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

MODULE TITLE | Mathematical Theory of Option Pricing | CREDIT VALUE | 15 |
---|---|---|---|

MODULE CODE | ECMM706 | MODULE CONVENER | Prof Mark Holland (Coordinator) |

DURATION: TERM | 1 | 2 | 3 |
---|---|---|---|

DURATION: WEEKS | 0 | 11 weeks | 0 |

Number of Students Taking Module (anticipated) | 32 |
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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.**

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.

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.

- 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.

Scheduled Learning & Teaching Activities | 33.00 | Guided Independent Study | 117.00 | Placement / Study Abroad |
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Category | Hours of study time | Description |

Scheduled learning and teaching activities | 33 | Lectures |

Guided independent study | 117 | Lecture and assessment preparation; wider reading |

Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
---|---|---|---|

Assessments x 4 | 10 hours each | All | Verbal |

Coursework | 20 | Written Exams | 80 | Practical Exams |
---|

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 |

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 |

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.

information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|

Set | Wilmott P., Howison S. & Dewynne J. | The Mathematics of Financial Derivatives: A Student Introduction | Cambridge University Press | 1995 | 000-0-521-49699-3 | [Library] | |

Set | Higham, D.J. | An Introduction to Financial Option Valuation - Mathematics, Stochastics and Computation | Cambridge Univeristy Press | 2004 | 0-521-54757-1 | [Library] | |

Set | Etheridge, A. | A Course in Financial Calculus | Cambridge University Press | 2002 | 0-521-89077-2 | [Library] | |

Set | Baxter, M. and Rennie, R. | Financial Calculus: An Introduction to Derivative Pricing | Cambridge University Press | 1996 | 0-521-55289-3 | [Library] |

CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
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PRE-REQUISITE MODULES | ECMM702 |
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CO-REQUISITE MODULES |

NQF LEVEL (FHEQ) | 7 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Thursday 06 July 2017 | LAST REVISION DATE | Wednesday 27 February 2019 |

KEY WORDS SEARCH | Option pricing; financial mathematics; financial derivatives; schochastic calculus; financial option valuation. |
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