MSc in Mathematical Finance

by Online Distance Learning

make options your future

make futures your option

**MODULE SPECIFICATIONS**

**Numerical and Computing Techniques in Finance (Online Version)**

Level M, Diploma Stage, 20 credits

Old code: 0571002 (until 2010/11)

New code: MAT00031M (from 2011/12)

**Pre-requisites:** Mathematical Methods of Finance (Online Version), Discrete Time Modelling and Derivative Securities (Online Version), Portfolio Theory and Risk Management (Online Version), or equivalent

**Aims and Distinctive Features:** The aim of the module is to provide programming skills required for the implementation of mathematical models in quantitative finance. The focus will be on the C++ programming language, which is widely accepted as the main tool amongst practitioners in the financial community. The implementation of a given model rarely narrows down to the pricing of a single particular financial instrument. Most often it is possible to devise general numerical schemes which can be applied to various types of derivatives. The code should be designed so that it easily integrates with the work of other developers and can be modified by other users. The student will learn such skills by writing C++ programs designed for pricing various types of derivatives, starting from the simplest discrete time models and finishing with continuous time models based on finite difference or Monte Carlo methods.

**Learning Outcomes:** By the end of the module, students should:

a) be able to write comprehensive C++ programs;

c) be familiar with functions and function pointers;

b) be familiar with classes and handle virtual functions, inheritance and multiple inheritance;

d) be able to implement non-linear solvers;

e) be familiar with data structures and dynamic memory allocation;

f) understand and have experience of using class and function templates;

g) be familiar with standard numerical methods (finite difference, Monte Carlo) for solving representative problems;

h) be able to price European and American options under the CRR model;

i) be able to price American options by means of finite difference methods under assumptions of the Black Scholes model;

j) be able to price barrier and Asian options by means of Monte Carlo simulation.

**Indicative Content:**

**Numerical techniques**

1. Modelling principles. Using data in mathematical modelling.

2. Pricing by backward induction.

3. Representative equations of Black-Scholes type and elementary finite difference approximations.

4. Basic concepts of consistency, convergence and stability.

5. Explicit and implicit difference methods.

6. Monte Carlo Methods for exotic options. Generating random variables with multidimensional normal distributions by the Box-Muller method. Generating sample paths of Brownian Motion.

7. Generating sample paths of solutions to stochastic differential equations. Application to pricing barrier options and Asian call options.

**C++ programming**

9. Basic features: syntax, variables and their types, arrays, pointers, conditional statements and loops.

10. Functions and function pointers.

11. Classes.

12. Dynamic memory allocation.

13. Inheritance.

14. Virtual functions and polymorphism.

15. Templates.

**Learning and Teaching Strategy:** Interactive presentations recorded on CD/DVD in lieu of lectures, equivalent to 30 one-hour lectures, and 10 one-hour one-to-one online tutorials scheduled at regular intervals covering the three core modules comprising the Diploma Stage of the programme. Individual feedback and advice will be offered to students during scheduled online tutorials and via a VLE discussion forum. The final online tutorial will include a recorded online viva.

**Arrangements for Revision and Private Study:** Students are expected to contribute a considerable amount (of the order of 180 hours) of individual study time, including interactive CD/DVD presentations in lieu of lectures, exercises and assessed coursework assignments, library and textbook work. The final week (for Fast Track students) or two weeks (for Standard Track students) of the Diploma Stage preceding the online viva will be devoted to revision and no new material will be covered during that period.

**Assessment:** Four assessed computer-based coursework assignments comprising 100% of the final mark followed by a recorded online viva to authenticate the work submitted for assessment. The weightings of individual coursework assignments to be advised prior to commencing the Diploma Stage of the online programme. Marking will be based exclusively on written work submitted electronically for each assignment, whereas the online viva scheduled at the end of the Diploma Stage will serve to authenticate the work submitted for assessment but will not otherwise affect the marks. Assessed work will be routinely screened using online tools for the detection of unfair means such as unacknowledged copying of material or collusion.

**Recommended Texts:**

1. K. Back, A course in Derivative Securities: Introduction to Theory and Computation.

2. D.J. Duffy, Introduction to C++ for Financial Engineers. An Object-Oriented Approach, John Wiley & Sons (2006).

3. P. Glasserman, Monte Carlo Methods in Financial Engineering.

4. M. Joshi, C++ Design Patterns and Derivatives Pricing, Cambridge University Press (2004).

5. D. Lamberton, B. Lapeyre Introduction to Stochastic Calculus Applied to Finance, Second Edition, Chapman & Hall/Crc Financial Mathematics Series.

6. P. Wilmott, Paul Wilmott Introduces Quantitative Finance, John Wiley & Sons, Chichester (2001).

7. D. Yang, C++ and Object-Oriented Numeric Computing for Scientists and Engineers.