Synopsis for Asset Pricing and Portfolio Theory

Further details

(Dr J Ruf - 8 lectures - MT - Core course)

Overview

The course imparts, in a simple setting of finite discrete markets, the main ideas underlying:

• The modelling a financial market with stock prices described by a vector stochastic process S on a probability space
• Preferences towards risk and return via mean-variance methods and utility functions;
• Optimal investment and consumption via utility maximisation;
• A simple model of equilibrium, exemplified by the Capital Asset Pricing Model (CAPM);
• Contingent claim pricing: what price to charge at time zero for a contract paying a non-negative random payoff Y at time T, where Y depends on the evolution of S; how to mitigate the risk from selling such a contract by judicious trading in S (hedging), distinguishing complete and incomplete markets;
• Equivalent martingale measures Q and their role in derivative pricing and computation of optimal investment rules;
• The connection between equivalent martingale measures and no-arbitrage (NA), culminating in simple versions of the First and Second Fundamental Theorems of Asset Pricing

Synopsis

Examples of probabilistic models of financial markets; discrete and continuous models; finite and infinite models; single period models; finite discrete-time markets; binomial models. Risk and return; Markowitz mean-variance framework for portfolio selection. Sharpe-Lintner Capital Asset Pricing Model - a simple equilibrium model. Utility functions and risk aversion; primal and dual methods for utility maximisation problems in single period models and finite discrete models; utility maximisation in a binomial model as an example; the role of equivalent martingale measures. Contingent claim pricing in finite discrete markets; perfect hedging; risk-neutral valuation; complete and incomplete markets. No-arbitrage and equivalent martingale measures; fundamental theorems of asset pricing in discrete time.

1. J Cvitanic J and F Zapatero, Introduction to the Mathematics and Economics of Financial Markets, MIT Press 2004

Alternative texts

1. SE Shreve, Stochastic Calculus for Finance I:The Binomial Asset Pricing Model, Springer-Verlag 2004
2. SE Shreve, Stochastic Calculus for Finance II: Continuous Time Models, Springer-Verlag 2004
3. H Foellmer and A Schied, Stochastic Finance: An Introduction in Discrete Time, 2nd Edition, Walter de Gruyter 2004
4. SR Pliska, Introduction to Mathematical Finance: Discrete Time Models, Blackwell 1997