uniformity of news arrival has a
seemingly intuitive ring. The specification of this model is, however, not based on directly
empirically observable temporal dependencies.
Like Heston, the popular usage of the Variance-Gamma model seems to be linked to the availability
of semi-analytical results for risk neutral option pricing. Like Heston, there is little in the literature
by way of documenting option hedge performance. So our criticisms of the risk-neutral application
of jump-diffusion models are equally applicable to the “risk-neutral” applications of the Variance-
Gamma process. What is the point of adding jumps and then continuing to pretend that there is a
unique derivative price associated with a perfect replicating strategy?
GARCH This model addresses basic aspects of volatility persistence and it is one of the oldest model
(Bollersev [1986]; Engle [1994]) that is a candidate for examining hedging slippage experienced by
the option trader-hedger. Most incarnations of GARCH specify the dependence of future volatility
on quadratic returns in the past. Those dependencies can be generalized further. Like the Heston
model, the GARCH model involves an evolution equation for the squared volatility process.
Compared to the approaches overviewed so far, GARCH provides a more credible framework –
insofar as it is somewhat empirically based, and generalizations can incrementally address realistic
features about asset returns. GARAM can be considered a generalization of the GARCH approach,
with the key difference that it directly addresses the squared return at some base-time-scale and uses
directly inferable correlations to specify the return process and its temporal aggregation. In
contrast, the volatility evolution specified in GARCH is ambiguous about the time scale over which
it is inferred. Instead of initial volatility (over what time scale?) being an input into GARAM, the
conditioning return vector is the input.
Multi-Fractal Cascades This class of models address basic empirical aspects of volatility persistence and asymmetry of
return distributions. The specification of these models invokes analogies from turbulence and the
mathematics of Fractals that has been applied to multiple physical and sociological phenomena.
Borland et al [2009] provide an overview of this category of models. Among the other key
contributions are Muzzy et al [2000] and Pochart & Bouchaud [2002].
Certainly, turbulence provides a more realistic paradigm for describing markets than Brownian
motion theory applied to a dilute gas. This is because the spatial-temporal scales of fluctuations in
turbulence are large enough for them to be vividly visible and sometimes measurable (to varying
degree) in the laboratory and in nature (e.g., a smoke plume in the natural environment, which is
very poorly described by merely a Brownian motion too). Perhaps, turbulence exhibits complexity
that begins to be comparable to financial markets. In contrast, a scale disparity is often successfully
General Auto-Regressive Asset Model
6
imposed in applying Brownian motion to understand basic physical continuum phenomena without
the richness and multi-scale complexity characteristic to turbulence.
The notion of auto-correlation and cross-correlation between quantities at different points in time
(i.e., time-lags) was originally applied in turbulence by G. I. Taylor [1921], and continues to be
applied to large-scale transport of mass and momentum in engineered and geophysical settings. The
GARAM specification was motivated by direct empirical observations of correlation functions of
return magnitude and sign. In identifying multiple time-scales of fluctuations of the return
magnitude, GARAM results in a phenomenological description of asset returns that have a greater
resemblance to turbulence than to kinetic theory of gases and Brownian motions! The direct
motivations for some of the multi-fractal cascade based models are systematic trading strategies and
option hedge performance – another common feature with the development of GARAM.
Organization
After communicating our motivation for developing GARAM and providing a general overview of
other financial time-series models in Section 1, we communicate empirical features of an equity
index in Section 2. These empirical features are addressed in the GARAM specification in Section
3, which also shows the results of unconditional and conditional simulation of returns. The more
technical aspects of parameter estimation are in Appendix-A. The methodology for simulation and
conditioning (filtering) return time series via GARAM are catalogued in Appendix-B. A
discussion of the implications of GARAM are provided in Section 4.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
7
2. Empirical Features of an Equity Index
SPX (Standard & Poor’s 500 Index) is a capitalization weighted index of 500 stocks, that is
designed to measure performance of the broad US economy representing all major industries. We
present empirical features of SPX and illustrate the applicability of GARAM to SPX in this work.
2008 SPX dropped 40% in 2008, with a couple of days of approximately 9% drops in late fall. There
were also upswings of more than 10% in that period (Figures 1a, b).
The return and volatility charts (Figure 1b & 1c) show a transition from a low volatility regime to a
high volatility regime. In the low volatility regime the return magnitude is seldom more than a few
percentage points - up or down. In the high volatility regime the return magnitude approached
double digits in 2008. The different volatility regimes appear persistent. In 2008, prior to
September, realized volatility, while fluctuating, was mostly well below 30%. For the later part of
the year, realized volatility was well above 40%, hitting as high as 80%.
Now imagine selling a forward starting put on S&P 500 in January 2008, with the option start date
Sept 1 2008 and maturity on Dec 31 2008? What should be the risk-capital of such a trade if you
sell the put contract and attempt to replicate by trading the underlying index? What stochastic
model of the SPX index would you employ to represent the risk-return of that attempted
replication?
Heteroskedasticity
The transition from modest to a persistent high volatility in 2008, while news-worthy, is not
qualitatively unprecedented. While the index returns themselves do not seem to exhibit significant
temporal correlation, the clustering of high volatility periods and low volatility periods is visible to
even the naked eye in Figures 2a, b, c which cover the period 1950 onwards.
The basic notion of heteroskedasticity is that one value of the return standard deviation, say inferred
from all the available observations, is not capable of describing the statistics of outcomes. In the
high volatility regime the return outcomes have a higher magnitude than consistent with the long-
term volatility. In the low volatility regime, the return magnitude outcome is smaller than that
consistent with the long-term volatility. So the notion of volatility of volatility, as well as the
temporal correlation between volatility over different lags jump out of Figures 2a, b, & c. Then
going one step further, the notion of correlation between return sign and volatility at different time
lags arises. Finally, in specifying a stochastic model for index returns, why drive the
phenomenological description by a volatility defined over an arbitrary time-scale – why not address
the fabric of the squared return stuff directly?
General Auto-Regressive Asset Model
8
Figure 1a. SPX level in 2008.
Figure 1b. SPX daily return in 2008.
Figure 1c. SPX daily return volatility in 2008.
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Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
9
Table 1. S&P500 daily return statistics (January 3, 1950 – June 2, 2009)
Figure 2a. SPX level (Jan 3, 1950 - June 2, 2009)
Figure 2b. SPX daily return (Jan 3, 1950 - June 2, 2009)
statistic return squared-return log-squared-return sign-indicator
mean 0.000272389 0.0000945 -11.091 0.0623
std. dev. 0.00971865 0.000534 2.302 0.9981
skewness -1.09 67.02 -0.84 -0.12
kurtosis 33.07 6283.78 4.24 1.02
pentosis -498.78 611817 -10.011 -0.2518
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General Auto-Regressive Asset Model
10
Figure 2c. SPX daily return volatility (Jan 3, 1950 - June 2, 2009)
Term-Structure of Return Skewness & Kurtosis The return over different time scales T (i.e., ( ) ( ) ( )[ ]TtststrT −≡ /ln ) exhibits varying departures
from Gaussianity. The term structure (i.e., T dependence) of kurtosis and skewness of returns are
shown in Figure 3 and Figure 4.
