### Brief Description

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.

### Aims

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).

### Syllabus

- Introduction to numerical computation. Numerical approximation and different methodologies. Discussion of errors, roundoff, truncation, discretisation.
- Introduction to numerical solution of ODE's using multi-step methods. Implicit/explicit schemes. Euler and Runge-Kutta methods. Newton linearisation. Solution using library routines. Treatment of initial and boundary value problems.
- 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.
- Binomial tree valuation of European/Vanilla call/put options. Assessment (and improvement) of accuracy. Application to early-exercise put options.
- Introduction to solution of PDE's using finite-difference methods. Discussion of stability, consistency and convergence. Brief introduction to error analysis. Methods for parabolicequations. 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).
- Advanced techniques: quadrature methods, body-fitted (free-boundary) coordinate systems.

### Intended Learning Outcomes

- (I)
translate mathematical problems (well defined systems of mathematical equations) into computational tasks,
- (II) to assess the accuracy of any numerical approximations, through numerical experimentation (and, when possible, by comparison with analytic solution),
- (III) to process numerical results into a comprehensible form (including the use of standard graphical plotting packages), for presentation in a report,
- (IV) to be able to give a critical assessment of the integrity of numerical methods and results..