Asymptotic analysis for stock loans near maturity

Abstract

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.