The slow decay of the kurtosis and skewness is readily visible in Figure 3 & 4. The kurtosis is
decaying almost monotonically, barring a local peak over 42 days. The negative skewness seems to
initially build-up (it also shows a bump down over 42 days) and then decay even slower than the
kurtosis. Neither moment achieves a close proximity with Gaussianity (skewness = 0; kurtosis = 3)
in even a two year time-scale.
The ‘base-time-scale” over which observation are made is taken as 1 day for this paper – although
that can be changed and GARAM is not limited to any specific observation time-scale – rather it
applies to time scales larger than the base time-scale. For example, a dataset of returns can in fact
exhibit positive skewness of returns at the base time-scale, and the negative skewness builds over
time and decays slowly like that shown for SPX. This points to the important consideration of the
temporal dynamics of asymmetry. Also, over smaller time-scales assets typically exhibit greater
kurtosis. This feature further emphasizes the lack of any rational foundation for the beliefs that
continuous time hedging results in perfect replication as per the dictates of the Brownian motion
driven risk-neutral derivative pricing “theory” – for over smaller time intervals, Brownian motion
is an even more inapplicable model!
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Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
11
Figure 3. SPX return kurtosis term-structure
(Jan 3, 1950 - June 2, 2009)
Figure 4. SPX return skewness term structure
(Jan 3, 1950 - June 2, 2009)
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General Auto-Regressive Asset Model
12
Stochastic modeling of equity returns to guide hedging and trading strategies is more demanding
than simply fitting the marginal return distribution at a base observation time scale. A choice of a
return distribution with positive or little skewness over say the base time-scale will not ever produce
significant negative skewness over a longer time-scale unless the temporal dependencies and
dynamics force it to. What better way of specifying these dynamics than an approach driven from
observable covariances of key return metrics? That is the approach taken in GARAM. The
covariances driving GARAM are shown next.
Temporal Correlation of Squared Return and Return Sign The autocorrelation of the return r (measured over the base-time-scale) with itself is denoted by
( )τρrr . We see that while there are some potentially interesting levels over a couple of days
(Figure 5), it does not show any long term memory and simply noisily oscillates around zero
rapidly. In contrast, the autocorrelation of the squared return r2 with itself, denoted by ( )τρ 22
rr,
decays slowly, and shows significantly higher levels, compared to ( )τρrr , even over 100 days
(Figure 5). The cross-correlation between the return sign indicator I (with I = 1 if r > 0 and I = -1
if r ≤ 0) and the squared return at a time τ in the future is denoted by ( )τρ 2Ir
( ) ( ) ( )2
2
)])([( 22
rIIr
rtrItIE
σσ
ττρ
−+−≡
This cross-correlation exhibits higher levels than the return auto-correlation. Specifically, there are
negative values of ( )τρ 2Ir at positive lags τ, that remain persistently negative and only decay slowly
with lag τ.
In making any central limit theorem type argument in support of Normality of return distributions,
one must contend with the relatively long ranging correlations exhibited by the squared return
(Figure 5) and the cross-correlation between the return indicator and the squared return. The term-
structure of the return kurtosis and skewness (Figures 3 & 4) reflect these correlations.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
13
Figure 5. Autocorrelation of daily SPX return, ( )τρrr and squared return, ( )τρ 22rr.
(January 3, 1950 – June 2, 2009)
Figure 6. Autocorrelation of daily SPX squared return, ( )τρ 22rr, & return sign indicator, ( )τρII ,
and cross-correlation between return sign indicator and squared return, ( )τρ 2Ir .
(January 3, 1950 – June 2, 2009)
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
-252 -210 -168 -126 -84 -42 0 42 84 126 168 210 252
au
to-c
orr
ela
tio
n
lag (days)
return
squared return
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
-252 -210 -168 -126 -84 -42 0 42 84 126 168 210 252
au
to-c
orr
ela
tio
n, c
ross
-co
rre
lati
on
lag (days)
squared return
indicator
indicator-squared return
General Auto-Regressive Asset Model
14
Leverage-Effect
The decay of skewness (Figure 4) is even slower than the excess kurtosis. This is associated with
the negative correlation between the return sign and the future return magnitude, ( )τρ 2Ir, τ ≥ 0, i.e.,
the leverage effect (Bouchaud, Matacz, and Potters [2001]). The original discussion of the leverage
effect is based on negative correlation between return and future squared return ( )τρ 2rr, τ ≥ 0. That
observation is qualitatively similar to the negative correlation between the sign of return and the
future squared return (Figure 6). From the perspective of model specification, casting the leverage
effect as the correlation between the indicator and squared return, ( )τρ 2Ir, enables specifying two
separate stochastic processes for the squared return and indicator. These separate processes are
linked via their cross-correlation, but they are separate insofar as the marginal specification of the
squared return process and indicator process can be made independently. In contrast, it is not
possible to make a marginal specification of the squared return independently of the marginal
specification for return.
In GARAM, a marginal specification of the squared return is made, and a marginal specification of
the return indicator is made. The empirical cross-covariance between these two processes results in
a complete specification. The marginal description of return is a model output that is controlled by
(1) the marginal description of the squared return (return magnitude); (2) the marginal description
of return sign indicator; and (3) the cross-correlation between the squared return and return sign
indicator.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
15
3. General Auto-Regressive Asset Model
The General Auto-Regressive Asset Model (GARAM) approach to equity modeling seeks to
address two main features about the returns:
(1) The temporal clustering of squared return & associated return kurtosis
(2) The temporal structure of the skewness & asymmetry of returns
These empirical features are central to understanding derivative trading risk-return and to designing
systematic investment approaches. The empirically observed long-term temporal persistence of
squared asset returns (or absolute value of asset returns) is the key empirical starting point for this
model. GARAM specifies a methodology that reproduces the term structure of persistence of
squared returns, and also a process for generating returns on a specified time-grid.
Motivated by the observed temporal persistence of the squared return (Figures 2 & 5), GARAM
directly models the squared return as a function of a classical auto-correlated second order
stationary stochastic process. The non-Normality of the marginal distribution of squared returns is
captured by this function. The temporal characteristics of the squared return are captured by the
temporal auto-covariance function. To model return, the return sign indicator is needed, in addition
to the squared return which quantifies the return magnitude. The temporal correlation of the return
indicator, and its co-variation with the return magnitude is modeled by another stationary stochastic
process that determines the return sign via a threshold.
Specification Based on the prominence of volatility and its temporal dynamics we start by modeling the squared
return over same base time scale. We start with the normalized logarithm of the squared return,
with the hope that log-Normality may be a not-too-bad starting point to develop a stochastic model
( )22/ln Rrx = (1)
The data exhibits modest deviations from log-normality for the squared return (Figure 7a). We
attempt to hammer out the visible deviations from log-normality by multiplying x by a
monotonically increasing function of x
( )[ ]( )[ ] 0;2/1 132
1
1 ≥+++= −ppxpTanpxy π
(2)
The functional form was motivated by a visual inspection of the deviations of x from Gaussianity –
where the left tail was fatter than Normal and the right tail a little thinner (Figure 7a). We choose
pi ( )31 ≤≤ i so that it is reasonable for y to be modeled as a classical stationary stochastic process
characterized until two moments as jointly Gaussian. R is a normalization parameter chosen to make
y to have a zero mean value. These parameters (p1, p2, p3, R) are chosen to set the skewness of y to
zero, its kurtosis to 3, and pentosis to zero – the success of this procedure is visible in Figure 7b.
