Pricing Vulnerable Options in the Bifractional Brownian Environment with Jumps

Panhong Cheng,Zhihong Xu,Antonio Di Crescenzo

doi:10.1155/2021/1451692

Abstract

In this paper, we study the valuation of European vulnerable options where the underlying asset price and the firm value of the counterparty both follow the bifractional Brownian motion with jumps, respectively. We assume that default event occurs when the firm value of the counterparty is less than the default boundary. By using the actuarial approach, analytic formulae for pricing the European vulnerable options are derived. The proposed pricing model contains many existing models such as Black–Scholes model (1973), Merton jump-diffusion model (1976), Klein model (1996), and Tian et al. model (2014).

Highlights

Options are popular financial derivatives that play important roles in financial markets
Following Tian et al [18], the discontinuous changes in the underlying asset prices and firm values of the counterparty both consist of two parts: idiosyncratic shocks for each asset price and common ones that have an impact on all asset prices. en, we show how to price European vulnerable options by actuarial approach in a jump bifractional Brownian environment
Based on the consideration of the jump risk of financial asset prices, the credit risk of the counterparty and the long-range dependence of financial asset prices in options trading, a more realistic pricing model for European vulnerable options is proposed in this paper

Summary

Introduction

Options are popular financial derivatives that play important roles in financial markets. Niu et al [21] studied the valuation of European vulnerable options where the underlying asset price was driven by a double exponential jump-diffusion process and the credit risk was described by the reducedform approach and obtained the pricing formulae of vulnerable options via the equivalent martingale measure technique and two-dimensional Laplace transform. E bifractional Brownian motion (in short, BFBM), known as a generalization of FBM, is a family of centered Gaussian processes that has the characteristics of self-similarity and long-range dependence and has the property of nonstationary increment and can be a semimartingale under appropriate conditions, and it can be applied to model the evolution of financial asset prices such as the underlying stock prices in options pricing.

Bifractional Brownian Motion

Pricing for Vulnerable Options in the Jump BFBM Environment

Financial Model Description

Concluding Remarks and Future Directions