Geometrical Approximation Method and Stochastic Volatility Market Models

We want to purpose and introduce a method that we name Geometrical Approximation (G.A.), by which it is possible to study the stochastic volatility market models (as Heston and SABR). The G.A. intends to be an alternative method useful to obtain the price of Vanilla options, this is less expensive than the other ones from computational point of view. There are many economic, empirical, and mathematical reasons for choosing a model with such a form (see Cont 2001 for a detailed statistical/empirical analysis). Empirical studies have shown that an asset's log-return distribution is non-Gaussian. It is characterised by heavy tails and high peaks (leptokurtic). There is also empirical evidence and economic arguments that suggest that equity returns and implied volatility are negatively correlated (also termed ‘the leverage effect'). This departure from normality is a plague of the Black-Scholes-Merton model. In contrast, Heston's model can imply different distributions