SERIE A: CONFERENCIAS, SEMINARIOS Y TRABAJOS DE MATEMATICA No. 17 AN INTRODUCTION TO BLACK-SCHOLES MODELING AND NUMERICAL METHODS IN DERIVATIVES PRICING
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چکیده
The technique of dynamic hedging, combined with the application of Ito calculus and the absence of arbitrage hypothesis, provides a methodology for the valuation of financial derivatives by models of partial differential equations of Black-Scholes type. This document is intended to summarize in a simple way the concepts and techniques used in this methodology up to get the prices of the more traditional products. The document is divided into two main parts: the models and numerical methods. Prior to both, the lognormal stochastic model for the underlying asset is briefly recalled. In the models section, first the dynamic hedging technique is described to deduce the European vanilla options pricing models and the popular Black-Scholes formula. The methodology is extended to the case of American options, Asian options and options on various assets. The modeling part concludes with the statement of bonds pricing models as an example of interest rate derivatives. Finite differences and finite elements numerical methods are first described for European and American options. Then, some indications are given about its application to Asian options and bond models. Finally, some basic ideas on the technique of Monte Carlo simulation for European options are presented. Resumen. La técnica de la cobertura dinámica, combinada con el cálculo de Ito y la hipótesis de ausencia de arbitraje, proporciona una metodoloǵıa para la valoración de derivados financieros mediante modelos de ecuaciones en derivadas parciales de tipo Black-Scholes. En este documento se pretende resumir de modo sencillo los conceptos y técnicas empleadas en esta metodoloǵıa hasta llegar a obtener los precios de los productos derivados más clásicos. El documento se divide en dos partes principales: los modelos y los métodos numéricos. Previamente a ambas, se recuerda brevemente el modelo estocástico lognormal para el subyacente. En el apartado de modelos, primero se describe la técnica de cobertura dinámica para deducir los modelos de opciones vainilla europeas y la popular fórmula de Black-Scholes. La metodoloǵıa se extiende al caso de las opciones americanas, asiáticas y sobre varios activos. Los parte de modelos termina con la valoración de bonos como ejemplo de derivados de tipos de interés. Los métodos numéricos de diferencias y elementos finitos se describen primero para opciones europeas y americanas. A continuación, se dan indicaciones sobre la aplicación a los modelos de opciones asiáticas y bonos. Finalmente, se dan algunas ideas básicas sobre la técnica de simulación de Monte Carlo para opciones europeas.
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تاریخ انتشار 2010