Appendix-A provides more details about parameter estimation for GARAM.
General Auto-Regressive Asset Model
16
To be sure, the parameterization (1) and (2) are not the only one possible. In fact, certain aspects of
return are not addressed at all in (1) and (2) – for instance the probability of zero returns. We
overlook that in favor of the familiarity of the nearly log-normal behavior of squared returns and the
minimal effect of discarding returns that are identically zero.
Based on (1) and (2) the normalized squared return can be expressed as a function of y
( ) ( )yFRr =2
/ (3)
The function F is found from (1) and (2) and is guaranteed to exist as y is monotone in x. Therefore
from a time series of return r we can infer the corresponding values of y. From the y time-series we
can assess its auto-correlation:
( ) ( )( ) ( )( )[ ] 2/ yyy ytyytyE σττρ −+−≡
The stationary zero mean stochastic process y, and likewise the squared return or return magnitude,
is now specified, by relationships (1) and (2) that relate y to r2.
To aid analyzing investment strategies and basic options , we want to be able to simulate return,
which requires specifying its sign in addition to its magnitude (which is provided by the direct
model of squared returns specified above). For that we pair y with the stationary stochastic process
z and specify an indicator function based on the value of z, to specify the sign of the return:
( )
≥−
<+=
I
I
zz
zztI
if 1
if 1 (4)
Further specification of z is provided by its auto-correlation
( ) ( )( ) ( )( )[ ] 2/ zzz ztzztzE σττρ −+−≡
The complete specification of y-z jointly stationary jointly Gaussian stochastic process also requires
the cross-correlation function
( ) ( )( ) ( )( )[ ] ( )yzzy ytyztzE σσττρ /−+−≡ .
The inference of ( )τρ zzand ( )τρ zy are shown in Figure 8. Note that negative values of ( )τρ 2
Ir
translate into positive values of ( )τρ zy due to the definition of the return sign indicator threshold in
(4). Further details of fitting parameters to empirical information is provided in Appendix-A. With
the specification of y and z, we have the GARAM specification for asset return
( ) ( )( ) ( )( )tyFRtzItr = (5)
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
17
(a)
(b)
Figure 7. Empirical probability density functions of x and y specified in (1) and (2) (centered &
normalized). The bin-size of 1/10th
the standard deviation (i.e., 0.1) were employed to assess the
probability density from data. The parameters relating the daily return r to x, and y are as follows:
p1 = 0.397652; p2 = 0.434011; p3 = 1.10068; R = 0.00461727. The other key parameters controlling
the marginal density of r are ( ) 0.04750,0.0954113,3.57133 === zyIy z ρσ
(January 3, 1950 – June 2, 2009)
0.0001
0.001
0.01
0.1
1
-6 -4 -2 0 2 4 6
pro
ba
bil
ity
de
nsi
ty
normalized x
Empirical
Normal
0.0001
0.001
0.01
0.1
1
-5 -4 -3 -2 -1 0 1 2 3 4 5
pro
ba
bil
ity
de
nsi
ty
normalized y
Empirical
Normal
General Auto-Regressive Asset Model
18
Figure 8. Correlation functions ( )τρ yy ( )τρ zz
( )τρ zy inferred from SPX daily return data.
(January 3, 1950 – June 2, 2009)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-25
2
-21
0
-16
8
-12
6
-84
-42 0
42
84
12
6
16
8
21
0
25
2
au
to-c
orr
ela
tio
n o
f y
lag (trading days)
-0.10
0.10.20.30.40.50.60.70.80.9
1
-25
2
-21
0
-16
8
-12
6
-84
-42 0
42
84
12
6
16
8
21
0
25
2
au
to-c
orr
ela
tio
n o
f z
lag (trading-days)
-0.05
-0.02
0.01
0.04
0.07
0.1
-84
-63
-42
-21 0
21
42
63
84
z-y
cro
ss-c
orr
ela
tio
n
lag (trading days)
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
19
Monte-Carlo Simulation
Unconditional Simulation The methodology of generating stationary stochastic processes with pre-specified second order
statistics is well established and is covered in Appendix-B. Using that methodology we directly
simulate the y and z processes that provide us with the return magnitude and sign. In doing so we
are able to capture the central tendency of returns, the dispersion of returns, the fat-tails and
asymmetry of returns. The autocorrelation of y embodies the term structure of the return kurtosis.
The cross-correlation of z and y is a measure of the term structure of return skewness. With these
correlation functions directly empirically prescribed, we capture the key features of the return
dynamics.
MC realization 1 MC realization 2
Figure 9a. Two Simulated 2 yr (504 trading days depicted on the horizontal axes) time series based
on GARAM fit to SPX
0.8
0.85
0.9
0.95
1
1.05
1.1
0 42 84 126 168 210 252 294 336 378 420 462 504
value
0.8
0.85
0.9
0.95
1
1.05
1.1
0 42 84 126 168 210 252 294 336 378 420 462 504
value
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0 42 84 126 168 210 252 294 336 378 420 462 504
return
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0 42 84 126 168 210 252 294 336 378 420 462 504
return
0
0.05
0.1
0.15
0.2
0.25
0 42 84 126 168 210 252 294 336 378 420 462 504
21-day realized vol
0
0.05
0.1
0.15
0.2
0.25
0.3
0 42 84 126 168 210 252 294 336 378 420 462 504
21-day realized vol
General Auto-Regressive Asset Model
20
MC realization 3 MC realization 4
Figure 9b. Two more simulated 2 yr (504 trading days depicted on the horizontal axes) time series
based on GARAM fit to SPX
The simulated return time-series in Figure 9 shows clustering of high and low volatility periods that
resembles empirical observations shown in Figure 1 & 2. Indeed, GARAM ensemble statistics are
fit to those inferred from empirical observations, as far as mean, standard deviation, and temporal
autocorrelation of squared return, return indicator, and cross covariance between squared return and
return sign indicator. We think that these characteristics span the necessary statistics for the model
to begin to be useful in understanding derivative trading risk-return, as well as designing related
investment strategies.
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
0 42 84 126 168 210 252 294 336 378 420 462 504
value
0.8
0.85
0.9
0.95
1
1.05
1.1
0 42 84 126 168 210 252 294 336 378 420 462 504
value
-0.06
-0.04
-0.02
0
0.02
0.04
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return
-0.03
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-0.01
0
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return
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 42 84 126 168 210 252 294 336 378 420 462 504
21-day realized vol
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 42 84 126 168 210 252 294 336 378 420 462 504
21-day realized vol
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
21
Term-Structure of GARAM Return Distribution
0.0000001
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
-6 -4 -2 0 2 4 6
normalized 1 day return
GARAM
Normal
Empirical
0.0000001
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
-6 -4 -2 0 2 4 6
normalized 10 day return
GARAM
Normal
Empirical
0.0000001
0.000001
0.00001
0.0001
0.001
0.01
0.1
1
-6 -4 -2 0 2 4 6
normalized 20 day return
GARAM
Normal
Empirical
Figure 10.
