## Module overview

Following an initial discussion of the assessment and measurement of investment risk, mean-variance portfolio theory is introduced and used to determine the risk and return for a portfolio of risky assets, the composition of the optimal such portfolio, and the location of the efficient frontier. Single- and multifactor models of asset returns are then introduced and, in conjunction with concepts from mean variance portfolio theory, lead to the establishment of equilibrium asset pricing models, such as the Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT). The various forms of the Efficient Markets Hypothesis are discussed against this background.

Attention then turns to stochastic models for security prices, such as geometric Brownian motion, and to the essential mathematical tool required for analysis and solution of the underlying stochastic differential equations, namely the Ito calculus, and, in particular, the Ito integral and the Ito formula. With such a stochastic model for the underlying random variable, it is possible to develop a model for the valuation of a derivative security whose price is contingent on this underlying random variable, and this is a central aspect of the module.

The approach to derivative security pricing, and, in particular, option pricing, is built up in stages: first, the discrete-time binomial lattice approach is used; next, the continuous-time Black-Scholes approach is used; and finally, following the introduction of concepts such as martingales and risk-neutral measures, the martingale approach, or, equivalently, the state-price deflator approach is used. Calculations of option prices are extended to the partial derivatives of such prices, the so-called Greeks, and the role of such partial derivatives in the risk

management of a portfolio of derivative securities is described.

Finally, the risk-neutral and state-price deflator approaches are applied to the pricing of zero-coupon bonds and interest rate derivatives for general single-factor diffusion models of the risk-free rate of interest, such as those of Vasicek, Coss, Ingersoll, and Ross, and Hull and White.

### Linked modules

Pre-requisite: MATH6131