Stock loan valuation under a stochastic interest rate model

Stock loans are loans collateralized by stocks. They are modern fnancial products designed for investors with large equity positions. Mathematically, stock loans can be regarded as American call options with a time-dependent strike price once established. This study focuses on stock loans under a stochastic interest rate framework. The partial diferential equation (PDE) governing the value of the stock loan is derived by portfolio analysis. Boundary conditions are then prescribed to close the PDE system. In particular, boundary conditions along the interest rate direction are the focus of our derivation. After simplifying the pricing system by a series of transformations, the predictor-corrector method is adopted to solve the transformed PDE system numerically. Moreover, we introduce the alternating direction implicit (ADI) method in the two-factor model to improve the computational effciency. To ensure the stability of the predictor-corrector method, a hybrid finite difference scheme is adopted. Numerical results suggest that the current method is reliable and the stochastic interest rate leads to a higher optimal exercise price of the stock loan in comparison with that calculated from the Black-Scholes model.