# Mathematics

## ECMM706 - Mathematical Theory of Option Pricing (2018)

MODULE TITLE CREDIT VALUE Mathematical Theory of Option Pricing 15 ECMM706 Prof Mark Holland (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 0 11 weeks 0
 Number of Students Taking Module (anticipated) 32
DESCRIPTION - summary of the module content

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.

For MSc students: ECMM702, ECMM703.

For other Level 7 students: ECM1705, ECM1707, ECM1709, ECM2702.

AIMS - intentions of the module

By taking this module, 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.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:
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.
Discipline Specific Skills and Knowledge:
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.
Personal and Key Transferable/ Employment Skills and  Knowledge:
8 demonstrate enhanced problem solving skills and the ability to use the sophisticated computer package Matlab.

SYLLABUS PLAN - summary of the structure and academic content of the module

- 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.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 33 117
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 33 Lectures Guided independent study 117 Lecture and assessment preparation; wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Assessments x 4 10 hours each All Verbal

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours - Summer Exam Period 1-7 In accordance with CEMPS policy
Coursework – assignment 1 10 10 hours 1-8 Written/tutorial
Coursework – assignment 2 10 10 hours 1-8 Written/tutorial

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August 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 50% 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.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk