Option Pricing under Double Stochastic Volatility Model with Stochastic Interest Rates and Double Exponential Jumps with Stochastic Intensity

Yiming Wang,Ying Chang

doi:10.1155/2020/2743676

Abstract

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

Highlights

An abundance of empirical studies show the existence of the asymmetric leptokurtic features and the volatility smile after Black and Scholes [1] did some experimental and pioneering work in European option pricing
Both the COS method and fast Fourier transform (FFT) takes less time, and they improve in computation speed. e price differences between the COS method and semi-closed form prices are negligible compared to the price differences between FFT and semiclosed form prices which means that the COS method shows higher accuracy than FFT
We examined the relative differences of call option prices with different grid points to compare the COS method and FFT in terms of the stability. e result demonstrates that the COS method shows higher stability than FFT

Summary

Introduction

An abundance of empirical studies show the existence of the asymmetric leptokurtic features and the volatility smile after Black and Scholes [1] did some experimental and pioneering work in European option pricing. Ey contributed to the existing literature by expanding Vasicek’s model that they added a term to the diffusion coefficient, and it maintains the mean-reverting and nonnegative properties that make it more proper for application than Vasicek’s model Since this model has been a benchmark to specify the dynamic change of the variance and the interest rate for decades. Grzelak et al [21] proposed option pricing combining stochastic volatility with Schobel–Zhu and CIR processes and stochastic interest rate with Hull–White process, and they used some techniques for obtaining the formula for discounted characteristic function.

The Model and Semi-Closed Form Formula

Conclusion

Proof of Lemma 1