Return probability density of
SPX: Empirical, GARAM, &
Normal Distribution.
The Normal distribution is
clearly unacceptable! The
build-up of asymmetry in
GARAM is accentuated by the
dashed line, that would be
horizontal for a symmetric
return distribution.
This build-up of asymmetry is
driven by the covariance
between return sign indicator
and future return magnitiude, in
the GARAM model. This has
been called the leverage-effect.
The temporal evolution of
kurtosis in GARAM is driven
by the auto-covariance of
return magnitude (squared
return).
(January 3, 1950 - June 2, 2009)
General Auto-Regressive Asset Model
22
GARAM not only provides a satisfactory description of fat tails of daily returns, it also realistically
captures the scaling of the return distribution with time-scale, as shown in Figure 10. The
departures from Gaussianity are not subtle! As the kurtosis decreases with increasing time-scales,
the skewness is getting relatively more pronounced. GARAM captures these features because the
empirical covariances of return magnitude and return indicator are the underpinnings of this
stochastic model.
The Fractal-Cascade models have addressed these time-aggregation aspects of financial time-series
return and made connections of financial return time-series with concepts in the study of turbulence
and fractals (see Borland et al [2009] for a comprehensive review). The contribution of the
discipline of turbulence research, to modern analysis, includes early utilization of the notion of
autocorrelation and auto-regressive models to recognizing multi-scale fluctuations. GARAM,
primarily grounded in empirical observations of the covariance structure of return magnitude and
return sign indicator, also results in a sufficient framework for describing the term-structure of Non-
Gaussianity of real returns.
Conditional Simulation As the squared returns exhibit significant temporal correlation, their historically realized values
need to be accounted for in simulating plausible future values. Also, as the return sign indicator
shows significant correlation with future squared returns, its historically realized values are
pertinent conditioning variable for future returns. We account for the historical realized return time-
series via conditional simulation, which involves one step beyond the unconditional MC simulation.
The methodology of conditional simulation is described in detail in Appendix-B. The impact of
conditioning on the simulated returns is illustrated in Figures 11a & b.
The conditioning in GARAM is based on return observations. This conditioning can be thought of
as a way of reflecting recent conditions in the stochastic description of the future returns. The
conditioning information input in GARAM adds clarity relative to GARCH models – where a
somewhat arbitrary starting volatility is an input. The time-scale of the input volatility in a GARCH
based model of asset returns is ambiguous.
In GARAM there is no intrinsic limitation on the amount of information to be used as conditioning
information. Information from a long time ago will have less impact on future simulations than
relatively recent information – as determined by the covariance functions in the GARAM
specification.
In Figure 11a we show the case where the conditioning information is a one year long low volatility
regime return time-series. We see the gradual increase in volatility with time, as the y
autocorrelation decreases with time-lag. Conversely, in Figure 11b, we gave a case where the
conditioning return data is in a high volatility regime – we see the gradual decrease in volatility with
time.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
23
Figure 11a. Four 252 day conditional realizations of SPX simulated via GARAM with 252 days of
conditioning on low volatility regime returns
Figure 11b. Four 252 day conditional realizations of SPX simulated via GARAM with 252 days of
conditioning on high volatility regime returns
-6%
-4%
-2%
0%
2%
4%
6%
0 42 84 126 168 210 252 294 336 378 420 462 504
da
ily
re
turn
day
conditioning observations 1 2 3 4
-15%
-10%
-5%
0%
5%
10%
15%
0 42 84 126 168 210 252 294 336 378 420 462 504
da
ily
re
turn
day
conditioning-observations 1 2 3 4
General Auto-Regressive Asset Model
24
Evolution of GARAM
GARAM can be compared with GARCH models. Both these models result in asset returns that
display excess kurtosis, and the squared returns exhibits serial correlation, i.e., clustering of
volatility. GARAM enables a direct specification of the serial autocorrelation function of the
squared return that is more flexible and straightforward than the GARCH model. The standard
GARCH model has no skewness, whereas GARAM incorporates that based on the observed
covariance of the return sign and the squared return.
GARAM utilizes the well developed theory of stationary stochastic process (see Appendix-B).
Therefore, the simulation, forecasting, and conditioning methodologies that have been developed
around stationary stochastic processes can be brought to bear via GARAM.
We emphasize that GARAM represents a general approach to stochastic characterization of asset
return time-series that addresses realistic features of asset returns – namely fat-tails, asymmetry, and
clustering of volatility. The specific choices can be modified and built upon. For example our
rendition of F(y) by (1) and (2) is simply an evolution of a log-Normal model for the squared return
marginal density, and the indicator function is the simple-most possible. The temporal covariance
and auto-covariance are empirically based.
We were lead to creating GARAM by the need to have a realistic objective measure model for
equity index returns, and the difficulty in accommodating the empirical auto-covariance of the
return magnitude (i.e., squared return) by the traditional GARCH and Heston models. The version
of GARAM described here is the third version of the model.
item\model version 1 version 2 version 3 (this paper)
r2 marginal density log-normal log-normal empirical alteration of
log-normal
r2 auto-covariance empirical empirical empirical
r sign auto-covariance none; assumed iid empirical empirical
r sign & r2 covariance none empirical empirical
Table 2. Evolution of GARAM
Subsequent developments in GARAM could also directly address the trading calendar and the
corresponding frequencies of fluctuations observed in the auto-covariance functions. While this
paper shows an application of GARAM using data on a daily frequency, the framework is
particularly flexible in representing multiple scales of information, and therefore intraday (high
frequency) fluctuations can also be accommodated.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
25
4. Discussion
Risk-Taking Cultures & Asset Return Descriptions The design of investment strategies and option trading strategies can be aided by a realistic
stochastic model of the underlying. To be practically useful, these stochastic models should not be
driven by the dogma or convenience of any prior hypothesis or theory (e.g., efficient market
hypothesis; risk neutral derivative pricing theory). Rather, one of the main purposes of the
stochastic model could be a realistic sizing of the risk capital of a trading strategy. This objective
whittles down the options of modeling the underlying asset to those that are closely tied to
observable empirical reality. It is not sufficient to get the return distribution over a specific time-
scale right – rather the dynamics of aggregation have to be addressed so that the probabilistic
structure over multiple time-scales is addressed simultaneously. Generalized GARCH models,
multi-fractal cascades, and the GARAM approach, advanced here, appear to be promising choices.
One needs to be quite humble about any stochastic model’s ability to capture the whole fabric of
market dynamics, that reflects human greed and fear during normal and abnormal times. However,
the differences in models used by different groups do differentiate their risk-taking cultures.
Clinging to unrealistic asset descriptions in the name of some idealized theory or legacy trading
system is a signature of an organization that does not measure risks and enters into transactions
without deliberating risk-return tradeoffs. Indeed, the market events of 2007-2008 have
differentiated market participants – Banks and Hedge funds - in terms of risk taking cultures.
