Abstract
Abstract A stock loan or securities lending is a loan which borrowers have a share of stock can use it as collateral. However, closed form solution is not available for perpetual America options, the valuation of stock loan is an optimal stopping problem related to a perpetual American option. As Xia and Zhou (2007) published their work about stock loans. The stock loan pricing problem has thus attracted a great deal of attention since their work. Our purpose is that under interest rate is stochastic to find an optimal exercise boundary. As a result, we focus on the stochastic process about stock and interest rate as well as the correlation between both. Finally, we use Variational Inequality to figure out the optimal stopping time which indicates the optimal exercise price.