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.

[1]  Alex W. H. Chan.  Merton, Robert C. , 2010 .

[2]  G. Barone-Adesi,et al.  Efficient Analytic Approximation of American Option Values , 1987 .

[3]  Steven Kou,et al.  A Jump Diffusion Model for Option Pricing , 2001, Manag. Sci..

[4]  Xiao Lan Zhang,et al.  Numerical Analysis of American Option Pricing in a Jump-Diffusion Model , 1997, Math. Oper. Res..

[5]  P. Forsyth,et al.  Robust numerical methods for contingent claims under jump diffusion processes , 2005 .

[6]  R. C. Merton,et al.  Option pricing when underlying stock returns are discontinuous , 1976 .

[7]  Kaushik I. Amin.  Jump Diffusion Option Valuation in Discrete Time , 1993 .

[8]  Francis A. Longstaff,et al.  Valuing American Options by Simulation: A Simple Least-Squares Approach , 2001 .

[9]  Qing Zhang,et al.  Valuation of Stock Loans with Regime Switching , 2009, SIAM J. Control. Optim..

[10]  Steven Kou,et al.  Option Pricing Under a Double Exponential Jump Diffusion Model , 2001, Manag. Sci..

[11]  Mark D. Schroder,et al.  Changes of Numeraire for Pricing Futures, Forwards, and Options , 1999 .

[12]  Ning Cai,et al.  On first passage times of a hyper-exponential jump diffusion process , 2009, Oper. Res. Lett..

[13]  R. Caflisch,et al.  Pricing and Hedging American-Style Options: A Simple Simulation-Based Approach , 2009 .

[14]  M. Dai,et al.  Optimal Multiple Stopping Models of Reload Options and Shout Options , 2008 .

[15]  Shek-Keung Tony Wong.  The generalized perpetual American exchange-option problem , 2008, Advances in Applied Probability.

[16]  Nick Webber,et al.  Valuing American Options by Simulation , 2012 .

[17]  M. Dai,et al.  A lattice algorithm for pricing moving average barrier options , 2010 .

[18]  Hui Wang,et al.  First passage times of a jump diffusion process , 2003, Advances in Applied Probability.

[19]  Min Dai,et al.  OPTIMAL REDEEMING STRATEGY OF STOCK LOANS WITH FINITE MATURITY , 2009, 0906.0702.