While models are being almost uniformly vilified in the popular press4, our anecdotal findings are
that organizations with trading & risk management personnel that are empowered to address the
mechanics of trades and the associated risk capital driven from realistic descriptions of the
underlying, have sailed through these events with less harm than others. Organizations where risk
taking is controlled by interested parties with conveniently held beliefs like perfect replication in a
Brownian motion driven world have done poorly. The collusion between interested parties and
uninformed models and modelers has been punished. The low standards in quantitative finance
models with respect to realism and the high level of dogmatism (feigned or genuine) are subject to a
painful Darwinism type elimination - that has been quite expensive to the larger society.
Volatility Trading The market demand-supply dynamics control prices of options and a market agent can decide to
participate at prices that suit their utility. While models do not “price” a traded option, models can
help specify a hedging strategy and quantify the range of P&L outcomes while following that
strategy. The reluctance of quantitative modelers to acknowledge that human reaction controlled
risk premiums have a role to play in option pricing is understandable – after all these risk premiums
are the stuff of psychology of greed and fear that the quantitative modeler finds overwhelming.
However, that is no excuse to cling to the falsehood that the value of an option is the unique price of
replicating it – because there is no perfect replication scheme!
4 The vilification of models is mostly justified, considering they cling to Brownian Motion and the ensuing risk-neutral
valuation model based perfect replication mendacity. See Triana [2009] for a broad and eloquent critique of
mathematical finance, and Taleb [2007] for debunking the mythology of Normal distributions in Finance.
General Auto-Regressive Asset Model
26
Replacing the unique price of replicating an option with the expected cost of attempting to hedge it
is a far cry from the risk neutral theory – however that is implicit in much of the “research” on
options that fails to shed light on risk-return tradeoffs and connect them to market dynamics.
We think that a more fruitful line of analysis is directly interpreting option risk premium by
assessing the attempted replicating strategies’ P&L distribution. Then the option value can be seen
to set a return on risk capital or a Sharpe ratio (or any other return per unit risk metric). To pursue
this line of analysis of options requires a realistic model of the underlying that can be conditioned
on observations. We believe that GARAM, coupled with the modern derivative trading analysis
approach (i.e., OHMC) can serve this purpose of describing option trading risk-premium dynamics.
Future Work The real-world trading calendar controls many important frequencies of information (seasonality
effects) embedded in the empirical covariance functions of return magnitude (or squared) and sign.
Therefore a stochastic model for the asset that is explicitly aware of the trading calendar and the
import of certain dates and time-scales would be useful in leveraging the information contained in
the empirical covariance functions. Also, more sophisticated ways of coupling the discrete return
sign to return squared (i.e., return magnitude) could also be explored.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
27
Appendix-A GARAM Parameter Estimation
We consider the asset values at discrete time steps denoted by tk. The asset values over successive
time-steps define its return r(tk)
( ) ( ) ( )[ ]1/ln −≡ kkk tststr
(A1)
The empirical characteristics of the return process r(tk) are inferred from asset value data assuming
this relationship between successive asset values and the return. GARAM directly addresses the
magnitude of returns and employs a return sign indicator to complete the stochastic description of
the asset. The return sign also pertains to the discrete return over the interval defined by tk-1 and tk:
( )
( )( )
≤−
>+=
0 if 1
0 if 1
k
k
ktr
trtI
(A2)
For notational simplicity we drop the discrete time subscripts, with the tacit understanding of the
discrete description of the problem, and the definition of return and return sign indicator to be
applicable over successive instances of a time-grid.
GARAM is based on mapping the squared return ( )tr2
and return indicator ( )tI to stationary
stochastic processes y(t) and z(t) respectively. The squared return is related to y as follows
( ) ( )[ ]( )[ ] 0;2/1 ;/ln 132
1
1
22 ≥+++== −ppxpTanpxyRrx π
( ) ( )( )tyFRtr22 =⇒ (A3)
The function F(y) is numerically computed, and the y-x relationship is chosen to address the
empirical marginal density of the squared returns. The functional form of y-x relationship was
motivated from empirical observations of departures of the marginal density of the squared return
from log-Normality as depicted in the main section Figure 7.
The sign of the return is specified by the return sign indicator function of z
( )( )( )( )
≥−
<+=
I
I
ztz
ztztzI
if 1
if 1 (A4)
The return model in terms of the two stochastic processes y and z is
( ) ( )( ) ( )( )tyFRtzItr = (A5)
General Auto-Regressive Asset Model
28
We specify the characteristics of y and z and the other parameters to fit the empirical characteristics
we deem central to equity derivative trading & risk management. The model parameters are
( ) ( )( ) ( ) ( )( ) ( ) ( )( )τρτρτρzzypppRτtytzτtztzτtytyIzy +++≥ , , , , , , , , , ,0 , 321 σσ
(A6)
The two stationary, jointly Gaussian stochastic processes (y and z), employed as building blocks to
address these features, are completely characterized by the auto and cross covariance functions via
a multi-Gaussian joint distribution, elaborated below.
Marginal density of y(t) & z(t)
( ) ( )( )
( )( )
y
y
t
yty
tyfσπ
σ
2
2exp
2
2
−−
=y ( ) ( )( )
( )( )
z
z
t
ztz
tzfσπ
σ
2
2exp
2
2
−−
=z
Joint density of ( ) ( )τ+tyty and
( ) ( ) ( ) ( )( )( )( )
( )( ) ( ) ( )( ) ( )( ) ( )( )
( )τρπσ
σ
τ
σ
ττρ
στρττ
22
2
2
22
2
2
)
12
2
12
1exp
,
yyy
yy
yy
yyy
tt
ytyytyytyyty
tytyf−
−+
+−+−
−−
−
−
=++yy
Joint density of ( ) ( )τ+tztz and
( ) ( ) ( ) ( )( )( )( )
( )( ) ( ) ( )( ) ( )( ) ( )( )
( )τρπσ
σ
τ
σ
ττρ
στρττ
22
2
2
22
2
2
)
12
2
12
1exp
,
zzz
zz
zz
zzz
tt
ztzztzztzztz
tztzf−
−+
+−+−
−−
−
−
=++zz
Joint density of ( ) ( )τ+tytz and
( ) ( ) ( ) ( )( )( )( )
( )( ) ( ) ( )( ) ( )( ) ( )( )
( )τρσπσ
σ
τ
σσ
ττρ
στρττ
2
2
2
2
2
2
)
12
2
12
1exp
,
zyyz
yyz
zy
zzy
tt
ytyytyztzztz
tytzf−
−+
+−+−
−−
−
−
=++yz
With this specification we are able to express key empirical return unconditional return statistics as
two dimensional integrals. Some of those two-dimensional integrals can be further simplified into
one dimensional integrals, as shown next.
