Stochastic Processes Applied to Finance

• Área: Matemática da Informação
• Código: MIF005
• Carga horária: 45 horas
• Créditos: 3
• Course language: English

Prerequisite

Good knowledge of probability.

Docente(s)

• Margaret Armstrong
• Alain Galli

Organisation

Each afternoon is split into 3 sessions with a 15 minute break in between:

• 2 x 1 hour lectures
• Computer practical session using Matlab
• Textbook: Options, Futures and other Derivatives by John Hull

Ementa

The course is split into three broad sections: finance, stochastic processes, and numerical methods.

After describing the main types of financial derivatives (futures, options, puts & calls, Amercian versus European options etc), we present the concept on “no arbitrage” and will use it to derive the put-call parity but in order to determine the value of a derivative, we need to know how the price of the underlying asset varies over time.

Different types of stochastic processes can be used to represent these fluctuations, but in finance the main ones are Itô diffusion processes such as geometric Brownian motion. On the mathematical side, the course will cover filtrations, martingales, Itô’s formula & Girsanov’s theorem.

There are two broad ways of evaluating (i.e. pricing options): by non-arbitrage and using the Feynman-Kac theorem. In some cases, these lead to closed form expressions which can be evaluated immediately, but in most cases, numerical methods such Monte Carlo simulations or binomial trees.

Several of the computer practical classes will be devoted to simulations: first generating random number, random variables and sample paths for the common stochastic processes. When using simulations it is important to know how to speed up their convergence. So the topics of variance reduction (control variates, antithetic sampling, stratified sampling, Latin hypercubes, etc), quasi- Monte-Carlo methods (Halton & Sobol sequences) and discretisation schemes will be treated. Finally the Longstaff-Schwartz method will be explained.

These numerical methods will be used to evaluate different types of options on interest rates, foreign exchange rates, and commodities. Another important concept that will be taught is delta hedging. Other topics such as the risk-free rate, bonds, term structure, rating agencies, historic versus implied volatility, stochastic volatility, variance swaps and credit risk derivatives will be treated briefly.

Bibliografia