École Nationale Supérieure de
Techniques Avancées

Abstract

This report is divided into two parts. In the first part, I revisit some fundamental results of financial mathema- tics, including continuous-time processes, martingale theory, the Black & Scholes model, the local volatility model and the Dupire formula. The aim of this section is to provide the basis for the project. The project itself consists of implementing an option pricing technique. Deriv already has a PDE options pricing engine based on the Black & Scholes model [2]. Although the Black Scholes model is quite powerful, it cannot accurately reproduce prices observed in the market. A more sophisticated model, such as the local volatility model [7], is better able to take into account observations of market-traded option prices. At Deriv, my task was to adapt the pricing engine by solving the Black & Scholes partial differential equation in order to integrate the local volatility model. In the second part of the report, I present the data used and the results obtained when pricing options using the local volatility model. This approach, based on the local volatility model, significantly improves the accuracy of calculated option prices, by bringing them closer to observed market prices. The results highlight the effectiveness of this technique, and pave the way for more advanced applications in the field of quantitative finance. In summary, this report presents an in-depth study of the fundamental concepts of financial mathematics, follo- wed by a practical implementation of the local volatility model in Deriv’s options pricing engine. The results confirm the relevance of this approach and open up new prospects for the company in the field of accurate option pricing.