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
29
Unconditional Moments of Return
( ) ( )( ) ( )( )tyFRtzItr
n
nnn 2=
( )[ ] ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )
( )( )
( )tdytdztytzftzItyFRtrEty tz
tt
n
n
nn
∫ ∫∞
−∞=
∞
−∞=
= ,2yz
even n:
( )( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( )( )tyftdztytzftzI t
tz
tt
n
yyz =∫∞
−∞=
,
( )[ ] ( )( ) ( ) ( )( ) ( )( )
,...6,4,2 ;2 == ∫∞
−∞=
ntdytyftyFRtrEty
t
n
nn
y
(A7a)
odd n:
( )( ) ( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )( ) ( )( )
( ) ( ) ( ) ( )( ) ( )( )∫∫∫∞
=−∞=
∞
−∞=
−=
I
I
ztz
tt
z
tz
tt
tz
tt
ntdztytzftdztytzftdztytzftzI ,,, yzyzyz
( ) ( )( )
( )( )( ) ( )( )
2012
0
12
tyf
ytyzz
Erft
zy
y
zy
z
I
y
−
−−
−
+=ρ
σρ
σ( ) ( )( )
( )( )( ) ( )( )
2012
0
12
tyf
ytyzz
Erft
zy
y
zy
z
I
y
−
−−
−
−−ρ
σρ
σ
( )( ) ( )( )
( )( ) ( ) ( )( ) ( ) ∫−=
−
−−
−
=u
t
t
zy
y
zy
z
I
dteuErftyf
ytyzz
Erf0
2
22 where
012
0
πρ
σρ
σy
( )[ ] ( )( )
( )( ) ( )( )
( )( ) ( ) ( )( ) ( )( )
,....5,3,1 ;012
0
2
2 =
−
−−
−
= ∫∞
−∞=
ntdytyf
ytyzz
ErftyFRtrEty
t
zy
y
zy
z
I
n
nn
y
ρ
σρ
σ
(A7b)
General Auto-Regressive Asset Model
30
The centered return statistics that are associated with describing the shape of the marginal
distribution, can be calculated from the prior expressions, and are enumerated here:
Mean: [ ]rEr ≡
Variance: ( )[ ] [ ] ( )2222 rrErrEr −=−≡σ
Skewness: ( )[ ] [ ] [ ] ( )
3
323
3
323
rr
r
rrErrErrE
σση
+−=
−≡
Kurtosis: ( )[ ] [ ] [ ] [ ]( ) ( )
4
42234
4
4364
rr
r
rrrErrErErrE
σσκ
−+−=
−≡
Pentosis: ( )[ ] [ ] [ ] [ ]( ) [ ]( ) ( )
5
5322345
5
5410105
rr
r
rrrErrErErrErrE
σσζ
+−+−=
−≡
Expected Value of Return Sign Indicator
( )( )[ ] ( )( ) ( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
( ) ( )( ) ( )( )
( ) ( ) ( )
−=
−−−
−+=
−== ∫∫∫∞
=−∞=
∞
−∞=
z
I
z
I
z
I
ztz
t
z
tz
t
tz
t
zzErf
zzErf
zzErf
tdztzftdztzftdztzftzItzIE
I
I
σσσ 221
2
1
21
2
1
zzz
(A8)
Covariance of Return Sign Indicator and Squared Return
( )( ) ( )[ ] ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( )( )
∫ ∫∞
−∞=
∞
−∞=
+ +++=+ty tz
tt tdztytzftzItdytyFRtrtzIE ττττ τ ,22
yz
( )( ) ( ) ( ) ( ) ( )( ) ( )( )
( )( ) ( )( )
( )( ) ( ) ( )( )ττρ
σ
ττρ
στ ττ +
−
−+−
−
=+ +
∞
−∞=
+∫ tyf
ytyzz
ErftdztytzftzI t
zy
y
zy
z
I
tz
tt yyz212
,
( )( ) ( )[ ] ( )( )
( )( ) ( )( )
( )( ) ( ) ( )( ) ( )( )∫∞
−∞=+
+ ++
−
−+−
−
+=+τ
τ τττρ
σ
ττρ
σττ
ty
t
zy
y
zy
z
I
tdytyf
ytyzz
ErftyFRtrtzIE y2
22
12
(A9)
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
31
Auto-Covariance of Return Sign Indicator
( )( ) ( )( )[ ] ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )( )( )
∫ ∫∞
−∞=+
∞
−∞=
+ +++=+τ
τ ττττtz tz
tt tdztztzftzItdztzItzItzIE ,zz
( )( )
( ) ( ) ( )( )
( )( ) ( ) ( )( ) ( )( )∫∞
−∞=+
+ ++
−
−+−
−
+=τ
τ τττρ
σ
ττρ
στ
tz
t
zz
z
zz
z
I
tdztzf
ztzzz
ErftzI z212
( )( ) ( )( )[ ]
( ) ( ) ( )( )
( )( ) ( ) ( )( ) ( )( )
( ) ( ) ( )( )
( )( ) ( ) ( )( ) ( )( )∫
∫
∞
=+
+
−∞=+
+
++
−
−+−
−
−++
−
−+−
−
=+
I
I
ztz
t
zz
z
zz
z
I
z
tz
t
zz
z
zz
z
I
tdztzf
ztzzz
Erf
tdztzf
ztzzz
ErftzItzIE
τ
τ
τ
τ
τττρ
σ
ττρ
σ
τττρ
σ
ττρ
στ
z
z
2
2
12
12
(A10)
Auto-Covariance of Squared Return
( ) ( )( )tyFRtr
22 =
( ) ( )[ ] ( )( ) ( )( ) ( ) ( ) ( ) ( )( ) ( )
( )( )
( )τττττ
τ +++=+ ∫ ∫∞
−∞=
∞
−∞=+
+ tdytdytytyftyFtyFRtrtrEty ty
tt ,422
yy
This is the only two-dimensional integral that needs to be numerically assessed to effect a
calibration to observations. The general non-linearity of F, embraced to capture the empirical
marginal density of squared returns, leads to this numerical computation requirement.
General Auto-Regressive Asset Model
32
GARAM Calibration Steps
GARAM can be calibrated by fitting empirical moments and correlations to model parameters. The
fitting directly encompasses the marginal density of the squared returns and the correlation structure
of the squared return and the return sign indicator. Here are the steps followed:
1. Set .0,0 ,1 === yzzσ
2. Fit R, p1, p2, and p3 to ensure that the empirical inferences from (A3) result in the marginal
distribution of y being close to Normal with a zero mean. This is accomplished by setting
03 ==−== yyyy ζκη .
3. Infer yσ from data, using fitted R, p1, p2, and p3.
4. Fit Iz and ( )0zyρ to empirical inference of [ ]rE and ( )02Ir
ρ .
5. Fit yσ to rσ .
6. Repeat 4 & 5 untill convergence is achieved.
7. Fit ( )τρ zy to empirical inference of ( )τρ 2Ir.
8. Fit ( )τρ zz to empirical inference of ( )τρ II .
9. Fit ( )τρ yy to empirical inference of ( )τρ 22
rr .
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
33
Appendix-B Stationary Stochastic Processes: Simulation & Filtering We built the General Auto-Regressive Asset Model - that addresses asymmetries and fat-tails of
returns - around classical autoregressive stochastic processes, to be able to leverage the well-
developed theory and techniques for simulation and conditioning (or filtering) while accounting for
the temporal memories exhibited by the return process. By capturing temporal correlations between
different facets of the return process, GARAM aids developing a systematic trading-risk
management system and also facilitates interrogating option risk-premiums. This appendix
elaborates the details of classical autoregressive processes underlying GARAM.
