Closed Form Solution for Heston PDE by Geometrical Transformations

One presents and discusses an alternative solution writeable in closed form of the Heston’s PDE, for which the solution is known in literature, up to an inverse Fourier transform. Since the algorithm to compute the inverse Fourier Transform is not able to be applied easily for every payoff, one has elaborated a new methodology based on changing of variables which is independent of payoffs and does not need to use the inverse Fourier transform algorithm or numerical methods as Finite Difference and Monte Carlo simulation. In particular, one will compute the price of Vanilla Options for small maturities in order to validate numerically the Geometrical Transformations technique. The principal achievement is to use an analytical formula to compute the prices of derivatives, in order to manage, balance any portfolio through the Greeks, that by the proposed solution one is able to compute analytically. The above mentioned numerical techniques are computationally expensive in the stochastic volatility market models and for this reason is usually employed the Black Scholes model, that is unsuitable, as it has been widely proven in literature to compute the sensitivities of a portfolio and the price of derivatives. The present article wants to introduce a new approach to solve PDEs complicated, such as, those coming out from the stochastic volatility market models, with the achievement to reduce the computational cost and thus the time machine; besides, the proposed solution is easy to be generalized by adding jump processes as well. The present research work is rather technical and one does wide use of functional analysis. For the conceptual simplicity of the technique, one confides which many applications and studies will follow, extending the applications of the Geometrical Transformations technique to other derivative contracts of different styles and asset classes.