Valuation of stock loans with jump risk

A stock loan is a special loan with stocks as collateral, which offers the borrowers the right to redeem the stocks on or before the maturity (Xia and Zhou, 2007; Dai and Xu, 2011). We investigate pricing problems of both infinite- and finite-maturity stock loans under a hyper-exponential jump diffusion model. In the infinite-maturity case, we derive closed-form formulas for stock loan prices and deltas by solving the related optimal stopping problem explicitly. Moreover, we obtain a sufficient and necessary condition under which the optimal stopping time is finite with probability one. In the finite-maturity case, we provide analytical approximations to both stock loan prices and deltas by solving an ordinary integro-differential equation as well as a complicated non-linear system. Numerical experiments demonstrate that the approximation methods for both prices and deltas are accurate, fast, and easy to implement.