**GUILLAUME ANDRIEUX - **Co-head of Volatility, Man AHL, London

**VINCENT TENA - **Maître de Conférence, Université Paris - Dauphine

The purpose of this course is to expose students to a rigorous understanding of dynamic stock models and derivative securities. It introduces mathematical concepts to learn the fundamentals of pricing and hedging both in discrete and continuous time, which are essential for future practitioners in many fields of market finance.

This course plans to be self-contained, and the first chapter presents the relevant concepts of probability theory used afterward. However, students are advised to have a basic knowledge of probability theory, and such preliminary material is not included in this course. Chapter 2 focuses on the rigorous mathematical definition of arbitrage opportunities, leading to basic financial derivative valuation techniques. After a study of option pricing in a single period and multiperiod binomial models (Chapter 3 and 4), we introduce in Chapter 5 stochastic calculus tools such as Brownian motion, stochastic integrals and Itô's formula. Chapter 6 presents the Black Scholes model, and Greeks computation.

A support class is offered to M1-level students. 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 in diverse financial applications to gain a better practical understanding of them.

The outline of the 'Derivatives pricing and stochastic calculus 1' course was followed to facilitate the synergies between both courses.

- 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

Back K. , A Course in Derivative Securities: Introduction to Theory and Computation. Springer Science Textbook, 356 pages. 2006. (Part 1)

Lamberton D. and B. Lapeyre, Introduction to Stochastic Calculus applied to finance, 256 pages. 2007. (Chapters 1, 3, and 4)

1 final exam

Catching up ... Read “All the Math you need” and the beginning of ”Elementary Stochastic Calculus” of “*Paul Wilmott Introduces quantitative Finance*”, Willmott P, 2nd Edition, Wiley. 2007