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+ | =====Course unit: Mathematical finance ===== | ||
+ | ==== Course metadata ==== | ||
+ | * Title in French: Mathématiques financières | ||
+ | * Course code: tba | ||
+ | * ECTS credits: 3 | ||
+ | * Teaching hours: 72h | ||
+ | * Type: specialized course | ||
+ | * Language of instruction: | ||
+ | * Coordinator: | ||
+ | * Instructor(s): | ||
+ | * //Last update 27/08/2021 by C. Pouet// | ||
+ | ==== Brief description ==== | ||
+ | The aim of the course is to provide students with mathematical methods that allow valuating financial assets. | ||
+ | |||
+ | This course unit is divided into three parts: | ||
+ | * ** Stochastic calculus and introduction to the Black-Scholes model ** (24 hours) taught by Sébastien Darses. | ||
+ | * ** Volatility models ** (24 hours) taught by Ismaïl Akil. | ||
+ | * ** Interest rate models ** (24 hours) taught by Abderrahim Ben Jazia. | ||
+ | |||
+ | ==== Learning outcomes ==== | ||
+ | |||
+ | * Understand stochastic calculus and know how to apply its main results | ||
+ | * Know how to apply stochastic methods to price financial products | ||
+ | * Understand the mathematical contexts under which the classical financial mathematics models hold | ||
+ | * Know and understand the relevance and limits of financial mathematics models | ||
+ | * Understand the impact of volatility on the profit and losses of a hedged position | ||
+ | * Know how to build numerical methods for pricing financial products | ||
+ | |||
+ | |||
+ | ==== Course content ==== | ||
+ | |||
+ | |||
+ | === Stochastic calculus and introduction to the Black-Scholes model=== | ||
+ | |||
+ | - Gaussian variable and stochastic processes | ||
+ | - Brownian motions | ||
+ | - Stochastic integration and semi-martingales | ||
+ | - Stochastic differential equations | ||
+ | - Parabolic partial differential equations and semigroups | ||
+ | - Measure change and Girsanov theorem | ||
+ | - Introduction to financial mathematics | ||
+ | |||
+ | === Volatility models==== | ||
+ | - Elementary financial mathematics notions | ||
+ | - PDE: Black Scholes and risk neutral measure | ||
+ | - Dupire’s local volatility: advantages and drawbacks | ||
+ | - Stochastic volatility (Heston and SABR) | ||
+ | - Tutorial: discretization of the Heston’s model | ||
+ | |||
+ | === Interest rate models=== | ||
+ | - A Mathematical Toolkit | ||
+ | - Interest rates, swaps and options | ||
+ | - One-factor Short-Rates Models | ||
+ | - Two-factor Short-Rates Models | ||
+ | - The Health-Jarrow-Morton (HJM) Model | ||
+ | - The change of numeraire | ||
+ | - Derivatives Pricing under the Libor Market Model | ||
+ | |||
+ | ==== Bibliography ==== | ||
+ | Check the availability of the books below at [[https:// | ||
+ | - Stochastic calculus | ||
+ | * Evans, L. (2010). An Introduction to Stochastic Differential Equation. American Mathematical Society. | ||
+ | * Le Gall, J.-F. (2006). Intégration, | ||
+ | - Volatility models | ||
+ | * El Karoui, N. (2004) Couverture des risques dans les marchés financiers. Ecole Polytechnique | ||
+ | - Interest rate models | ||
+ | * Brigo, D., & Mercurio, F. (2007). Interest rate models-theory and practice: with smile, inflation and credit. Springer Science & Business Media | ||
+ | * Privault, N. (2012). An elementary introduction to stochastic interest rate modeling. World Scientific. |