Asymptotic analysis for stock loans near maturity

In this paper, we derive the asymptotic expressions of the scaled value function and the optimal redemption boundary of stock loan with dividend-paying near maturity. Using the equation satisfied by the derivative of the value function at the exercise boundary, we set up the asymptotic expression for the boundary. When the risk-free rate r is smaller than the loan rate ?, i.e., r < ?, the boundary tends to Ke?T0 in parabolic-logarithm form, this case is the main result. For the case r ? ?, the corresponding problem returns back to a usual American call option with interest-free rate r ? ? and the existing results can be utilized to make proper adjustments for the stock loan. The matched expansion for the value function is performed with a small parameter. Numerical examples are provided to demonstrate the effectiveness of the proposed method.