### Stochastic Analysis and Mathematical Finance

The lecture covers the following topics:

**Financial markets**Financial markets, derivatives, call and put options, arbitrage, put-call parity, model free arbitrage bounds.

**Asset pricing in finite economies**Numeraire, self-financing portfolios, discrete stochastic integral, equivalent martingale measures, fundamental theorems of asset pricing, law of one price, replication, risk neutral pricing.

**Stochastic analysis**Brownian motion, predictability, finite variation processes, Stieltjes integral, square integrable martingales, local martingales, quadratic variation, semimartingales, Ito isometry, stochastic integral, Ito’s formula.

**Continuous time financial markets**

Trading strategies, wealth processes, arbitrage, admissible strategies, equivalent martingale measures, replication, efficient strategies, risk neutral pricing, complete markets.

**The Black-Scholes model**Levy’s characterization, Girsanov’s theorem, Novikov’s condition, market price of risk, martingale representation, Black-Scholes model, Delta hedging, implied volatility, stochastic differential equations, diffusion markets, Feynman-Kac representation.

#### Resources

#### Simulations

QuadraticVariationBM.m: Discretized Total Variation (black) and Quadratic Variation (red) of a Brownian Motion (blue).

QuadraticVariationStochInt.m: Discretized Total Variation (black) and Quadratic Variation (red) of a stochastic Integral (blue) of a Fractional Brownian Motion (blue dotted) integrated with respect to a Brownian Motion. Requires the function **fbm1d**.

SimpleIntegral.m: Stochastic Integral (red) of a Simple Integrand (black) with respect to a Brownian Motion (blue).