Objective

The objective of this course is to present the mathematical concepts used in the modeling and valuation of derivatives in finance. The presentation of the course will be mathematically rigorous, but some stochastic calculus results will be admitted, to be demonstrated later in the 3rd year stochastic calculus course. After obtaining a mathematical definition of the notion of arbitrage on a financial market, we will study in the first part of the course discrete finite state space models, which give good intuitions for the study of models in continuous time. Using the theory of stochastic calculus, we will present in the second part of the course the valuation of assets within the framework of the Black & Scholes model. It is advisable to follow the Introduction to Stochastoc Processes course to better assimilate the notions of stochastic calculus and to follow the Simulation course in parallel.

At the end of the course, the student will be able to :

• Give the definition of an arbitrage on a financial market, find the parity formula Call Put in AOA.
• Understand the link between the absence of arbitrage and the existence of a risk-neutral probability.
• Mathematically represent an option price and its hedging strategy in binomial and Black Scholes models.
• Define the Brownian motion and state its main properties.
• Apply Ito's formula to a process of dimension 1, thus verifying for example that a process is a martingale.

Planning

1. Probability and Arbitrage. The financial market as a random environment. Mathematical definition of arbitrage. Consequences of the absence of arbitrage opportunities. Applications: Valuation of a Forward contract and Call Put Parity Formula.
2. Discrete one-period model. Assumptions on the financial market. Absence of arbitrage and market completeness. Valuation under risk neutral probability. Portfolio optimization and notions of economic equilibrium.
3. Discrete n-period model. The market model. Replication portfolio. Risk neutral valuation of options. Binomial model with n periods. The Black Scholes model as a limit.
4. Stochastic calculus. Process, Filtration and Martingale. Brownian motion: definition and properties. Quadratic variation, stochastic integral, Ito formula and Ito process. Introduction to the notion of Stochastic Differential Equation.
5. Financial markets in continuous time. Black-Scholes model. Price dynamics and discounting. Girsanov's Theorem and Risk Neutral Valuation, Feynman-Kac Representation and EDP Pricing. Delta hedging. Limitations of the Black-Scholes model: the volatility smile. Application: valuation of a European Call (closed formula, tree, EDP, Monte Carlo). Continuous time portfolio optimization.

Références

LAMBERTON D. & LAPEYRE B., (1997) : Introduction au calcul stochastique appliqué à la finance, Ellipses [17 LAM 00 A]SHREVE S. (1997) : Stochastic calculus and finance, Lecture notes [78 SHR 00 A]DANA & JEANBLANC (1998) : Marchés financiers en temps continu – Valorisation et équilibre , Economica [78 DAN 00 A]OKSENDAL B. (1998) : Stochastic differential equations, Springer [17 OKS 00 A]