Option Valuation Under a Double Regime‐Switching Model

• Author: Yang Shen,Tak Kuen Siu,Kun Fan
• Date: 01 May 2014
• Published date: 01 May 2014
• DOI: http://doi.org/10.1002/fut.21613

OPTION VALUATION UNDER A DOUBLE

REGIME-SWITCHING MODEL

YANG SHEN, KUN FAN and TAK KUEN SIU*

This paper is concerned with option valuation under a double regime-switching model, where

both the model parameters and the price level of the risky share depend on a continuous-time,

finite-state, observable Markov chain. In this incomplete market set up, we first employ a

generalized version of the regime-switching Esscher transform to select an equivalent

martingale measure which can incorporate both the diffusion and regime-switching risks. Using

an inverse Fourier transform, an analytical option pricing formula is obtained. Finally, we apply

the fast Fourier transform method to compute option prices. Numerical examples and empirical

studies are used to illustrate the practical implementation of our method. © 2013 Wiley

Periodicals, Inc. Jrl Fut Mark 34:451–478, 2014

1. INTRODUCTION

Regime-switching models are one of the most popular and practically useful models in

econometrics and finance. The history of the regime-switching models may be traced back to

the early works of Quandt (1958), Goldfeld and Quandt (1973), and Tong (1978, 1983).

Hamilton (1989) popularized applications of regime-switching models in economics and

finance. One of the main features of these models is that model dynamics are allowed to

change over time according to the state of an underlying Markov chain, which is also called a

modulating Markov chain. This provides us with a natural and convenient way to describe the

effect of structural changes in economic conditions, which may be attributed to changes in

economic fundamentals or financial crises, on price series.

Since the last decade or so, there has been an interest on studying option valuation

problems in regime-switching models. Switches may occur in the model parameters (e.g., the

appreciation rate and the volatility) and the price level of the risky share whenever transitions

in the modulating Markov chain occur. The literature on option valuation can be divided into

two categories in terms of the regime-switching models. The former includes Guo (2001),

Buffington and Elliott (2002), Elliott et al. (2005), Liu et al. (2006), Boyle and Draviam

Yang Shen is a PhD candidate at the Department of Applied Finance and Actuarial Studies, Faculty of

Business and Economics, Macquarie University, Sydney, NSW, Australia. Kun Fan is a joint PhD candidate at

the School of Finance and Statistics, East China Normal University, Shanghai, China and the Department of

Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney,

NSW, Australia. Tak Kuen Siu is a Professor of Actuarial Science in the Faculty of Actuarial Science and

Insurance at the Cass Business School, City University London, London, United Kingdom. We thank the

editor and an anonymous referee for their helpful comments. Tak Kuen Siu would like to acknowledge the

Discovery Grant from the Australian Research Council (ARC) (project no.: DP1096243).

*Correspondence author, Cass Business School, City University London, London, United Kingdom. Tel: þ44-20-

7040-0998, Fax: þ44-20-7040-8572, e-mail: ken.siu.1@city.ac.uk, ktksiu2005@gmail.com

Received September 2012; Accepted February 2013

The Journal of Futures Markets, Vol. 34, No. 5, 451–478 (2014)

© 2013 Wiley Periodicals, Inc.

Published online 28 March 2013 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21613

(2007), Siu (2008), Yuen and Yang (2010) and others, where the regime-switching models

can only describe the switches of model parameters. The latter includes Naik (1993), Yuen

and Yang (2009), and Elliott and Siu (2011), where not only the model parameters but also the

price level of the share may switch whenever a regime switch occurs. To differentiate these

two kinds of models, we call them the single regime-switching model and the double regime-

switching model, respectively. Numerous works focus on option valuation under the single

regime-switching models, while relatively little attention has been paid to that under

the double regime-switching models. However, the double regime-switching models provide a

more flexible way than their single regime-switching counterpart to describe stochastic

movements of the risky share due to the fact that a jump in the share price level occurs in the

former, but not in the latter, when there is a regime switch. Regime switches caused by

transitions in the modulating Markov chain are often interpreted as structural changes in

macro-economic conditions and in different stages of business cycles. These changes are

inevitable in a long time span. They may cause not only shifts in the mean and volatility levels

of the share price, but also sudden jumps in the share price level (see Naik, 1993; Yuen and

Yang, 2009; Elliott and Siu, 2011).

It appears that Naik (1993) was a n early attempt on option prici ng under the double

regime-switching models, wh ere a martingale method was employe d for the pricing of a

European option under a two-st ate, double regime-switchi ng model. Yuen and Yang

(2009) extended the model of Naik (1993) to a multi-regi me case and adopted the extended

model for pricing of a Europe an option, an American optio n and other exotic options us ing

a trinomial tree method. Elliot t and Siu (2011) considered a risk- based approach for

pricing an American continge nt claim under a multi-state, doub le regime-switching

model. Almost all of the works on op tion valuation under the sing le regime-switching

models do not incorporate the r egime-switching risk in the selection of a pric ing kernel. Most

of the existing works focus on capturi ng regime-dependent risk. Com pared with the single

regime-switching models, the d ouble regime-switching mod els allow us to “naturally”price

the regime-switching risk whe n one changes the real-world measur e to an equivalent

martingale measure. Howeve r, like a single regime-swi tching model, a financial market

described by a double regime-sw itching model is also incom plete. Consequently, not all

contingent claims can be perf ectly hedged by continuous ly trading primitive securi ties and

there is more than one pricing ker nel, or equivalent martingale measure. A primal pro blem is

how to select an equivalent mart ingale measure in such a market set up. In Naik (1993) and

Yuen and Yang (2009), equival ent martingale measures were selected by either ignor ing the

regime-switching risk or taki ng an exogenous regime-switch ing risk. Neither of them

determines the regime-swi tching risk endogenousl y from their double regime- switching

models.

In this paper, we consider option valuation under a double regime-switching model.

More specifically, the model parameters, including the risk-free interest rate, the appreciation

rate and the volatility rate, are modulated by a continuous-time, finite-state, observable

Markov chain. In addition, when a regime switch occurs, there is a jump in the price level of

the risky share. Consequently, the dynamics of the share is a discontinuous process. The jump

component of the share price is modeled by the jump martingale related to the modulating

Markov chain. We first apply a generalized version of the regime-switching Esscher transform

to select an equivalent martingale measure, which takes into account both the diffusion risk

from the Brownian motion and the regime-switching risk from the chain. Furthermore, the

(local)-martingale condition and the model dynamics of the share are obtained under this

equivalent martingale measure. Then we use the inverse Fourier transform to derive an

integral pricing formula of a European call option. The fast Fourier transform (FFT) method is

adopted to discretize the integral pricing formula. Using the FFT method, we provide the

452 Shen, Fan, and Siu