Convertible bond pricing in a mixed fractional Brownian motion environment

We establish a convertible bond (CB) pricing model. The underlying stock price is described by a stochastic differential equation driven by a mixed fractional Brownian motion with the Hurst parameter H satisfying 1/2<H<1, which characterizes the serial autocorrelation implicating sort of a memory of the stock price. The interest rate follows the Vasicek process. The default risk is addressed by the reducedform approach. The existence of the risk neutral valuation relationship (RNVR) of the convertible bond is obtained by the employment of risk preference attitude of the investors and restrictions on investors' utility function. An explicit pricing formula for the convertible bond with default risk is derived. As a result, the derivative of the convertible bond price with respect to the Hurst parameter H is also shown explicitly. The results show that the influence of the Hurst parameter on the CB price is based on its influence on the integrated volatility of the conditional underlying stock price.