An analytical perturbative solution to the Merton–Garman model using symmetries

• Author: Nathaniel Wiesendanger Shaw,Xavier Calmet
• Published date: 01 January 2020
• Date: 01 January 2020
• DOI: http://doi.org/10.1002/fut.22061

© 2019 The Authors. The Journal of Futures Markets published by Wiley Periodicals, Inc.

J Futures Markets. 2020;40:3–22. wileyonlinelibrary.com/journal/fut

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3

Received: 30 May 2019

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Accepted: 2 September 2019

DOI: 10.1002/fut.22061

RESEARCH ARTICLE

An analytical perturbative solution to the Merton–Garman

model using symmetries

Xavier Calmet

1

|

Nathaniel Wiesendanger Shaw

1,2

1

School of Mathematical and Physical

Sciences, University of Sussex, Brighton,

United Kingdom

2

Business School, University of Sussex,

Falmer, Brighton, United Kingdom

Correspondence

Xavier Calmet, School of Mathematical

and Physical Sciences, University of

Sussex, Brighton BN1 9QH, United

Kingdom.

Email: x.calmet@sussex.ac.uk

Funding information

Science and Technology Facilities

Council, Grant/Award Number: ST/

P000819/1

Abstract

In this paper, we introduce an analytical perturbative solution to the

Merton–Garman model. It is obtained by doing perturbation theory around

the exact analytical solution of a model which possesses a two‐dimensional

Galilean symmetry. We compare our perturbative solution of the Merton–Gar-

man model to Monte Carlo simulations and find that our solutions perform

surprisingly well for a wide range of parameters. We also show how to use

symmetries to build option pricing models. Our results demonstrate that the

concept of symmetry is important in mathematical finance.

KEYWORDS

Merton–Garman model, option pricing model, perturbation theory

JEL CLASSIFICATION

C02; G10

1

|

INTRODUCTION

Calculating the price of an option is an important challenge in mathematical finance. The first attempts in that direction are

attributed to Louis Bachelier who during in his Doctoral thesis, Théorie de la spéculation, published in 1900, considered a

mathematical model of Brownian motion and its use for valuing options. This study provided the foundations for the

Black–Scholes model (Black & Scholes, 1973). However, although the Black–Scholes model was a breakthrough in the field, it

is widely accepted that it has limitations. In particular, the volatility is treated as a constant which is not very realistic.

Since the seminal works of Black and Scholes (1973) and Merton (1973), more sophisticated models with a

time‐dependent volatility have been proposed. For example, the affine Heston model (see Heston, 1993), which

assumes a time‐dependent volatility, with a stochastic process involving the square‐root of the stochastic

volatility, and a leverage effect, has been implemented in a large number of empirical studies (Andersen,

Benzoni, & Lund, 2002; Bakshi, Cao, & Chen, 1997; Bates, 2000, 2006; Chernov, Gallant, Ghysels, & Tauchen,

2003; Eraker, 2004; Huang & Wu, 2004; Pan, 2002, to name a few). Such models have however limitations and are

often modified artificially by combining them with models of jumps in returns and/or in volatility (such as

Benzoni, 2002; Jones, 2003). As a consequence, there is a substantial strand of literature devoted to nonaffine

volatility models, which note that the popular square‐root stochastic volatility model is not very realistic (see,

e.g., Aït‐Sahalia & Kimmel, 2007; Chourdakis & Dotsis, 2011; Christoffersen, Jacobs, & Mimouni, 2010; Duan &

Yeh, 2010; Eraker, Johannes, & Polson, 2003; Kaeck & Alexander, 2012, to name a few). However, the issue with

such models is a general lack of closed form characteristic function, which makes pricing much more

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This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided

the original work is properly cited.

challenging. As stated in Chourdakis and Dotsis (2011) when regarding the place of nonaffine models and the

debate of their tractability against affine models: “Does analytically tractability come at the cost of empirical

misspecification?”It is a useful endeavor to study the nonaffine model as we propose in this paper, if an

analytical solution for the option pricing formula can be found.

