A Simple Model for Option Pricing with Jumping Stochastic Volatility

Abstract

This paper proposes a simple modification of the Black–Scholes model by assuming that the volatility of the stock may jump at a random time τ from a value σa to a value σb. It shows that, if the market price of volatility risk is unknown, but constant, all contingent claims can be valued from the actual price C0, of some arbitrarily chosen "basis" option. Closed form solutions for the prices of European options as well as explicit formulas for vega and delta hedging are given. All such solutions only depend on σa, σb and C0. The prices generated by the model produce a "smile"-shaped curve of the implied volatility.