Option Pricing with Threshold Mean Reversion

• Author: Hoi Ying Wong,Fangyuan Dong,Zeyu Chi
• Date: 01 February 2017
• DOI: http://doi.org/10.1002/fut.21795
• Published date: 01 February 2017

Option Pricing with Threshold

Mean Reversion

Zeyu Chi, Fangyuan Dong, and Hoi Ying Wong*

Mean reversion and regime switching are well-known features of commodity prices. Recent

empirical research additionally documents the time variation of the mean reversion rate and

volatility.This paper considers the option pricing framework for an underlying commodity price

with mean reversion rate and volatility change according to a self-exciting regime switching

model. Weoffer empirical evidence for the proposed model and derive analytic pricing formulas

for the European and barrier options. Numerical examples demonstrate the application and

the ability of the proposed model in capturing volatility smile and regime-switching in the mean

reversion rate, simultaneously.©2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:107–131, 2017

1. INTRODUCTION

Evidence on the mean reversion in commodities is abundant while derivatives pricing models

with mean reversion have gained popularity in the past two decades. Most popular mean re-

version models are inspired by the Ornstein–Uhlenbeck (OU) process, such as Cox-Ingersoll-

Ross (1985)and Vasicek (1977). The OU process randomly oscillates around the long-term

mean level with a constant mean reversion rate so that it permits serial dependence.

Sorensen (1997) advocates mean reversion for currency exchange rate and find that

the mean-reverting feature significantly affects the American currency options price. Ekvall,

Jennergren, and Naslund (1997) give several reasons for mean-reverting exchange rates using

an equilibrium model and derive the closed-form solution for pricing European currency

options. Hilliard and Reis (1998) investigate how the price of commodity futures and future

options are affected by stochastic convenient yield under the two-factor model of Schwartz

(1997). It is also noticed that if the convenient yield is a deterministic function of asset price,

two-factor model can be reduced to mean-reverting process. Hui and Lo (2006) and Wong

and Lau (2008) employ a mean-reverting log-normal model (MRL) to value barrier options.

Wong and Lo (2009) introduce a stochastic volatility to the MRL model. Fusai, Marena, and

Roncoroni (2008) show the importance of mean reversion for commodity derivatives and

derive an analytic solution to discrete Arithmetic Asian options. Chung and Wong (2014)

further generalize it to include jumps.

The aforementioned research, however, concentrates on constant mean reversion rate

and volatility, except for Wong and Lo (2009) who take stochastic volatility into account

but still assume a constant mean reversion rate. When we empirically estimate the time

Zeyu Chi, Fangyuan Dong, and Hoi Ying Wong are at the Department of Statistics, The Chinese University

of Hong Kong, Shatin, N.T., Hong Kong.

*Correspondence author,Department of Statistics, The Chinese University of Hong Kong, Shatin, N.T., Hong

Kong. Tel: +852 39438520, Fax:+852 26035188, e-mail: hywong@cuhk.edu.hk

Received June 2015; Accepted April 2016

The Journal of Futures Markets, Vol. 37, No.2, 107–131 (2017)

©2016 Wiley Periodicals, Inc.

Published online 30 May 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21795

108 Chi, Dong, and Wong

series models for commodity prices, the mean reversion rate is found to be time varying

and quite depends on the commodity price itself. In time series analysis, Tong (1983, 1990)

describes this feature through the self-exciting threshold autoregressive model (SETAR). The

SETAR model considers coefficients in an autoregression to take different values depending

on whether the previous asset value is above or under a certain threshold, thus exhibiting

regime switching dynamics.

The other model reflecting the regime-switching pattern is called Markov-modulated

regime-switching (MMRS) model. Christoforidou and Ewald (2015a, 2015b) develop a class

of analytical formulas for various option prices on commodities under multi-factor OU pro-

cess with the volatilities controlled by a Markov process. It belongs to a different category

in the reason that the SETAR model views regime-switching as an endogenous event deter-

mined by the asset value itself, whereas the MMRS model regards it exogenously governed

by a Markov chain.

A natural extension of the discrete-time SETAR model is its diffusion limit which turns

out to be a stochastic differential equation (SDE) with a piecewise linear drift term. Freidlin

and Pfeiffer (1998) use the Brownian motion with a drift that switches between positive and

negative to construct a diffusion threshold model with an unknown upper threshold and a

zero lower threshold. Decamps, Goovaerts, and Schoutens (2006) study this class of thresh-

old processes for interest rate and obtain semi-analytic expressions for the transition density

of self-exciting threshold (SET) diffusion. When the mean reversion speed always stays pos-

itive, the SET diffusion is known as an ergodic threshold OU (TOU) process. Kutoyants

(2012) renders parameter estimation scheme for several ergodic TOU processes.

This paper develops a new option pricing framework where the underlying asset price

is modeled by a generalized TOU process. The generalized process allows both mean rever-

sion rate and volatility to change once the asset price crosses certain threshold parameters,

whereas classical TOU process assumes a constant volatility. Therefore, the associated SDE

has piecewise linear drift and volatility terms. We call this generalized TOU process the

threshold mean reversion (TMR).

We apply this model to the commodity derivatives market because our data analysis

shows that commodities exhibit SETAR feature when they are fit to time series models. In

addition, their volatilities show different values in different regimes separated by the esti-

mated threshold. As commodities are not traded directly on exchange, the classical dynamic

hedging using the underlying asset no longer works in this situation so that the drift term

of the asset value process cannot be reduced to the risk-free interest rate under the pricing

measure in general. In practice, investors often manage the commodity risk using futures

contracts so that the drift term of the asset value process under the pricing measure should

match or is calibrated to the observed futures term structure. Therefore, this paper allows the

log-asset value to follow the TMR under the pricing measure as well. New pricing formulas

are then needed for European and path-dependent options when the underlying commodity

price follows a TMR process.

We derive the analytic option pricing formulas for European options and some popular

path-dependent options, including barrier options and lookback options. The formulas enable

us to investigate the impact of the mean reversion and regime-switching on these options.

More delightfully, they are consistent with the existing pricing formulas proposed in Hilliard

and Reis (1998) when there is only one regime and the two factors in the model of Hilliard

and Reis (1998) are perfectly correlated.

The remainder of the paper is organized as follows. Section 2 introduces the TMR

process using empirical data. Section 3 derives the analytic formulas of vanilla call option

which will be useful for pricing barrier options. Section 4 presents solution to the valuation

of exotic options such as barrier and lookback options. Section 5 gives several numerical