GARAM Model

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


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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*

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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].


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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!


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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


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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.


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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?


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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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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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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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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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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.


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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)

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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


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.


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.


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)


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


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)


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


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

0.06

0 42 84 126 168 210 252 294 336 378 420 462 504

return

-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

0.3

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


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


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


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)


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.


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


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.


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.


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.


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)


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.


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)


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)


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.


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 .


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)


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)


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 ,, =++


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)


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


ihg

n

j

jihg

gg

j

n

j

jihh

gh

jgh

i

g,2,2,2

1

0

1

0

2

ˆ=+=

∂

∂∑∑

−

=

−

=

λλλ

σ


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)


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).


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]

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