Variance Swap Pricing under Markov-Modulated Jump-Diffusion Model

Shican Liu,Yonghong Wu,Yu Yang,Hu Zhang

doi:10.1155/2021/9814605

Abstract

This paper investigates the pricing of discretely sampled variance swaps under a Markov regime-switching jump-diffusion model. The jump diffusion, as well as other parameters of the underlying stock’s dynamics, is modulated by a Markov chain representing different states of the market. A semi-closed-form pricing formula is derived by applying the generalized Fourier transform method. The counterpart pricing formula for a variance swap with continuous sampling times is also derived and compared with the discrete price to show the improvement of accuracy in our solution. Moreover, a semi-Monte-Carlo simulation is also presented in comparison with the two semi-closed-form pricing formulas. Finally, the effect of incorporating jump and regime switching on the strike price is investigated via numerical analysis.

Highlights

Risk, often measured by the variance of a specific underlying asset’s return, has always been a major concern for the investors in financial markets. e variance changes sarcastically over the investment period, providing practitioners opportunities to speculate on the spread between the realized variance and the implied variance, as well as the motivation to hedge against the variance risk
As the stochastic volatility is always investigated together with a stochastic interest rate, a model incorporating these two factors is investigated in the paper Cao et al [13] where it is proved that the effect of the interest rate is not as vital as that of the volatility on the variance swap price
Since the main contribution of this paper is integrating both the regime switching and jump diffusion in the variance swap pricing problem, we examine the effect of both the Merton-type and Kou-type jumps under the regimeswitching model in numerical analysis

Summary

Introduction

Often measured by the variance (volatility) of a specific underlying asset’s return, has always been a major concern for the investors in financial markets. e variance (volatility) changes sarcastically over the investment period, providing practitioners opportunities to speculate on the spread between the realized variance (volatility) and the implied variance (volatility), as well as the motivation to hedge against the variance risk. As the stochastic volatility is always investigated together with a stochastic interest rate, a model incorporating these two factors is investigated in the paper Cao et al [13] where it is proved that the effect of the interest rate is not as vital as that of the volatility on the variance swap price Along another line, the regime-switching model, where the parameters for the dynamics of the underlying asset’s price are modulated by an observable Markov chain that represents the general varying market states, is widely accepted as economically reasonable and has been applied to a large number of financial models such as those related to financial time series and derivative pricing (see Buffington and Elliott [14], Liu et al [15], Yao et al [16], Boyle and Draviam [17], Yuen and Yang [18], and Costabile et al [19]).

Model Description

Pricing Variance Swap

Numerical Examples

Conclusion

Derivation of the Continuous Model

Algorithm of Our Pricing Formula