Exponential time differencing schemes for pricing American option under the Heston model

Abstract

[EN] The classic Black-Scholes model makes assumptions that are not empirically valid. The model is widely employed as a useful approximation to reality, but proper application requires understanding its limitations and constant volatility of the stock returns is one of them. In fact, this assumption is one of the biggest source of weakness, because the variance has been observed to be non-constant leading to models, such as GARCH, to model volatility changes. There are other approaches to model the asset volatility, as consider that follows a random process or, in other words, consider the volatility as a stochastic process. This point of view lead us to a Partial Differential Equation (PDE) different from the classic Black-Scholes, now there are involved two different variables, apart of the time: asset level S and variance . Deal with this PDE and the presence of cross-derivatives is a challenging task. It is even more difficult to deal with American options which allows to exercise the option at any time before the expiration date. But the solution to this problem is of great interest to the financial markets.