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Simulation in the Heston Model

Abstract

All the methods we have encountered so far for pricing options under the Heston model have been analytic, in the sense that the option price is expressed in closed-form and involves one or more complex integrals that must be evaluated numerically. This is one standard approach for many option pricing methodologies. Another approach is to use simulation, which we describe in this chapter.

Monte Carlo simulation in the context of the Heston model refers to a set of techniques to generate artificial time series of the stock price and variance over time, from which option prices can be derived. There are several choices available in this regard. The first choice is to apply a standard method such as the Euler, Milstein, or implicit Milstein scheme, as described by Gatheral (2006) and Kahl and Jäckel (2006), for example. The advantage of these schemes is that they are easy to understand, and their convergence properties are well-known. The other choice is to use a method that is better suited, or that is specifically designed for the model. These methods include the IJK scheme of Kahl and Jäckel (2006), the quadratic-exponential scheme of Andersen (2008), the transformed volatility scheme of Zhu (2010), the scheme of Alfonsi (2010), or the moment-matching scheme of Andersen and Brotherton-Ratcliffe (2005). These schemes are designed to have faster ...

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