MTH3251 · Financial mathematics

Overview

You will use the concept of random variables and their uses as models of uncertain future payoffs. An important concept for analysis is the conditional expectation. Special attention is given to normal distribution and multivariate normal distribution, in which explicit calculations are possible. Systems evolving in time encorporating uncertainty are modelled as stochastic (random) processes. Examples of such in discrete time are Random Walk and Martingales. As an application we look at the Risk model in insurance and obtain the bound on Ruin probability. Models in continuous time are based on Brownian motion. Stochastic analysis uses the novel concepts of Ito integral and Ito's formula. Applications in finance include the Black-Scholes model and the Ornstein-Uhlenbeck process. Simple stochastic differential equations are introduced. Another application to interest rates is Vasicek's stochastic differential equation. A new mathematical technique Change of probability measure is introduced. Girsanov theorem gives the change of measure for Brownian motion and related processes. To manage financial risks the Fundamental theorems of Asset pricing are stated and applied to various models, such as the . Binomial and Black-Scholes models. Important concepts of arbitrage, replicating portfolios are used for pricing and hedging options and other financial contracts.

Areas of study: Applied mathematics Financial and insurance mathematics Mathematical statistics Mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Analyse discrete-time models in finance by applying random walks, martingales, conditional expectation, and stopping times, and using these to study applications such as insurance and ruin probabilities;
2. Interpret and model continuous-time processes including Brownian motion and diffusions, and use stochastic calculus tools such as Ito’s formula and stochastic differential equations to solve problems in financial mathematics;
3. Apply measure-change and asset-pricing frameworks by using the Equivalent Martingale Measure, implementing the Binomial, multi-period models, and the Black–Scholes models, and applying the fundamental theorems of asset pricing to problems of pricing and hedging;
4. Communicate mathematical reasoning and results effectively through clear oral and written explanations, and collaborate in small groups to solve problems in financial mathematics.

Workload

• Three 1-hour seminars; • One 2-hour applied class (in weeks 2-12) and • 7 hours of independent study per week.

Student experience, not official rules. Nothing here changes what the Handbook says.