Pricing long-maturity equity and FX derivatives with stochastic interest rates and stochastic volatility

Pricing long-maturity equity and FX

derivatives with stochastic interest rates

and stochastic volatility

Alexander van Haastrecht1 2, Roger Lord 3,

Antoon Pelsser4 and David Schrager5.

First version: January 10, 2005

This version: September 22, 2008

Abstract

In this paper we extend the stochastic volatility model of Sch¨obel and Zhu (1999) by including

stochastic interest rates. We allow all driving model factors to be instantaneously correlated with

each other, i.e. we allow for a general correlation structure between the instantaneous interest

rates, the volatilities and the underlying stock returns. By deriving the characteristic function

of the log-asset price distribution, we are able to price European stock options eciently and in

closed-form by Fourier inversion. Furthermore we present a Foreign Exchange generalization of

the model and show how the pricing of forward starting options can be performed. Finally, we

conclude.

Keywords: Stochastic volatility, Stochastic interest rates, Sch¨obel-Zhu, Hull-White, Foreign

Exchange, Equity, Forward starting options, Hybrid products.

1 Introduction

The OTC derivative markets are maturing more and more. Not only are increasingly exotic structures

created, the markets for plain vanilla derivatives are also growing. One of the recent advances in

equity derivatives and exchange rate derivatives is the development of a market for long-maturity

European options6. In this paper we develop a stochastic volatility model aimed at pricing and risk

managing long-maturity equity and exchange rate derivatives.

We extend the models by Stein and Stein (1991) and Sch¨obel and Zhu (1999) to allow for Hull

and White (1993) stochastic interest rates as well as correlation between the stock price process, its

1Netspar/University of Amsterdam, Dept. of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The

Netherlands, e-mail: a.vanhaastrecht@uva.nl

2Delta Lloyd Insurance, Risk Management, Spaklerweg 4, PO Box 1000, 1000 BA Amsterdam

3Rabobank International, Financial Engineering, Thames Court, 1 Queenhithe, London EC4V 3RL, e-mail:

roger.lord@rabobank.com

4Netspar/University of Amsterdam, Dept. of Quantitative Economics, Roetersstraat 11, 1018 WB Amsterdam, The

Netherlands, e-mail: a.a.j.pelsser@uva.nl

5ING Life Japan, Variable Annuity Market Risk Management, e-mail: 02037 schrager@ing-life.co.jp

6The implied volatility service of MarkIT, a financial data provider, shows regular quotes on a large number of major

equity indices for option maturities up to 10-15 years.

1

stochastic volatility and interest rates. We call it the Sch¨obel-Zhu Hull-White (SZHW) model. Our

model enables to take into account two important factors in the pricing of long-maturity equity or

exchange rate derivatives: stochastic volatility and stochastic interest rates, whilst also taking into

account the correlation between those processes explicitly. Because it is hardly necessary to motivate

the inclusion of stochastic volatility in a derivative pricing model. The addition of interest rates as

a stochastic factor is important when considering long-maturity equity derivatives and has been the

subject of empirical investigations most notably by Bakshi et al. (2000). These authors show that

the hedging performance of delta hedging strategies of long-maturity options improves when taking

stochastic interest rates into account. Interest rate risk is not so much a factor for short maturity

options. This result is also intuitively appealing since the interest rate risk of equity derivatives,

the option’s rho, is increasing with time to maturity. The SZHW model can further be used in the

pricing and risk management of a range of exotic derivatives. One can think of

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