Jump without tears: A new splitting technology for barrier options
A. Itkin 1, P. Carr 21 Department of Finance and Risk Engineering, New York University Politechnic Institute, Brooklyn, New York, 11201, USA
2 Morgan Stanley and New York University
Received by the editors February 4, 2011 and, in revised form, May 31, 2011.
The market pricing of OTC FX options displays both stochastic volatility and stochastic skewness in the risk-neutral distribution governing currency returns. To capture this unique phenomenon Carr and Wu developed a model (SSM) with three dynamical state variables. They then used Fourier methods to value simple European-style options. However pricing exotic options requires numerical solution of 3D unsteady PIDE with mixed derivatives which is expensive. In this paper to achieve this goal we propose a new splitting technique. Being combined with another method of the authors, which uses pseudo-parabolic PDE instead of PIDE, this reduces the original 3D unsteady problem to a set of 1D unsteady PDEs, thus allowing a significant computational speedup. We demonstrate this technique for single and double barrier options priced using the SSM.AMS subject classifications: 60J75, 35M99, 65L12, 65L20, 34B27, 65T50
Key words: Barrier options, pricing, stochastic skew, jump-diffusion, finite-difference scheme, numerical method, the Green function, general stable tempered process.