Résumé:

Financial stocks are often modeled by stochastic differential equations. These equations describe the behavior of the underlying asset and some of the model's parameters. Nowadays, the models retaining our attention have a stochastic volatility term. They are known to allow for a better calibration to market data; they also efficiently capture the smile of the volatility observed in financial securities. Until recently, the Black-Scholes-Merton framework (Black and Scholes (1973) and Merton (1973)) was widely used. Semi-closed formulas exist for this model, which makes it very attractive from a practical point of view. However, the BSM model includes very coarse assumptions such as a constant volatility and a deterministic constant interest rate. These shortfalls combined to several financial crashes and the introduction of new complex products, have forced the financial analysts to develop new models. Heston (1993) proposed a model based on the square root process with mean reversion to express variance and based on the geometric Brownian motion for the stock price. This model became very popular and is widely used in practice since Heston managed to derive a semi-closed formula for the price of a European call option. Moreover, the variance process (square root process) is widely applied in finance since many analytic results are known about this SDE. For a concrete example, you can refer to the Cox et al. (1985) short rate model. In this presentation, we introduce the model. Moreover, an exhaustive review of the most popular simulation schemes for this model is made. We conclude this presentation by a numerical comparison of these schemes.