Valuation of stock loans using exponential phase-type Lévy models

Stock loans are collateral loans with stocks used as the collateral. This paper is concerned with a stock loan valuation problem in which the underlying stock price is modeled as an exponential phase-type Lévy model. The valuation problem is formulated as the optimal stopping problem of a perpetual American option with a time-varying exercise price. When a transformation is applied to the perpetual American option, it becomes a perpetual American call option in an economy with a negative interest rate, thus causing standard Wiener-Hopf techniques to fail. We solve this optimal stopping problem using a variational inequality approach.