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# Computational Finance - MAT00069M

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• Department: Mathematics
• Module co-ordinator: Dr. Eric Dykeman
• Credit value: 10 credits
• Credit level: M
• Academic year of delivery: 2018-19

## Module summary

Stochastic Processes can also be used as a pre-requisite for this module, if you have not studied Mathematical Methods of Finance.

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## Module will run

Occurrence Teaching cycle
A Autumn Term 2018-19 to Spring Term 2018-19

## Module aims

The module aims to provide students with knowledge and skills to implement (in Matlab) and find the numerical solutions of the models arising in the area of mathematical finance.

The module is going to introduce a number of different numerical techniques suitable to the models arising in mathematical finance, including, but not limited to, trees, finite difference, Fourier transform and Monte Carlo simulation approaches. All numerical techniques will be introduced and tested in either risk management or derivatives pricing context. Calibrating models to market data will be introduced at the end of the course as well.

## Module learning outcomes

At the end of the module you should be able to...

• Understand and be able to construct Binomial and Trinomial Trees to approximate a continuous time stochastic process which may be described by a stochastic differential equation;
• Understand and be able to implement finite difference schemes, including explicit, implicit and Crank-Nicolson schemes to solve the Black-Scholes partial differential equation without and with time or state dependent coefficients;
• Understand and be able to compute the characteristic function of a random variable and be able to apply FFT and Fourier Cosine Expansion method;
• Understand and be able to implement Monte Carlo simulation scheme starting with generating random numbers from certain distributions, also being able to discretize and simulate stochastic differential equations;
• Be able to apply variance reduction methods in Monte Carlo Simulation, such as antithetic variates, control variates and importance sampling;
• Be able to compute the prices and hedge ratios of European options under Black-Scholes framework using each of the above approaches;
• Be able to compute the prices and hedge ratios of American options under Black-Scholes framework using tree and finite difference schemes;
• Understand the calibration technique and be able to calibrate the Black-Scholes model to market option prices.

• Master a number of different numerical techniques specific to models in finance and also learn to use Matlab as a computing language. Those skills are of course transferable to different areas of applied mathematics.

## Module content

Syllabus:

• Binomial and Trinomial Trees to approximate a continuous time stochastic process which may be described by a stochastic differential equation.
• Finite difference schemes, including explicit, implicit and Crank-Nicolson schemes, to solve the Black-Scholes partial differential equation without and with time or state dependent coefficients;
• The characteristic function of a random variable and FFT and Fourier Cosine Expansion method;
• Monte Carlo simulation scheme including generating random numbers from certain distributions, discretizing and simulating stochastic differential equations and a number of variance reduction methods, such as antithetic variates, control variates and importance sampling;
• Computation of the prices and hedge ratios of European options under Black-Scholes framework using each of the above approaches;
• Computation of the prices and hedge ratios of American options under Black-Scholes framework using tree and finite difference schemes;
• Calibration technique and the calibration of the Black-Scholes model to market option prices.

## Assessment

Task Length % of module mark
Practical
Matlab practical project
N/A 40
University - closed examination
Computational Finance
2 hours 60

### Special assessment rules

None

Only the failed components of the module shall be reassessed.

### Reassessment

Task Length % of module mark
Practical
Matlab practical project
N/A 40
University - closed examination
Computational Finance
2 hours 60

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

A. Hirsa: Computational Methods in Finance, Chapman & Hall/CRC Financial Mathematics Series, 2012

M. Aichinger, A. Binder: A Workout in Computational Finance, Wiley, 2013

L. Clewlow, C. Strikland: Implementing Derivatives Models, Wiley Series in Financial Engineering, 1998

H. Wang: Monte Carlo Simulation with Applications to Finance, Chapman & Hall/CRC Financial Mathematics Series, 2012

C. Chiarella, B Kang, G Meyer: The Numerical Solution of the American Option Pricing Problem Finite Difference and Transform Approaches, World Scientific, 2014

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.