Aim of this course is to cover some of the most important topics of Mathematical Finance in continuous time involving techniques related to Stochastic Calculus and dynamical optimization.
Expected learning outcomes
Pricing and hedging using probabilistic/analytic methods, of financial derivatives in complete/incomplete markets, described by diffusion time-continuous processes. Resolution of some problems concerning dynamic optimization, using optimal control/stopping methods.
I Stochastic Processes On the Brownian Motion (BM) and continuous time martingales. The exponential martingale. The BM paths are not of finite variation. Quadratic variation of the BM. The stochastic integral w.r.to the BM and the martingale property. Ito processes and Ito formula. Novikov Lemma and Girsanov Theorem. The predictable representation property. Examples of stochastic differential equations.
II Continuous time stochastic models and the Black and Scholes model Description of the BS model and its analysis in the light of the general principles of asset pricing. Verification of the existence of an equivalent martingale measure. The BS formula and its derivation (probabilistic method and analytical method). The BS partial derivatives equation and boundary conditions. Contingent claims valuation in incomplete markets in continuous time. Evaluation with non-negotiable financial instruments: the partial differential equation and the market price of risk. The problem of calibration: implied volatility and "smile" effect. Limits of the BS model. Outlines of models with local volatility and stochastic volatility.
III Introduction to stochastic optimization problems. Examples of deterministic/stochastic optimization problems in Finance. Mathematical formulation of the problem. Admissible controls, the Dynamic Programming Principle, the Hamilton-Jacobi-Bellman equation, the Verification Theorem. Applications to Finance. Merton's problem over a finite horizon, Linear Quadratic Gaussian control and Systemic Risk, optimal investment-consumption with random time horizon (hints).
IV Pricing of exotic and american options Lookback options and Asiatic options. American Opzions: optimal stopping problem. Pricing of American Perpetual Put Options: probabilistic approach. Pricing of American Put Options and American Call Options with finite expiration.