# Mathematical Theory of Option Pricing

## Module description

On this module you will be expected to study stochastic models of finance including the Black Scholes option pricing model.  You will have the opportunity to study numerical methods in order to solve partial differential equations. The module applies the mathematical and computational material in ECMM702 and ECMM703 to a central problem in finance, that of option pricing.

Prerequisite module: ECMM702 and either of ECMM703, ECMM712 or equivalent

## Full module specification

Module title: Mathematical Theory of Option Pricing ECMM706 M 2014/5 Dr Mark Holland - Convenor 15 7.5 Prerequisite module: ECMM702 and either of ECMM703, ECMM712 or equivalent Duration (weeks) - term 1: 0Duration (weeks) - term 2: 11 weeksDuration (weeks) - term 3: 0

## Module aims

You will gain an understanding of the theoretical assumptions on which the mathematical models underlying option pricing depend, and of the methods used to obtain analytic or numerical solutions to a variety of option pricing problems.

## ILO: Module-specific skills

• 1. comprehend the mathematical theories needed to set up the Black-Scholes model;
• 2. understand the role of Ito's calculus in the Black-Scholes PDE;
• 3. transform the Black-Scholes PDE to the heat diffusion equation;
• 4. analyse and derive the solution of the Black-Scholes PDE for the standard European put/call options.

## ILO: Discipline-specific skills

• 5. derive rigorously a quantitative model from a set of basic assumptions;
• 6. use the solution to the mathematical model to predict the behaviour of the option price;
• 7. relate and transform one PDE to a simpler type.

## ILO: Personal and key skills

• 8. demonstrate enhanced problem solving skills and the ability to use the sophisticated computer package Matlab.

## Learning activities and teaching methods (given in hours of study time)

Scheduled Learning and Teaching ActivitiesGuided independent studyPlacement / study abroad
331170

## Details of learning activities and teaching methods

CategoryHours of study timeDescription
Scheduled learning and teaching activities 33Lectures/tutorials
Guided independent study117Lecture and assessment preparation; wider reading

## Formative assessment

Form of assessmentSize of the assessment (eg length / duration)ILOs assessedFeedback method
Assessments x 410 hours eachAllVerbal

## Summative assessment (% of credit)

CourseworkWritten examsPractical exams
20800

## Details of summative assessment

Form of assessment% of creditSize of the assessment (eg length / duration)ILOs assessedFeedback method
Written exam  closed book802 hours1-7In accordance with CEMPS policy
Coursework  assignment 11010 hours1-8Written/tutorial
Coursework  assignment 21010 hours1-8Written/tutorial

## Details of re-assessment (where required by referral or deferral)

Original form of assessmentForm of re-assessmentILOs re-assessedTimescale for re-assessment
All aboveWritten exam (100%)AllAugust Ref/Def period

## Re-assessment notes

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.
If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

## Syllabus plan

- financial concepts and assumptions: risk free and risky asset;
- options;
- no arbitrage principle;
- put-call parity;
- discrete-time asset price model: option pricing by binomial method;
- interpretation in terms of risk-neutral valuation;
- continuous time stochastic processes: Brownian motion, stochastic calculus;
- Ito’s lemma and construction of the Ito integral;
- Black-Scholes theory: geometric Brownian motion;
- derivation of the Black-Scholes PDE; transformation to the heat equation;
- explicit formulae for vanilla European call and put options;
- extensions, e.g. to dividend-paying assets and American options;
- numerical methods: finite difference schemes for the Black-Scholes PDE, including comparison of stability and accuracy;
- overview of risk-neutral valuation approach in continuous time.

## Indicative learning resources - Basic reading

ELE – http://vle.exeter.ac.uk

1. The Mathematics of Financial Derivatives: A Student Introduction,Wilmott P., Howison S. & Dewynne J.,,Cambridge University Press,1995,332.632 WIL,000-0-521-49699-3,
2. An Introduction to Financial Option Valuation - Mathematics, Stochastics and Computation,Higham, D.J.,,Cambridge Univeristy Press,2004,332.645 HIG,0-521-54757-1,
3. A Course in Financial Calculus,Etheridge, A.,,Cambridge University Press,2002,332.6322 ETH,0-521-89077-2,
4. Financial Calculus: An Introduction to Derivative Pricing,Baxter, M. and Rennie, R.,,Cambridge University Press,1996,332.6322 BAX,0-521-55289-3,

Yes

19/11/12

19/11/12