تسعير ة الخيارات ﺑﺈستعمال نماذج كسرية

Abstract

Different models for stochastic volatility for choice evaluate have been presented to catch the volatility effect since a long time, from which the Heston model has been award more importance because of its reliability to get volatility, being positive with mean-reverting, and especially give a closed-form solution for European options. Few works has been achieved using the Heston Model evaluate options, because of the reality that a closed-form solution is not exist and that an additional computational needs compared to the standard relative model increases the complexity contained in the calibration. This led to obstruct using pricing models in the choice markets. For the Heston model, For both cases European and American options, we can use an implicit alternative direction method to provide an active solution. Related to pricing according to the Black Scholes model, a limited difference method joint with a projected successive over-relaxation model is applied. After knowing how to price the American and European option according to the Heston model and the Black-Scholes model, we present the calibration for both models. present a twosteps appreciation method. Firstly, an indirect deduction method is applied to assess the structural parameters in the Heston model that impact the asset return allocation. Secondly is to assess the immediate variance of the risk-value by reduction the space between the market prices and the model prices depending on the least squares method. We will handle the connection between the option price and spreading parameters. Therefore, the volatility of the European or American option according to the previous models will be explained to estimate the performance of the two models. Since the Heston model is more realistic compared with the Black-Scholes model. Performing his superiority in general, we can conclude that the stochastic volatility model is better than the constant model. Although, the Heston model has aimed to reduce the value of the in-the-money option and increase the price of the out-of-the-money option. While, the Black- Scholes model is Tends to reduce the price of both in the money and out of the money of an option . In the first chapter of this work, we introduce the fundamental calculation tools required by our methodology, especially some models for pricing options and then the fractional calculus. In the second chapter, we introduce some pricing options models under Partial Differential Equations respectively : Black-Scholes Model, Heston Model, and Vasicek Model . In the third chapter, we present pricing options models under fractional partial differential equations respectively : Black-Scholes Model, Heston Model, and Vasicek Model. The above one presents our contribution and main result.