## MAS6053 Financial Mathematics

 Semester 1, 2019/20 10 Credits Lecturer: Dr Dimitrios Roxanas uses MOLE Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

The discovery of the Capital Asset Pricing Model by William Sharpe in the 1960's and the Black-Scholes option pricing formula a decade later mark the beginning of a very fruitful interaction between mathematics and finance. The latter obtained new powerful analytical tools while the former saw its knowledge applied in new and surprising ways. (A key result used in the derivation of the Black-Scholes formula, Ito's Lemma, was first applied to guide missiles to their targets; hence the title `rocket science' applied to financial mathematics). This course describes the mathematical ideas behind these developments together with their applications in modern finance.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

## Outline syllabus

• Introduction, arbitrage, forward and futures contracts
• Options, binomial trees, risk-neutral valuation
• Brownian motion and share prices, the Black-Scholes analysis
• Portfolio theory, the Capital Asset Pricing Model.

## Office hours

Tuesday 14.00-15.00 in H13. If this time is not convenient, e-mail me to set an appointment.

## Aims

• To introduce students to the mathematical ideas and methods used in finance.
• To familiarise students with financial instruments such as shares, bonds, forward contracts, futures and options.
• To familiarise students with the notion of arbitrage and the notion of no-arbitrage pricing.
• To introduce the binomial tree and geometric Brownian motion models for stock prices.
• To familiarise students with the Black-Scholes option pricing method.
• To introduce the Capital Asset Pricing Model.

## Learning outcomes

• understand basic mathematical/statistical ideas and methods used in Finance
• familiarise with financial instruments such as shares, bonds, forward contracts, futures and options; understand the notion of portfolio and its value as a function of time
• understand the notion of arbitrage and the notion of no-arbitrage pricing; be in a position to price simple financial instruments using no-arbitrage arguments
• understand the time value of money, this includes being comfortable with "present value" and discount factors, as well as applications such as using the bootstrapping method to calculate yields (spot interests)
• understand the principle of risk-neutral valuation and use binomial trees to price simple derivatives
• understand the modelling ideas behind using geometric Brownian motion as a model for stock prices; understand the notion of an Ito Process and be able to apply Ito's formula
• understand the derivation of the Black-Scholes partial differential equation and how it is applied to option pricing method; understand properties of the PDE (especially linearity) and how they come in pricing portfolios;
• get a feeling of how to quantify uncertainty in a financial setting; understand the basics of Modern Portofolio Theory, and the Capital Asset Pricing Model. Be able to compute expected returns and variances of portfolios, covariances and correlations between assets. Be able to explain the notions of indifference curves, efficient frontier, market portfolio, beta of an asset, capital market line, and use algebraic and geometric arguments to construct optimal portfolios.

## Teaching methods

In addition to lectures and office hours, we will have an (optional) tutorial every few weeks, on Wednesday 16.00-17.00, in LT07. Tentative dates: 2/10, 16/10, 30/10, 27/11, 11/12, 18/12.

20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination [100%]. Format: 4 questions from 4. One of the questions will be on additional material that will be given to the students for self-learning.

## Full syllabus

Interest rates, bonds and yield curves. (2 lectures)

Forward and Futures contracts. (3 lectures)
Options. (3 lectures)
Binomial trees and risk neutral valuation. (2 lectures)
Review of probability. (1 lecture)
The stochastic process followed by stock prices. (2 lectures)
The Black-Scholes pricing formulas. (2 lectures)
Portfolio theory. (2 lectures)
The Capital Asset Pricing Model. (3 lectures)