Valuation of the Vulnerable Option Price Based on Mixed Fractional Brownian Motion

Yanmin Ouyang,Jingyuan Yang,Shengwu Zhou

doi:10.1155/2018/4047350

Yanmin Ouyang, Jingyuan Yang + Show 1 more

Open Access

https://doi.org/10.1155/2018/4047350

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Abstract

The pricing problem of a kind of European vulnerable option was studied. The mixed fractional Brownian motion and the jump process were used to characterize the evolution of stock prices. The closed-form solution to European option pricing was obtained by applying martingale measure transformation method. At the end of this paper, some numerical experiments were adopted to compare the new pricing formula introduced in this paper with the classical Black-Scholes pricing formula. The result showed that the new pricing formula conformed to the actual financial market. In fact, the option value is positively correlated with the underlying asset price and the company’s asset price and the jump process has significant influence on the value of option.

Highlights

Vulnerable European option is a kind of option with credit risk, which refers to the possibility of one party suffering losses as a result of the other party’s default on the contract
In the case of the underlying interest rates following a mean reverting square-root process, the analytical solutions to defaultable bonds were derived by Cathcart (2003) [10]
The problem that a new model of credit risk was proposed combining structural information with the reduced-form approach was studied by Ballestra (2014) [13] and a closed-form approximate solution was derived by perturbation approach and the Laplace transform

Summary

Introduction

Vulnerable European option is a kind of option with credit risk, which refers to the possibility of one party suffering losses as a result of the other party’s default on the contract. Klein’s option pricing model was expanded by Ammann (2001)[8] by applying structured method They derived the closed-form solution to vulnerable option price in the environment where interest rate and default intensity complying the stochastic differential equation. Xiaonan Su (2012) [25] supposed that the intensity of default is driven by a jump diffusion process and got vulnerable option pricing formula in a reduced-form by using martingale method. Assuming that the stock price obeyed the jump-diffusion model, Chao Wang (2015) [27] got analytic solution to vulnerable European option based on Xiaonan Su. Adopting the Ito’s formula of mixed fractional Brownian motion, Zhiguang Li (2016) [28] obtained the European option pricing formula with short-term interest rate obeying Vasicek model. For the reader’s convenience, the last part of this paper is an appendix to the concrete proof process of vulnerable option pricing

Hybrid Model

Vulnerable Option Pricing Model

Numerical Test

Conclusion

Derivation Process of III