Awell‐known example of such models is the Merton–Garman model (Garman, 1976; Merton, 1973) which is indeed a

more realistic model as it allows for a time‐dependent volatility and it is not restricted to an affine model for the volatility.

However, solving nonaffine models is time consuming, as it involves numerical methods. Thus, many practitioners are still

using the Black–Scholes formula to obtain a fast, albeit not necessarily very reliable, price quote for an option.

The aim of our study is twofold. We will derive an analytical approximative solution to the partial differential

equation describing the Merton–Garman model, which enables one fast calculation of option prices. This requires us to

identify a “symmetric”version of the model, which can easily be solved analytically. One can then reintroduce the

symmetry breaking terms of the original Merton–Garman differential equation and do perturbation theory around the

symmetric solution thereby obtaining an approximative but analytical solution to the original Merton–Garman

differential equation. We then propose a new approach to model building in option pricing based on the concept of

symmetry groups and representation theory. This concept has been extremely successful in modern physics. It is at the

origin of all successful models in physics, for example, in particle physics, cosmology, or solid‐state physics. We note

that perturbation theory has been used in option pricing models (Aguilar, 2017; Baaquie, 1997; Baaquie, Coriano, &

Srikant, 2003; Blazhyevskyi & Yanishevsky, 2011; Kleinert & Korbel, 2016; Utama & Purqon, 2016) but here we

organize perturbation theory around a very specific solution, namely that of the symmetrical model which we will

introduce in this paper.

This paper is organized as follows. In Section 2, we derive the partial differential equation, which describes the

Merton–Garman model. In Section 3, we explain how to reduce the original Merton–Garman model to a simple,

symmetrical, model. We present an exact analytical solution to the symmetrical model. We then restore the original

Merton–Garman model by re‐introducing the symmetry breaking terms and provide an analytical perturbative solution to

the Merton–Garman model. In Section 4, we compare our solutions to different numerical solutions found in the literature.

In Section 5, we propose a new approach to model building in mathematical finance. Finally, we conclude in Section 6.

2

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THE MERTON–GARMAN MODEL

In the Merton–Garman model, the price of an option is dependent on the time t, the price of the underlying S, and the

volatility V. Both Sand Vare taken to be time‐dependent functions, and thus the Merton–Garman model has the

potential to provide a more accurate calculation of an option price than, for example, the Black–Scholes model.

We start from the stochastic differential equations (SDEs) for the price of the underlying Sand for the volatility V:

d

S rSdt V SdW=+

,

S

(1)

d

Vκθ Vdt ξVdW=(−)+ ,

αV

(2)

which resembles a stochastic, mean reverting, volatility regime. Here,

ξ

is the standard deviation of the volatility and

κ

is the speed of mean reversion to the long‐run variance θ. The interest rate ris assumed to be constant. The model

described by Equations (1) and (2) covers many well‐known stochastic volatility models, for instance, setting

α

=

1

and

1/2 recovers the Hull and White (see Hull & White, 1987) and Heston models, respectively. However, we do not

constrain ourselves to either of these worlds. Here,

α

can take arbitrary values. We will denote the correlation between

the two Brownian motions

WS

and

W

V

by

ρ

.

We shall first consider a call option, but our results can be extended to a put option in a straightforward manner. Our

first step is to find the associated partial differential equation, which describes this model. We do so by applying the

Feynman–Kac formula (see, e.g., Hull, 1997), which states that the price of a call option, as defined by the model

dynamics in Equations (1) and (2), is given by

∑∑

∂

∂

∂

∂

∂

∂∂

C

tμtx C

xρσ txσtx C

xx rC+(,)+

1

2(, ) (, ) −=0

,

i

iiij

ij ij

ij

=1 , =1

2

(3)

4

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CALMET AND WIESENDANGER SHAW