Option Pricing Using Adomian Decomposition Method under Stochastic Interest Rates and Rough Heston Model

In the temporal domain, following the ideas of (Gonzalez-Gaxiola et al., 2015), the Adomian Decomposition Method (ADM) is used to solve the non-linear pricing PDE arising from a market model encompassing non-linear effects such as illiquidity or transaction costs. A first simple non-linear introductory example illustrates the interest of the ADM (conceptually simple, potentially robust and fast-converging numerical scheme) as an alternative to Monte-Carlo (MC) simulations or Finite Differences (FD). However, the classical ADM appears quickly limited whenever it has to deal with non-smooth payoff functions or with more risk factors (e.g. stochastic interest rates). Singularities in the payoff function can possibly be treated under a non-classical modification of the ADM. Yet, the latter extension is more involved on the mathematical and numerical points of view. It is only applicable for very specific problems and becomes intractable when it has to envision more risk factors (e.g. stochastic interest rates), especially in a non-linear context. The application of the ADM in the frequency domain leads to more promising results. Through the ADM, the Moment Generating Function (mgf) of the log return can be inferred under a rough Heston model. In turn, it can feed a fast pricing algorithm for European call options, over-performing a crude MC simulation.