Evaluation de dérivés et calcul stochastique 1

Les objectifs du cours

The purpose of this course (24h, Zhenjie REN) is to present the basics of probability theory and continuous time processes, allowing for a rigorous understanding of the standard dynamic stock models. Due to the motivations of the course, the mathematical concepts will be taught, considering concrete financial applications.
After a review of basic probability notions, the course will focus on the rigorous mathematical definition of arbitrage opportunities, leading to basic financial derivative valuation techniques. After a study of option pricing in multiperiod binomial models, we introduce stochastic calculus tools such as Brownian motion, stochastic integral and Ito formula. With these instruments, a complete study of the classical Black Scholes model will be performed: option valuation, PDE price characterization and Greeks computation. If time permits, the last chapter will deal with Monte Carlo simulations and practical calibration and estimation methods for the coefficients of the Black Scholes model.

A support course is offered to M1-level students (6h, Guillaume ANDRIEUX). The purpose of this support course is to illustrate and apply the notion studied in the ‘Derivatives pricing and stochastic calculus 1’ course. After a quick review of the key concepts presented in class, the course will focus on using these concepts on diverse financial applications to gain a better practical understanding of them.
The outline of ‘Derivatives pricing and stochastic calculus 1’ course was followed in order to facilitate the synergies between both courses.

Plan du cours

• Basics on probability : Random variable, expectation, variance, law, density, Gaussian law, ...
• Arbitrage : No arbitrage opportunity conditions, Call Put parity, American call, Forward contracts.
• A toy example: the Binomial pricing model. Model dynamics, Risk neutral pricing, Market completeness.
• Introducing dynamical strategies: multi-period binomial models. Portfolio strategies, asymptotic behavior with the number of periods
• Modelling the market randomness: the Brownian motion. Characterizations, quadratic variation,Geometric Brownian Motion, Merton vs. Bachelier
• Introducing portfolio dynamics: stochastic integration and Ito formula. Ito calculus, continuous trading strategies, stochastic differential equations
• The reference in continuous time: Black & Scholes,Model dynamics, Risk neutral pricing, Greeks computation.The price as the solution of a partial differential equation, Volatility calibration and estimation.Monte Carlo methods vs. Numerical PDE approximation.

Outlines for the support course:

• Arbitrage: NAO, different kind of forward contracts, Call-spread, Butterfly
  Binomial model: Exotic options, a first step to delta hedging
• Modelling the market randomness: analysis of random processes and their covariance structure
  Introducing portfolio dynamics: applying Ito formula to diverse processes
• The reference in continuous time: Black & Scholes and change of probability measure
   

Bibliographie

Comets F. and T. Meyre, Calcul Stochastique et modèle de diffusion, 324 pages. 2006.
Lamberton D. and B. Lapeyre, Introduction to Stochastic Calculus applied to finance, 256 pages. 2007.

Examen

1 final exam