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 satised 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 <beta , the boundary tends to Kexp(beta T0)
in parabolic-logarithm form, this case is the main result. For the case r >=beta ; the
corresponding problem returns back to a usual American Call option with interest-free
rate r-beta   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.