Syllabus
1. Basic financial contracts (options and futures), assets, price of an asset relative to another reference asset. Portfolio, portfolio value and development of self-financing portfolio.
2. Arbitrage, martingales and martingale measures and the First Fundamental Theorem of Asset pricing. Change of the numeraire.
3. Binomial model of price evolution, valuation and replication of financial contracts in a binomial model.
4. Diffusion models. Stochastic integration. Geometric Brownian motion. Stochastic differential equation.
5. Girsanov's theorem and martingale measures in diffusion models. Completeness of the market, Second Fundamental Theorem of Asset Pricing.
6. Representation of a continuous martingale by a stochastic integral, hedging and replication.
7. Black-Scholes formula. Valuation of options. Feynman-Kac formula, BS equation, replication strategy for simple claims.
8. Applications to real financial data. Automatic processing of financial data, valuation of contracts in real time.
9. Exchange rates, contracts on currencies.
10. Interest rate contracts. LIBOR, forward LIBOR, floorlets, caplets, swaps, swap rate and swaptions.
11. Forward interest rate, Heath-Jarrow-Morton model. Immediate interest rate application (Vašíček, Cox-Ingersoll-Ross).
Annotation
This course covers modern finance theory based on the no-arbitrage principle. In order to prevent an existence of a risk-free profit for any market agent, the prices must be martingales under the probability measures corresponding to the reference asset.
As a consequence, the prices of financial contract must satisfy certain partial differential equations in the case of diffusion models. The course illustrates these results for the most common financial contracts in various markets, such as stock, interest rate and exchange rate markets.
Examples of real data analysis using Python are given.