An iterative method for pricing American options under jump-diffusion models

Abstract

We propose an iterative method for pricing American options under jumpdiffusion models. A finite difference discretization is performed on the partial integro-differential equation, and the American option pricing problem is formulated as a linear complementarity problem (LCP). Jump-diffusion models include an integral term, which causes the resulting system to be dense. We propose an iteration to solve the LCPs efficiently and prove its convergence. Numerical examples with Kou's and Merton's jump-diffusion models show that the resulting iteration converges rapidly.

Reduced order models for pricing European and American options under stochastic volatility and jump-diffusion models

“…The integrals can be discretized using a second-order accurate quadrature formula. Here the linear interpolation is used for w between grid points and exact integration; see [11], for details. Under the Merton model the discretization of the integral leads to a full matrix while under the Bates model it leads to full diagonal blocks.…”

“…A penalty method together with FFT based fast method for evaluating the jump integral was used in [9]. An iterative method was proposed for LCPs with full matrices in [11]. An implicit-explicit (IMEX) method was proposed in [10] to treat the integral term explicitly and the same approach was studied in [13].…”

Double Discretization Difference Schemes for Partial Integrodifferential Option Pricing Jump Diffusion Models

“…The authors in 20 derive an analytical formula for the price of European options for any model including local volatility and Poisson jump process by using Malliavin calculus techniques. Various authors apart of 14 used finite difference schemes for PIDEs in [21][22][23][24][25][26][27] . Discretization of the integral term leads to full matrices due to its nonlocal character.…”

Finite Difference Schemes for Option Pricing under Stochastic Volatility and Lévy Processes: Numerical Analysis and Computing

“…Many authors used the finite difference (FD) schemes for solving these PIDE problems [2,4,5,17,25,54,74,75,82,85,87]. Dealing with FD methods for such PIDEs, the following challenges should be addressed.…”

“…Tavella and Randall in [82] use an implicit time discretization and propose a stationary rapid convergent iterative method to solve the full matrix problem quoted above, but with poor numerical analysis. A generalization of their iterative method to price American options is proposed in [75].…”