Exchange options and spread options with stochastically correlated underlyings

ABSTRACT

This paper investigates the valuation of exchange options and spread options with stochastically correlated underlying assets. Specially, the correlation is determined by market betas of underlying assets and depends on the level of the variance of the market index. Based on the closed form of the moment generating function of the log-return vector, we obtain pricing formulae for exchange options and spread options. Finally, numerical analysis illustrates the impacts of the market risk factor on option prices.

KEYWORDS:

• Exchange options
• spread options
• GARCH models
• stochastic correlation
• stochastic volatility

JEL CLASSIFICATION:

• G13

Acknowledgments

The author would like to thank the anonymous referees and the editor, Mark Taylor, for their helpful comments and valuable suggestions that led to several important improvements. All errors are my own responsibility.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ Alternatively, we can begin with the models in physical measures and then derive the risk-neutral dynamics by measure change techniques. Suppose the following dynamics under physical probability measure,

$\left\{\begin{array}{cc}ln{M}_t=ln{M}_{t-1}+r+({\lambda }_m-\frac{1}{2})h_{m,t}+\sqrt{h_{m,t}}{Z}_{m,t}^{\ast },& \hspace{0.17em}\\ h_{m,t}=w_m+b_m{h}_{m,t-1}+a_m{(Z_{m,t-1}^{\ast }-(c_m-{\lambda }_m)\sqrt{h_{m,t-1}})}^2,& \hspace{0.17em}\end{array}\right.$

where $Z_{m,t}^{\ast }$ is a standard normal disturbance under physical probability measure and ${\lambda }_m$ denotes the market price of risk. Following Heston and Nandi (2000), Christoffersen, Jacobs, and Ornthanalai (2012) and Wang (2018), we can assume the affine structure of the pricing kernel and derive the corresponding risk-neutral dynamics in (2.1). We refer the interested readers to Heston and Nandi (2000), and Wang (2018) for more details.

² The option values do not depend on the level of the market index.

Additional information

Funding

This study was supported by the National Natural Science Foundation of China [No. 11701084].