An efficient variable step-size method for options pricing under jump-diffusion models with nonsmooth payoff function

Abstract

We develop an implicit-explicit midpoint formula with variable spatial step-sizes and variable time step to solve parabolic partial integro-differential equations with nonsmooth payoff function, which describe the jump-diffusion option pricing model in finance. With spatial differential operators being treated by using finite difference methods and the jump integral being computed by using the composite trapezoidal rule on a non-uniform space grid, the proposed method leads to linear systems with tridiagonal coefficient matrices, which can be solved efficiently. Under realistic regularity assumptions on the data, the consistency error and the global error bounds for the proposed method are obtained. The stability of this numerical method is also proved by using the Von Neumann analysis. Numerical results illustrate the effectiveness of the proposed method for European options under jump-diffusion models.

A variable step‐size extrapolated Crank–Nicolson method for option pricing under stochastic volatility model with jump

“…In [34] and [35], the Merton model and the Kou model are studied by using the variable step‐size IMEX BDF2 method and the variable step‐size IMEX MP method, respectively. In this paper, we also applied these two methods to the Bates model and compared with the extrapolated CN method proposed in this paper.…”

“…To illustrate the effectiveness of the variable step-size extrapolated CN scheme, in this subsection, we also use the variable step-size IMEX BDF2 scheme [34] and the variable step-size IMEX MP scheme [35] to price the Bates model under European put options.…”

“…The authors of [34] have only recently used the variable step‐sizes IMEX two‐step backward differentiation formula (BDF2) method to solve the jump diffusion option pricing model, and obtained the stability and convergence by using the energy method based on the $$$ {L}_2 $$$ time regularity of the solution. We in [35] also proposed the variable step‐sizes IMEX midpoint (MP) formula to solve the Merton and Kou models, and proved the stability and convergence by using Von Neumann analysis. In this paper, a variable step‐sizes extrapolated CN method is proposed to solve the Bates model, and the stability and the upper bound of the $$$ {L}_2 $$$ error are derived based on the $$$ {L}_2 $$$ time regularity of the solution.…”

Deep learning numerical methods for high-dimensional fully nonlinear PIDEs and coupled FBSDEs with jumps

“…Consequently, one usually resorts to numerical methods to solve the two kinds of equations, and a volume of work has been performed on their numerical solutions. IMEX time discretizations combined with finite difference method, finite element method, or spectral method, have been used to solve low-dimensional PIDEs (see, for example, [1,31,26,36,37,28]), and multistep and prediction-correction schemes have been used to low-dimensional FBSDEJs (see, for example, [41,40,15]).…”