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MMath&Phys Mathematics and Physics / Course details
Year of entry: 2024
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Course unit details:
|Unit level||Level 4|
|Teaching period(s)||Semester 2|
|Available as a free choice unit?||No|
The course will be continually assessed via a series of miniprojects. The project material will cover a range of topics including the solution of nonlinear ODE's, SDEs, lattice (tree) methods, to the solution of the nonlinear partial differential equations, and will require students to write a series of computer programs to solve a specified problem.
|Unit title||Unit code||Requirement type||Description|
|Introduction to Financial Mathematics||MATH20912||Pre-Requisite||Compulsory|
Students are not permitted to take, for credit, MATH40082 in an undergraduate programme and then MATH60082 in a postgraduate programme at the University of Manchester, as the courses are identical.
The unit aims to introduce students to scientific computing (specifically computational finance) by means of a variety of numerical techniques, through the use of high-level computing languages. Students will use a combination of writing their own codes, together with the use of scientific libraries (such as NAG).
To familiarise students with modern numerical approaches and techniques (and capabilities).
Students should be able to:
- translate mathematical problems (well defined systems of mathematical equations) into computational tasks,
- to assess the accuracy of any numerical approximations, through numerical experimentation (and, when possible, by comparison with analytic solution),
- to process numerical results into a comprehensible form (including the use of standard graphical plotting packages), for presentation in a report,
- to be able to give a critical assessment of the integrity of numerical methods and results.
1.Introductory C++ course, extra 2hr lab session will be scheduled in the first week 
2.Introduction to numerical computation. Numerical approximation and different methodologies. Discussion of errors, roundoff, truncation, discretisation.
3.Monte Carlo simulations; generation of random numbers (including use of antithetic variables). Pricing of European/Vanilla call/put options. Simple path-dependency options (but NO early exercise examples). Assessment of advantages and disadvantages of simulation approach.
4.Binomial tree valuation of European/Vanilla call/put options. Assessment (and improvement) of accuracy. Application to early-exercise put options.
5.Introduction to solution of PDE's using finite-difference methods. Discussion of stability, consistency and convergence. Brief introduction to error analysis. Methods for parabolic equations. CFL condition. Discussion of methods of solution including iterative methods: Jacobi, Gauss-Seidel, SOR, Line relaxation and PSOR methods. Solution of European call/put options using Crank-Nicolson method. Solution of early exercise put options (using PSOR).
6.Advanced techniques: quadrature method, Monte-Carlo Least Squares, body-fitted (free-boundary) coordinate systems.
Teaching and learning methods
The independent study hours will normally comprise the following. During each week of the taught part of the semester:
- You will normally have approximately 60 minutes of video content. Normally you would spend approximately 2 hrs per week studying the notes and videos independently
- You will attend a review class discussing the course notes or assignments and demonstrating code - 1hr
- You will attend a lab session demonstrating code and working on problems in class - 2hr
- You will normally have assignments you can be working on - 4hr
Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit. Extra work may be required during the semester given that there is no end-of-semester assessment.
Lab Classes will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Project reports will be marked up with detailed individual feedback for students. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturers office hour.
|Written assignment (inc essay)||50%|
Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding. Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback. Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
G.D. Smith, 'Numerical Solution of Partial Differential Equations', Clarendon Press, Oxford, 1978.
P. Wilmott, S. Howison & J. Dewynne, 'The mathematics of financial derivatives', Cambridge 1995.
J.C. Hull, 'Options, Futures, and Other Derivatives', Sixth Edition, Prentice Hall 2005.
D.J. Higham, 'An introduction to financial option valuation', Cambridge 2004.
For Information and advice on Link2Lists reading list software, see:
|Scheduled activity hours|
|Practical classes & workshops||22|
|Supervised time in studio/wksp||2|
|Independent study hours|
|Paul Johnson||Unit coordinator|