Probabilistics methods in finance

Objectifs : Option pricing in discrete and continuous time, with martingales use and first steps of stochastic calculus.

Contenu du cours :

Chapter I. Preliminaries
   1. Derivative products, description and use: Forward/Future contracts, Options
   2. Rates and discounting
   3. Arbitrage methods

Chapter II. Forward contracts pricing (reminder, in tutorial)

Chapter III. Mathematical tools
   1. Conditional expectation, martingale.

Chapter IV. Option pricing in discrete time
   1. N periods binomial model (Cox-Ross-Rubinstein); self-financing strategies,
   2. risk-neutral probability, martingale property of the discounted price process,
   3. option pricing, delta hedging.

Chapter V. Option pricing in continuous time: Black-Scholes model
   1. Brownian motion and Ito processes.
   2. Quadratic variation of the Brownian motion,
   3. Ito integral for a simple process,
   4. Extension to the computation of ∫B_tdB_t, 
   5. Ito lemma (heuristic proof).
   6. Black-Scholes model
   7. Partial differential equation approach, hedging from that equation.
   8. Probabilistic approach for European options,
   9. Girsanov theorem (particular case),
 10. Black Scholes formula, delta computation, use.

Références :

• Hull, Options, futures, and other derivative securities, Prentice-Hall (2018: 10th ed).
• Baxter, M. and Rennie, A., Cambridge University Press, 1996.
• Kwok, Y.K., Mathematical models of financial derivatives, Springer, 2nd edition, 2008 (3 first chapters). 
• Jacod, J., Protter, P. (2000) Probability Essentials. Springer.