The Option Pricing With Default Risk

The foundation of financial transactions lies in the credit. So the possibility ofdefault, namely credit risk, has an important impact on financial activities. However,little attention has been paid to the study of impacts of credit risk on the pricing ofderivatives. Since the1990s, with the structural transform of global finance system,especially the development of over-the-counter derivatives, credit risks have attracted lotsof attention among market participants and academic researches. The majority of therelated literature on credit risk deal with the pricing of debt instruments, such as bonds,or mortgaged loans, and had generally neglected the impact of counterpart default risk forOTC financial derivatives. In fact, counterpart default risk was not only one of the mainculprits that lead to the financial crisis in2008, but also the main reason that lead to thehuge losses for some of the financial institutions around the world. Therefore, studyingthe impact of counterpart default on the pricing of derivatives is very important.In this thesis, we apply the risk-neutral valuation framework developed by Black-Scholes to study the pricing of options when the underwriter has default risks. To describedefault event, we assume the asset value of the underwriter follows a geometric Brownianmotion. If its asset value falls below a pre-specified level, then default will occur, and inthis case the option holder will sufer a loss. We first assume that the underlying assetis a stock, whose price follows a geometric Brownian motion. We studied how the priceof such options can be computed. We also studied the case when the underlying stockis for a firm whose asset value follows a geometric Brownian motion. In this case, theunderlying stock itself is a call option on its asset value. In order to investigate the impactof default risk on the option, we also derived pricing formulae for compound options.