Forward starting options pricing under a regime-switching jump-diffusion model with Wishart stochastic volatility and stochastic interest rate

Abstract

Vanilla options become effective immediately after they are entered, while some exotic options will only come to effective some time after they are bought or sold. Forward starting options are one kind of such exotic options started actually at some pre-specified future date. This paper presents an extension of regime-switching jump diffusion model, in which the parameters are driven by a continuous time and stationary Markov chain on a finite state space, by introducing the Wishart process into the instantaneous variance-covariance matrix of the risky asset price and stochastic interest rate. We derive the discounted conditional joint characteristic function and the forward characteristic function of the log-asset price and its the instantaneous variance-covariance process, and thereby the price of forward starting options are well evaluated by the probabilistic approach combined with the Fourier-cosine (COS) method. We also provide efficient Monte Carlo simulation of this proposed model, and simulated solutions to forward starting options pricing within a two-state regime switching framework. Numerical results show that the COS method is accurate and efficient for pricing forward starting options. Finally, we analyse impacts of some main parameters (especially, parameters in the Wishart stochastic volatility) in this proposed model on option prices and Δ values. Also, we consider the forward implied volatility. Furthermore, the forward starting options under the regime-switching jump diffusion model with Wishart stochastic volatility and stochastic interest rate which we derived are more generalized than those recently appeared in the derivatives pricing literature, and thus have wider application.

KEYWORDS:

• Forward starting options
• regime-switching jump-diffusion model
• Wishart stochastic volatility
• stochastic interest rate
• Fourier-cosine expansion

2010 AMS SUBJECT CLASSIFICATIONS:

• 90A09
• 91B28
• 91B24
• 93E20

JEL codes:

• G12
• G13

Acknowledgements

The authors are also grateful to the editors and the anonymous referee for theirs useful help and suggestions which have significantly improved this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability

All data used to support of the findings for this study are included within this article.

Notes

1 Unlike the multi-factor Heston's model, this paper allows M and Q to be non-diagonal to enrich the dynamic interaction among components of ${\mathrm{\Sigma }}_t$. An illustrative example using a 2-dimensional Wishart process can be found in [20].

2 In this paper, we assume that the interest rate $R_t$ is designated as a function of economic variables ${\mathrm{\Sigma }}_t$ with regime-switching, which can describe the correlation movements and structural changes of market, and can offer the analytic tractability, see [23].

3 The MSVSIRSJ model in this paper is in a Markov regime-switching framework, the integral interval in (19(19) $C(t,S)\approx P(t,T){\int }_a^{b}v(S_T)\sum _{k=0}^{N-1}{}^{\mathrm{\prime }}{F}_k\cos \left(\mathrm{k\pi }\frac{Y_T-a}{b-a}\right)\mathrm{d}{Y}_T,$(19) ) is then regime-dependent. To eliminate this influence of the Markov regime-switching, we set $[a,b]$ shown in (21(21) $a=\underset{1\le j\le \mathrm{\aleph }}{min}\{c_{j,1}-L\sqrt{\left|c_{j,2}\right|}\},b=\underset{1\le j\le \mathrm{\aleph }}{max}\{c_{j,1}+L\sqrt{\left|c_{j,2}\right|}\},$(21) ) (see [71]).

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant number 11461008] and the Guangxi Natural Science Foundation [grant number 2018GXNSFAA281016].