Robust numerical valuation of European and American options under the CGMY process

Abstract

We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Lévy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semi-Lagrangian scheme. The resulting Partial Integro-Differential Equation (PIDE) is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.

Pricing early-exercise and discrete barrier options by fourier-cosine series expansions

“…For this set of CGMY parameters it is now well-known that PIDE-based methods have convergence difficulties [2,44]. The reference value is found to be 28.829781986 .…”

“…They represent the state of the art for pricing options under the local volatility process. Generally speaking, however, the computational process with PIDE is rather expensive, especially for the infinite activity Lévy processes we are interested in, because they give rise to an integral in the PIDE with a weakly singular kernel [2,27,44].…”

A Fast Method for Pricing Early-Exercise Options with the FFT

“…To value Bermudan options, one can recursively call (13) and (4) backwards in time: First recover the option values on the last early-exercise date; then feed them into (13) and (4) to obtain the option values on the second last early-exercise date; · · ·, continue the procedure till the first early-exercise date is reached; for the last step, feed the option value on the first early-exercise date into (13) and there we obtain the option values on the initial date.…”

Application of Fast N-Body Algorithm to Option Pricing under CGMY Model

“…An efficient approach for computing the integral term is to use the FFT [7] [8] [9] [10], but this requires two FFT operations per time-step and direct use of FFT requires either a uniform grid or an additional interpolation scheme [11]. Also, an efficient scheme which transforms the PIDEs to pseudo-differential equations and applies FFT to the equations has been suggested [12], but this also requires two FFT operations per time-step for path-dependent option pricing.…”

“…In the present paper, we apply another efficient approach, called the Cartesian treecode [13], to the CGMY model ( [11] [22]). To our best knowledge, this is the first paper applying treecode to option pricing modeling.…”