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.



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).