43 Mathematics of finance and valuation (half unit)

Prerequisites

If taken as part of a BSc degree, 05a Mathematics 1 and 05B Mathematics 2 and
116 Abstract mathematics.

Aims and objectives

The unit is designed to introduce the main mathematical ideas involved in the
modelling of asset price evolution and the valuation of contingent claims (such as
call and put options) in a discrete and a continuous framework.
Learning outcomes
Having followed this unit, students should have
? knowledge, understanding and formulation of the principles of risk-neutral
valuation including some versions of the No-Arbitrage Theorem
? knowledge of replication and pricing of contingent claims in certain simple
models (discrete and continuous)
? knowledge of the derivation of the Black-Scholes equation, its solution in
special cases, the Black-Scholes formula.
Syllabus
This is an introduction to an exciting and relatively new area of mathematical
application. It is concerned with the valuation (pricing) of ‘financial derivatives’.
These are contracts which are bought or sold in exchange for the promise of some
kind of payment in the future, usually contingent upon the share-price then
prevailing (of a specified share, or share index).
The unit reviews the financial environment and some of the financial derivatives
traded on the market. It then introduces the mathematical tools which enable the
modelling of the fluctuations in share prices. Inevitably these are modelled by
equations containing a random term. It is this term which introduces risk; it is
shown how to counterbalance the risks by putting together portfolios of shares and
derivatives so that risks temporarily cancel each other out and this strategy is
repeated over time. As this procedure resembles hedging a bet – that is, betting
both ways - one talks of dynamic hedging. The yield of a temporarily riskless
portfolio is equated to the rate of return offered by a safe deposit bank account
(that is a riskless bank rate) which is assumed to exist; this equation assumes that
the market which values shares and derivatives actually is in equilibrium and hence
eliminates the opportunities of ‘arbitrage’ (such as making sure profit from, say,
buying cheap and selling dear).
The no-arbitrage approach implies in the continuous time model that the price of a
derivative is the solution of a differential equation. One may either attempt to solve

the differential equation by standard means such as numerical techniques or via
Laplace transforms, though this is not always easy or feasible. However, there is an
alternative route which may provide the answer: a calculation of the expected
payment to be obtained from the contract by using what is known as the synthetic
probability (or the risk-neutral probability. One proves that, regardless of what an
investor believes the expected growth rate of the share price to be, the dynamic
hedging acts so as to replace the believed growth rate by the riskless growth rate.
Though this may seem obvious in retrospect it does require some careful reasoning
to justify.
The unit considers two approaches to risk-neutral calculation, using discrete time
and using continuous time. Continuous time requires the establishment of a
second-order volatility correction term when using standard first-order
approximation from calculus. This leads to what is known as Ito’s Rule. Finite time
arguments need some apparatus from Linear Algebra like the Separating
Hyperplane Theorem. We enter the subject from the discrete time model for an
easier discussion of the main issues.
Essential reading
Hull, J.C. Options, Futures and other Derivatives. (Prentice Hall, 2005) sixth
edition [ISBN 978-0131499089].
Pliska, S.R. Introduction to Mathematical Finance: Discrete Time Models
(Blackwell, 1997) [ISBN 978-1557869456].
Roman, S. Introduction to Mathematicsof Finance. (Undergraduate Texts in
Mathematics, 2004, Springer) [ISBN 978-0387213644].
All information in this document is subject to confirmation in the Programme Regulations for degrees and
diplomas in Economics, Management, Finance and the Social Sciences that are reviewed annually.