Pricing black-scholes options with correlated credit risk and jump risk

Abstract

This article follows the framework of Klein (1996) to present an improved method of pricing vulnerable options under jump diffusion assumptions about the underlying stock prices and firm values which are appropriate in many business situations. In contrast to Klein’s (1996) model, jumps allow not only for sudden changes in stock prices and firm values, but also for a firm to default instantaneously because of an unexpected drop in its value. Therefore, our model is able to provide sufficient conceptual insights about the economic mechanism of vulnerable option pricing. In particular, an analytical pricing formula for vulnerable European options under jump diffusion model is derived. The numerical results show that a jump occurrence in firm values can increase the likelihood of default and reduce the vulnerable option prices.

Keywords:

• derivatives
• jump diffusion model
• default
• credit risk

JEL Classification:

• C5
• C6
• G1

Notes

¹ There are two classes of models in the existing literature that capture the credit default risks. One class, including Black and Scholes (1973), Merton (1974), Black and Cox (1976), Longstaff and Schwartz (1995) and others, models the credit default risk by structural approach. The other class of models adopts the reduced-form approach, including Duffie and Singleton (1996), Jarrow et al. (1997), Madan and Unal (1998) and others.

² We consider option pricing with jump diffusions similar to the approach starting with Merton (1974), and now using by Eraker (2004), Eraker et al. (2003), Zhang and Zhou (2013), Li et al. (2013), etc.

³ This derivation is available from the author upon request.