stochastic finance

After this course, the student will be familiar with... .... the basics of probability theory and stochastic calculus .... no‑arbitrage pricing techniques .... elementary portfolio optimization techniques Learning goals of the course: Learning goal 1: The student is familiar with the basics of probability theory and stochastic calculus. Learning goal 1a: The student is able to work with several standard probability distributions. Learning goal 1b: The student is able to use Itoʹs lemma and the associated rules. Learning goal 1c: The student is able to identify and to manipulate stochastic differential equations. Learning goal 2: The student is familiar with no‑arbitrage pricing techniques. Learning goal 2a: The student understands vanilla European style contracts and its different elements (strike, maturity, price). Learning goal 2b: The student is able to price vanilla European style derivatives using the binomial model. Learning goal 2c: The student is able to price vanilla European style derivatives using the Black‑Scholes model. Learning goal 3: The student is familiar with elementary portfolio optimization techniques. Learning goal 3a: The student is able to use the Markowitz approach for portfolio optimization. Learning goal 3b: The student is able to formulate the portfolio optimization problem with various constraints (minimum return, risk, shortfall, etc). Learning goal 3c: The student is able to formulate similar portfolio optimization problems in continuous time.

Course content The course introduces the fundamental stochastic tools for derivative asset pricing and portfolio theory. The ability to price and hedge derivative products and to properly manage asset portfolios is of paramount importance in the financial industry. The existing techniques to carry this out require a good command of the concepts in stochastic calculus that will be presented in this course. Qualifications associated to the course: The material that will be presented is the first step that needs to be taken by any student interested in quantitative finance. This course is a good educational block both for future practitioners in the financial industry and also for those who want to pursuit further studies in the applications of stochastic methods in finance and in financial econometrics. Methods applied in the course: The two main mathematical methods used in this course are probability theory and stochastic calculus. These tools allow for a mathematically rigorous formulation of the hypotheses underlying the no‑arbitrage approach to asset pricing and that yield explicit quantitative results that can be easily used in practice. Course structure 1. Probability Theory Introduction Distribution functions Normal distribution Multivariate normal distribution Lognormal distribution Binomial distribution 2. Pricing and No‑arbitrage Binomial model Fundamental asset pricing theorem 3. Itoʹs lemma and Stochastic Integrals Random walk and Brownian motion Ito processes and Ito lemma Derivative pricing Partial differential equations Stochastic differential equations 4. Risk Neutral Valuation Discrete model Lognormal model Extensions 5. Markowitz Portfolio Theory Markowitz approach Asset liability approach Shortfall constraint 6. Arbitrage Pricing Theory Model Discussion of the model Mathematics properties of the model 7. Portfolio Theory in Continuous Time

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