Analytically Pricing a Vulnerable Option under a Stochastic Liquidity Risk Model with Stochastic Volatility

This paper considers the valuation of a vulnerable option when underlying stock is subject to liquidity risks. That is, it is assumed that the underlying stock is not perfectly liquid. We establish a framework where the stock price follows the stochastic volatility model and the option contains the default risk of the option issuer. In addition, we assume that liquidity risks are caused by stochastic market liquidity, and the default occurs at the first jump time of a stochastic Poisson process, which has a stochastic default intensity process consisting of both idiosyncratic and systematic components. By employing a change of measure, we derive an analytical formula for the value of a vulnerable option. Finally, we present several numerical examples to illustrate the sensitivity of significant parameters.