Firm Value Model with Gaussian Interest Rates

In this paper, we apply the firm value model to study pricing problems for Euro-pean options for derivatives with counterparty default risk. We derive a closed-form formula for Black-Scholes type options with Gaussian interest rates. The results are base on Ammann’s theory (refer to [1]), but generalize the drift ì and volatility ó from constants to functions of time t. We provide complete and detailed proofs in the paper. By getting closed-form formula for Black-Scholes type options, we first give the price of derivatives and the firm’s assets. Then we will build a forward martingale measure to get a price model for credit-risky option. Finally, we derive a closed-form formula for Black-Scholes type options under this price model. In proofs, we use the following mathematics tools: Ito’s formula, martingale, Brownian motion, Girsanov Theorem (Girsanov Theorem for correlated Brownian Motions), etc. 1. The price of derivatives and the firm’s assets In this chapter, we consider the pricing problem for European type options with counterparty default risk, in which the price of derivatives and the firm’s assets are stochastic and correlated, given by (1.1) Then we assume that the liability is a constant D with v0>D. However, in this 94 paper, we assume that the market has a zero-coupon bond with maturing time T, the price process P(·, T) of which is given by (1.2) Whereã(·)is the instantaneous forward rate, which follows a family of Ito’s processes: (1.3) We assume that the coefficient ì and ó are all {