# COMPG005 - Numerical Analysis for Finance

This database contains the 2017-18 versions of syllabuses. Syllabuses from the 2016-17 session are available here.

Note: Whilst every effort is made to keep the syllabus and assessment records correct, the precise details must be checked with the lecturer(s).

Code COMPG005 MSc Students not on MSc CF, FRM, FM or EF should check with the module tutor. 1 Guido Germano (100%) An introduction to numerical/computational methods and techniques with code examples in Matlab and an emphasis on applications in finance. Demonstrable skills in turning mathematical equations into working Matlab code; application of numerical methods and programming proficiency in Matlab to solve practical problems in mathematical finance.

# Content

1. Introduction: Bibliography, Matlab, its command window and editor, basic use and important instructions: assignment, arrays, control structures (iteration i.e. for-loop, conditional execution i.e. if-then-else, etc.), online help and documentation, plot command, etc.
2. Fundamental probability distributions: Normal, exponential, log-normal, chi square, etc; plot of the probability distribution function, sampling with pseudo-random numbers, histograms, transformation from uniform to other distributions by inversion of the cumulative distribution function.
3. Random numbers: Linear congruential generators, requirements and statistical tests, pathologic cases, more advanced generators; inversion and transformation in one and more dimensions, acceptance-rejection method, Box-Muller method for normal deviates, polar method by Marsaglia, Ziggurat algorithm by Marsaglia and Tsang, correlated normal random variates, quasi-random numbers.
4. Monte Carlo methods: General considerations, risk-neutral valuation of options, Euler-Maruyama method for the numerical solution of a stochastic differential equation (SDE), approximation error, strong and weak solution.
5. Important stochastic differential equations: Arithmetic and geometric Brownian motion, Ornstein-Uhlenbeck process and the Vasicek model, Cox-Ingersoll-Ross process, constant elasticity of variance processes, Brownian bridge, Heston model of stochastic volatility.
6. Stochastic processes with jumps: Poisson process, normal compound Poisson process, Gamma process, jump-diffusion processes (Merton, Kou), time-changed Brownian motion (variance Gamma process), Lévy processes.
7. Partial differential equations: Classification, second-order PDEs, notable examples of elliptic, parabolic and hyperbolic PDEs, diffusion equation, Black-Scholes equation, Feynman-Kac theorem and relationship with SDEs.
8. Fourier methods: Definitions, inverse transform, notable examples (double exponential, normal, Dirac delta function), fast Fourier transform, Laplace transform, transform of the derivative, solution of the standard diffusion equation by Fourier transform and in Fourier-Laplace space, fractional derivatives, space-time fractional diffusion equation and its solution in Fourier-Laplace space, characteristic function, moment-generating function, cumulant-generating function, Lévy processes, correlation/convolution theorem, auto/cross-covariance and correlation, Parseval/Plancherel theorem, use in option pricing.
9. Exotic options: Fourier methods for the numerical pricing of discretely monitored path-dependent options like barrier and lookback.

# Method of Instruction

30 hours of lectures plus homework and assignments.

# Assessment

The course has the following assessment components:

• Coursework (30%)
• Written examination (2.5 hours, 70%)

To pass this course, students must:

• Obtain an overall pass mark of 50% for all sections combined.

# Resources

Reading list available via the UCL Library catalogue.