SIAM Journal on Numerical Analysis
SIAM Journal on Scientific and Statistical Computing
Penalty methods for American options with stochastic volatility
Journal of Computational and Applied Mathematics
Quadratic Convergence for Valuing American Options Using a Penalty Method
SIAM Journal on Scientific Computing
Finite Element Error Estimates for a Nonlocal Problem in American Option Valuation
SIAM Journal on Numerical Analysis
On modified Mellin transforms, Gauss-Laguerre quadrature, and the valuation of American call options
Journal of Computational and Applied Mathematics
A comparison study of ADI and operator splitting methods on option pricing models
Journal of Computational and Applied Mathematics
| Hi-index | 7.29 |
We develop adaptive @q-methods for solving the Black-Scholes PDE for American options. By adding a small, continuous term, the Black-Scholes PDE becomes an advection-diffusion-reaction equation on a fixed spatial domain. Standard implementation of @q-methods would require a Newton-type iterative procedure at each time step thereby increasing the computational complexity of the methods. Our linearly implicit approach avoids such complications. We establish a general framework under which @q-methods satisfy a discrete version of the positivity constraint characteristic of American options, and numerically demonstrate the sensitivity of the constraint. The positivity results are established for the single-asset and independent two-asset models. In addition, we have incorporated and analyzed an adaptive time-step control strategy to increase the computational efficiency. Numerical experiments are presented for one- and two-asset American options, using adaptive exponential splitting for two-asset problems. The approach is compared with an iterative solution of the two-asset problem in terms of computational efficiency.