Stock loan valuation based on the Finite Moment Log-Stable process

The empirical test suggests that the log-return series of stock price in US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. In this paper, we investigate the stock loan problem under the assumption that the return of stock follows the finite moment log-stable process (FMLS). In this case, the pricing model of stock loan can be described by a space-fractional partial differential equation with time-varying free boundary condition. Firstly, a penalty term is introduced to change the original problem to be defined on a fixed domain, and then a fully-implicit difference scheme has been developed. Secondly, based on the fully-implicit scheme, we prove that the stock loan value generated by the penalty method cannot fall below the value obtained when the stock loan is exercised early. Thirdly, the numerical experiments are carried out to demonstrate differences of stock loan model under the FMLS and the standard normal distribution. Optimal redemption strategy of stock loan has been achieved. Furthermore the impact of key parameters in our model on the stock loan evaluation are analyzed, and some reasonable explanation are given.