The auto-correlation of return variance measures is central to derivative hedge performance and
systematic trading strategies. Such auto-correlation is empirically observed for return magnitude
(see main section) and can be connected to the fat-tails of the asset return distribution. The
utilization of an auto-correlation function of a stochastic process appears to have been first invoked
by G. I. Taylor in the study of turbulence and associated mass & momentum transport phenomenon
(Taylor [1921]). It was later further elaborated in signal processing applications (see Kolmogorov
[1941] & Weiner [1949]) and is now commonly used in a myriad of theory and applications.
The techniques described here are applicable to non-stationary stochastic processes too, although
we restrict ourselves to stationary stochastic processes. For stationary stochastic processes there are
spectral techniques to simulate unconditional realizations of the stochastic process that are more
efficient than the technique employed here. However, even with the restriction to stationary
stochastic processes, the processes conditioned on observations are in general non-stationary. We
address filtering and Monte-Carlo simulation conditioned on observations too.
B.1 One Stochastic Process
Simulation of Unconditional Realizations
Here we address the simulation of a stationary zero mean jointly Normal stochastic process ( )( 0th ,
)( 1th , . . . . . , )( 1−nth ) at specific points in time. The auto-covariance of the process hhCov is
specified:
( ) ( ) ( )[ ] ( )ijhhhjijihh ttththEttCov −≡≡ ρσ 2
,
The simulation of the process is accomplished by first generating identically and independently
distributed standard Normal variates ( )( 0tu , )( 1tu , . . . . . , )( 1−ntu ), and then transforming them to
effect the auto-covariance function of the target stochastic process, employing the square-root of the
auto-covariance matrix. The square root of the covariance matrix ( )jihhij ttCovCov ,≡ denoted by
ijB can be found by performing a Cholesky decomposition:
ij
T
kjik CovBB = (B1)
General Auto-Regressive Asset Model
34
The correlated stochastic process can then be simulated by
h(ti) = ijB u(tj) (B2)
A summation over repeated indices is implied in (B2) (the index j) and elsewhere in this paper.
Simulation of Conditional Mean: Filtering
Given observations )( 0th , )( 1th , . . . . . , )( 1−nth of a zero mean stochastic process, we consider the
problem of estimating the value of h(t). The estimator is denoted as )(ˆ th . This classic estimation
problem has different names in the different disciplines it is employed in (e.g., filtering, Kriging,
smoothing, etc.).
Let us look for a linear estimate )()(ˆ1
0
i
n
i
i thth ∑−
=
= λ that is the “best” in the sense the estimation
variance ( ) ( ) ])()(ˆ[2
2
ˆ ththEth
−=σ is as small as possible. The variance of the estimate is
( ) ( )
−
−=−= ∑∑
−
=
−
=
1
0
1
0
22ˆ )()()()(])()(ˆ[
n
j
jj
n
i
iihththththEththEt λλσ
(B3)
( )
−+= ∑ ∑∑
−
=
−
=
−
=
1
0
1
0
21
1
)()(2)()(n
i
n
i
ii
n
j
jiji thththththE λλλ
Assuming that h(t) is second order stationary with a covariance function at time lag τ denoted by
Covhh(τ), the variance of the estimate can be written as
( ) ( ) ( ) ( )∑ ∑∑−
=
−
=
−
=
−−+−=1
0
1
0
1
0
2
ˆ 20n
i
n
i
ihhihh
n
j
ijhhjihttCovCovttCovt λλλσ . (B4)
The jλ that result in the minimum variance follow
( ) )(1
0
ihh
n
j
ijhhj ttCovttCov −=−∑−
=
λ ; 10 −≤≤ ni (B5)
The variance of the estimate then is given by substituting (B5) into (B4):
( ) ( )∑−
=
−−=1
0
2ˆ 0)(
n
i
ihhihhhttCovCovt λσ (B6)
If h(ti) are jointly Normally distributed, then the best linear unbiased estimate is also the conditional
mean:
)](),.......,(),(|)([)(ˆ 110 −= nththththEth (B7)
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
35
Simulation of Conditional Realizations
Given observations )( 0th , )( 1th , . . . . . , )( 1−nth , here are the steps to describe a possible outcome
of the stochastic process hc(t) that takes on observed values at the points of observation.
1. Generate an unconditional realization hu(t) for all t’s spanning the times of observations obsΤ and
the times for which the conditional realization is to be simulated simΤ .
2. Generate the conditional mean )(ˆ thu, for all simt Τ∈ , that attains values of the unconditional
realization at observation points obsΤ , i.e., 10 ),()(ˆ −≤≤= nithth iuiu
3. Generate the conditional mean function )(ˆ th , for all simt Τ∈ , that attains observed values at
observation points obsΤ , i.e., 10 ),()(ˆ −≤≤= nithth ii
4. hc(t) = )(ˆ th + (hu(t)- )(ˆ thu)
By repeating the process for a different unconditional realization, hu, the sub-ensemble of the
process h that pass through observations can be created. These conditional realizations are equally
likely, and their mean is )(ˆ th and their variance around )(ˆ th is the estimating variance 2
hσ given in
(B6).
B.2 Two Correlated Stochastic Process (2-D)
Simulation of Unconditional Realizations
Here we address the simulation of two jointly stationary zero mean jointly Normal stochastic
process ( ( ){ }00 ),( tgth , ( ){ }11),( tgth , . . . . . , ( ){ }11),( −− nn tgth ) at specific points in time. The auto-
covariance and cross-covariance of the processes are specified:
( ) ( ) ( )[ ]kikihh ththEttCov ≡,
( ) ( ) ( )[ ]kikigg tgtgEttCov ≡,
( ) ( ) ( )[ ]kikigh thtgEttCov ≡,
( ) ( ) ( )[ ]kikihg tgthEttCov ≡,
To facilitate the simulation we store this information in a composite correlation matrix Covij which
is 2nx2n in size:
10 −≤≤ ni 10 −≤≤ nj
( )
jihhji ttCovCov ,, =
( )jihgnji ttCovCov ,, =+ (B8)
( )
jighjni ttCovCov ,, =+ ( )
jiggnjni ttCovCov ,, =++
General Auto-Regressive Asset Model
36
The simulation of the process is accomplished by first generating 2n identically and independently
distributed standard Normal variates ( )120 −≤≤ nju j and then transforming them to effect the
auto-covariance function of the target stochastic process.
