From 2021.11.01 to 2022.01.12

Place: TBA

Teaching Assistant: Tao Xuan

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

- pricing of financial assets, such as options and other derivatives;
- optimal portfolio choice;
- modern risk management;

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.

- Blackboard lecture in English.
- Every week, a lecture note with the content of the lecture will be provided.
- Every week, a homework sheet has to be handed by group of 5 to the teaching assistant and will be graded.
- Every week, the teaching assistant will give a tutorial (correction of the homework sheet, review of some theoretical points of the lecture, etc.).
- Mid term and final exam.
- Prerequisite: A good knowledge of advanced calculus, linear algebra and basic probability theory.

- Introduction to mathematical finance.
- 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.

- Optimization.
- Unconstrained optimization.
- Constrained optimization, Lagrangian.
- Applications, implementation.

- Risk Management.
- Mean-Variance.
- Value at risk.
- Monetary risk measures.
- Optimized certainty equivalent, average value at risk.
- Applications, implementation.

- 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.

- 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.