# Stochastic modelling and derivatives

H. Pham et M. Garcin

This course provides an overview of stochastic modeling of financial markets. The aim is to understand the foundations of arbitrage theory for the valuation of derivatives, and to introduce the concepts of financial risk management.

Organization on the course website

I - **Introduction**

- Derivatives: futures, European and American options
- The role of models in finance
- Static arbitrage, dominance principle, put-call parity, American call and put price bounds.
- Principle of option valuation. Example of the binomial model.

II - **Discrete-time model**

- Financial assets model
- Martingale and arbitrage
- Valuation of contingent assets
- Complete markets
- Cox-Ross-Rubinstein model and its limit in continuous time: Black-Scholes formula

III - **Modeling in continuous time**

A. **Black-Scholes model and properties**

- Geometric Brownian model
- Black-Scholes formula
- Sensitivity and Greeks
- Formula robustness
- Discrete-time coverage
- Delta-Gamma coverage
- Formula robustness
- Historical and implied volatility

B. **General principles**

- Wealth process
- Arbitrage and risk-neutral probability
- Complete/incomplete market
- Arbitrage valuation and hedging

C. **Valuation of exotic options**

- Change of numeraire
- Multi-underlying options, exotic options, quanto options
- Static replication, variance swap, VIX

IV - **Alternatives to lognormal model**

- Implicit distribution
- Implied volatility
- Local volatility models
- Dupire formula
- CEV model
- Stochastic volatility models
- Heston model, Fourier pricing
- SABR model

V - **Market data and statistics**

- Market data, order book
- Reference price vs. index
- Robust statistics
- Bid-ask spread for options
- Bid-ask spread
- Hedging with transaction costs

VI - **Derivatives on crypto-markets**