The square root of the covariance matrix ikCov denoted by ijB can be found by performing a
Cholesky decomposition:
ij
T
kjik CovBB = (B9)
The correlated stochastic process can then be simulated:
10 −≤≤ ni ; ( ) jjii uBth ,= ; ( ) jjnii uBtg ,+= (B10)
Simulation of Conditional Mean: Filtering
We extend the filtering formulation to the two dimensional case. Given observations ( ){ }ii tgth ),(
( )10 −≤≤ ni of zero mean stochastic processes, we consider the problem of estimating the values of
( ){ }tgth ),( . The estimator is denoted as ( ){ }tgth ˆ),(ˆ . As before, let us look for linear estimates
( )[ ]∑−
=
+=1
0
)()(ˆn
i
i
hg
ii
hh
i tgthth λλ
( )[ ]∑−
=
+=1
0
)()(ˆn
i
i
gg
ii
gh
i tgthtg λλ (B11)
The variance of the estimates are ( ) ( ) ( )( ) ]ˆ[2
2
ˆ ththEth
−≡σ , and ( ) ( ) ( )( ) ]ˆ[22
ˆ tgtgEtg −≡σ . These are
written as
( ) ( )[ ] ( )
−+= ∑
−
=
21
0
2ˆ )( thtgthEt
n
i
i
hg
ii
hh
ihλλσ
( ) ( ) ( )
( ) ( )∑ ∑
∑∑∑∑∑∑−
=
−
=
−
=
−
=
−
=
−
=
−
=
−
=
−−
+++=
1
0
1
0
21
0
1
0
1
0
1
0
1
0
1
0
,2,2
,2,,
n
i
n
i
igh
hg
iihh
hh
i
h
n
i
n
j
jihg
hg
j
hh
i
n
i
n
j
jigg
hg
j
hg
i
n
i
n
j
jihh
hh
j
hh
i
ttCovttCov
ttCovttCovttCov
λλ
σλλλλλλ
(B12)
( ) ( )[ ] ( )
−+= ∑
−
=
21
0
2
ˆ )( tgtgthEtn
i
i
gg
ii
gh
ig λλσ
( ) ( ) ( )
( ) ( )∑ ∑
∑∑∑∑∑∑−
=
−
=
−
=
−
=
−
=
−
=
−
=
−
=
−−
+++=
1
0
1
0
21
0
1
0
1
0
1
0
1
0
1
0
,2,2
,2,,
n
i
n
i
igg
gg
iihg
gh
i
g
n
i
n
j
jihg
gg
j
gh
i
n
i
n
j
jigg
gg
j
gg
i
n
i
n
j
jihh
gh
j
gh
i
ttCovttCov
ttCovttCovttCov
λλ
σλλλλλλ
(B13)
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
37
To find the minimum variance estimator ( )th we have to find the weights hh
iλ and hh
iλ such that
( ) ( )0,0
2ˆ
2ˆ
=∂
∂=
∂
∂hg
i
h
hh
i
htt
λ
σ
λ
σ
(B14)
Now
( ) ( ) ( ) ( )ttCovttCovttCovt
ihh
n
j
jihg
hg
j
n
j
jihh
hh
jhh
i
h ,2,2,21
0
1
0
2
ˆ−+=
∂
∂∑∑
−
=
−
=
λλλ
σ
( ) ( ) ( )ttCovttCovttCov igh
n
j
jigg
hg
j
n
j
jigh
hh
jhg
i
h ,2,2,21
0
1
0
2
ˆ−+=
∂∑∑
−
=
−
=
λλλ
σ
therefore
( ) ( ) ( )ttCovttCovttCov ihh
n
j
jihg
hg
j
n
j
jihh
hh
j ,,,1
0
1
0
=+∑∑−
=
−
=
λλ (B15)
( ) ( ) ( )ttCovttCovttCov igh
n
j
jigg
hg
j
n
j
jigh
hh
j ,,,1
0
1
0
=+∑∑−
=
−
=
λλ
To find the minimum variance estimator ( )tg we have to find the weights gh
iλ and gg
iλ such that
( ) ( )0,0
2
ˆ
2
ˆ=
∂
∂=
∂
∂gg
i
g
gh
i
g tt
λ
σ
λ
σ
(B16)
Now
( ) ( ) ( ) ( )ttCovttCovttCovt
ihg
n
j
jihg
gg
j
n
j
jihh
gh
jgh
i
g,2,2,2
1
0
1
0
2
ˆ=+=
∂
∂∑∑
−
=
−
=
λλλ
σ
( ) ( ) ( ) ( )ttCovttCovttCovt
igg
n
j
jigg
gg
j
n
j
jigh
gh
jgg
i
g,2,2,2
1
0
1
0
2
ˆ=++=
∂
∂∑∑
−
=
−
=
λλλ
σ
therefore
( ) ( ) ( )ttCovttCovttCov ihg
n
j
jihg
gg
j
n
j
jihh
gh
j ,,,1
0
1
0
=+∑∑−
=
−
=
λλ
( ) ( ) ( )ttCovttCovttCov igg
n
j
jigg
gg
j
n
j
jigh
gh
j ,,,1
0
1
0
=+∑∑−
=
−
=
λλ (B17)
General Auto-Regressive Asset Model
38
Substituting (B15) into (B12) and (B17) into (B13) provides more compact expressions of
estimates of the mean squared error around the estimates ( )th , ( )tg :
( ) ( ) ( )[ ]∑−
=
+−=1
0
22
ˆ ,,n
i
igh
hg
iihh
hh
ihhttCovttCovt λλσσ (B18)
( ) ( ) ( )[ ]∑−
=
+−=1
0
22
ˆ ,,n
i
igg
gg
iihg
gh
igg ttCovttCovt λλσσ (B19)
Given the methodology described above, to simulate unconditional realizations of a pair of
correlated stochastic processes, and conditional simulation (2D Filtering), the method to create
conditional realizations of a pair of processes closely follows that for a single stochastic process.
For completeness we provide a stepwise description next.
Simulation of Conditional Realizations
Given observations of a pair of stochastic processes, ( ){ }00 ),( tgth , ( ){ }11),( tgth , . . . . . ,
( ){ }11),( −− nn tgth , here are the steps to describe a possible outcome of the pair of stochastic process
( ){ }tgth cc ),( that takes on observed values at the points of observation.
1. Generate an unconditional realization of the pair ( ) ( ){ }tgth uu , for all t’s spanning the times of
observations Tobs and the times for which the conditional realization is to be simulated Tsim.
2. Generate the conditional mean of the processes ( ) ( ){ }tgth uuˆ,ˆ , for all simt Τ∈ , that attain values of
the unconditional realization at observation points Tobs, i.e., 10 ),()(ˆ −≤≤= nithth iuiu
10 ),()(ˆ −≤≤= nitgtg iuiu
3. Generate the conditional mean functions ( ){ }tgth ˆ),(ˆ , for all simt Τ∈ , that attain observed values at
observation points Tobs, 10 ),()(ˆ −≤≤= nithth ii 10 ),()(ˆ −≤≤= nitgtg ii
4. ( ) ( ) ( ) ( )( )thththth uucˆˆ −+= ; ( ) ( ) ( ) ( )( )tgtgtgtg uuc
ˆˆ −+=
By repeating the process for a different unconditional realization, ( ) ( ){ }tgth uu , the sub-ensemble of
the processes ( ) ( ){ }tgth , that pass through observations can be created. These conditional
realizations are equally likely, and their mean values are ( ){ }tgth ˆ),(ˆ and the variance around
( ){ }tgth ˆ),(ˆ are the estimating variance 2
hσ and 2
gσ given in equations (B18) and (B19).
Jiaxin Wang, Andrea Petrelli, Ram Balachandran, Olivia Siu, Jun Zhang, Rupak Chatterjee, & Vivek Kapoor*
39
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Acknowledgements
Discussions with Santa Federico are gratefully acknowledged.
The views expressed here are those of the authors, and do not necessarily represent those of their employers. *corresponding author contact email: [email protected]