Financial Mathematics


From 2016.11.21 to 2017.01.06


Time: from 13:30 to 16:30

Place: TBA

Teaching Assistant: Yujie Jiao - 焦毓杰

Course Objective

The objective of this course is to introduce the mathematics needed for

Stochastics as a mathematical field developed parallel to the emergence of the finance industry (first insurance, then stock markets, derivatives, etc…). This lecture aims at being an introduction to the mathematics needed in modern finance. We cover the arbitrage theory, pricing of financial assets such as European and American options as well as the optimal portfolio selection. We also address modern risk management theory such as risk measures and risk allocations which are nowadays of paramount importance in the financial industry. We illustrate the application of these theories by implementing some classical numerical pricing methods.

Practical Organization

Content of the lecture

  1. Introduction to mathematical finance.
  2. Financial model in the static case.
    • Model (Stocks, Bonds, strategies, probability theory).
    • No Arbitrage theory, fundamental theorem of asset pricing.
    • Pricing of derivatives, law of one price, complete markets.
    • Applications, implementation.
  3. Optimization.
    • Unconstrained optimization.
    • Constrained optimization, Lagrangian.
    • Applications, implementation.
  4. Risk Management.
    • Mean-Variance.
    • Value at risk.
    • Monetary risk measures.
    • Optimized certainty equivalent, average value at risk.
    • Applications, implementation.
  5. Dynamical Financial market.
    • Model (filtration, conditional expectation, adaptiveness of stocks and predictability of strategies).
    • Fundamental theorem of asset pricing.
    • Dynamic pricing of european contingent claims.
    • Cox-Ross-Ingersoll binomial model, pricing, hedging.
    • Application, implementation.
  6. American option pricing and ruin probability in CRR model.
    • American options model.
    • Snell Envelope.
    • Optimal stopping exercise for buyer (stopping time, Doob’s theorem).
    • Application, implementation.
    • Ruin probability, unbiased, biased, in CRR and link to stopping times.
    • Application, implementation.
  7. Continuous time financial markets, Black and Scholes Formula.
    • Presentation on how the discrete time model converges to Brownian motion.
    • Stochastic integral intuitively (without proof).
    • Ito’s formula, stochastic differential equations.
    • Continuous financial model, Girsanov, link to PDE.
    • Pricing of european contingent claims, Black and Scholes formula.