Stochastic Financial Models(D)

STOCHASTIC FINANCIAL MODELS (D)
Methods, Statistics, Probability and Measure, and Markov Chains are desirable.

Utility and mean-variance analysis

Utility functions; risk aversion and risk neutrality. Portfolio selection with the mean-variance criterion; the eﬃcient frontier when all assets are risky and when there is one riskless asset. The capital-asset pricing model. Reservation bid and ask prices, marginal utility pricing. Simplest ideas of equilibrium and market clearning. State-price density. [5]

Martingales

Conditional expectation, deﬁnition and basic properties. Conditional expectation, deﬁnition and basic properties. Stopping times. Martingales, supermartingales, submartingales. Use of the optional sampling theorem. [3]

Dynamic models

Introduction to dynamic programming; optimal stopping and exercising American puts; optimal portfolio selection. [3]

Pricing contingent claims

Lack of arbitrage in one-period models; hedging portfolios; martingale probabilities and pricing claims in the binomial model. Extension to the multi-period binomial model. Axiomatic derivation. [4]

Brownian motion

Introduction to Brownian motion; Brownian motion as a limit of random walks. Hitting-time distributions; changes of probability. [3]

Black-Scholes model

The Black-Scholes formula for the price of a European call; sensitivity of price with respect to the parameters; implied volatility; pricing other claims. Binomial approximation to Black-Scholes. Use of ﬁnite-diﬀerence schemes to compute prices [6]

Appropriate books

J. Hull Options, Futures and Other Derivative Securities. Prentice-Hall 2003
J. Ingersoll Theory of Financial Decision Making. Rowman and Littleﬁeld 1987
A. Rennie and M. Baxter Financial Calculus: an introduction to derivative pricing. Cambridge University Press 1996
P. Wilmott, S. Howison and J. Dewynne The Mathematics of Financial Derivatives: a student introduction. Cambridge University Press 1995