Pricing European-Style Options in General Lévy Process with Stochastic Interest Rate

Xiaoyu Tan,Shuyi Wang,Shenghong Li

doi:10.3390/math8050731

Abstract

This paper extends the traditional jump-diffusion model to a comprehensive general Lévy process model with the stochastic interest rate for European-style options pricing. By using the Girsanov theorem and Itô formula, we derive the uniform formalized pricing formulas under the equivalent martingale measure. This model contains not only the traditional jump-diffusion model, such as the compound Poisson model, the renewal model, the pure-birth jump-diffusion model, but also the infinite activities Lévy model.

Highlights

European-style options such as vanilla call and put options are some of the most widely traded contracts among financial instruments and were first successfully priced in 1973 by Black and Scholes [1]
In the complete probability space (Ω, F, P), if the stock price satisfies the general Lévy process in Equation (3) and interest rate satisfies the Vasicek stochastic interest rate model in Equation (4), the market model can be expressed as ST = St exp
As seen from this example, once the distribution function or expression of νx and the specific form of Jx, we can get the corresponding option pricing formula according to formula Equations (13)

Summary

Introduction

European-style options such as vanilla call and put options are some of the most widely traded contracts among financial instruments and were first successfully priced in 1973 by Black and Scholes [1]. Merton (1975) [2] introduced the European-style option pricing formula with stock prices following the jump-diffusion model. Kou (2002) [3] proposed the double exponential jump-diffusion model and derived the analytical solutions to European call and put options. Nowak and Pawłowski (2019) [10] used a Lévy process of jump–diffusion type for description of an underlying asset and derived analytical option pricing formulas using the minimal Lq equivalent martingale measure. Tan, Jiang and Chen (2020) [11] proposed a European option-pricing model with stochastic volatility and stochastic interest rates and pure-jump Lévy processes.

Preliminaries

The Market Model

Option Pricing under General Lévy Process Model with Stochastic Interest Rate

Some Special Cases

Numerical Experiment

Conclusions