香港六合彩

Aims:

An introduction to numerical/computational methods with programming/exercises (Matlab, Python, C++) and an emphasis on applications in finance (derivatives pricing with Monte Carlo and Fourier transform methods, model calibration, etc.).

Intended learning outcomes:

On successful completion of the module, a student will be able to:

1. Demonstrate programming proficiency and skills in turning mathematical equations and models into working code.
2. Demonstrate capacity to solve practical problems in financial mathematics applying modern numerical techniques.

Indicative content:

The following are indicative of the topics the module will typically cover:

• Introduction: Programming languages, programming basics, data types, operators, expressions, control structures (sequential, conditional and iterated execution), vector/array operations, input/output, plots, style, floating-point representation of real numbers, numerical errors.
• Fundamental probability distributions: Normal, exponential, log-normal, chi square, plot of their PDF and CDF, sampling with pseudo-random numbers, histograms, transformation from uniform to other distributions using the quantile function.
• 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.
• Monte Carlo: Diffusion equation, Fokker-Planck equation, Kolmogorov backward equation, stochastic differential equation, Ito鈥檚 formula, Feynman-Kac theorem, Black-Scholes-Merton equation, risk-neutral valuation of options, Euler-Maruyama algorithm for the numerical solution of a stochastic differential equation, approximation error, strong and weak solution, Milstein algorithm.
• Drift-diffusion processes: Arithmetic and geometric Brownian motion, Ornstein-Uhlenbeck process and Vasicek model, Feller square-root process and Cox-Ingersoll-Ross model, constant elasticity of variance processes, Brownian bridge, Heston stochastic volatility model.
• Jump-diffusion processes: Poisson and normal compound Poisson process, finite-activity jump-diffusion processes (Merton, Kou), time-changed Brownian motion (variance gamma, normal inverse Gaussian, CGMY), L茅vy processes.
• European options: A simple program that prices European calls and puts with the analytical solution, the Fourier transform, and Monte Carlo.
• Fourier transform: Definitions, inverse transform, properties, notable transform pairs (double-sided exponential/Lorentzian, Dirac delta/1, rectangular pulse/sinc, Gaussian/Gaussian), discrete and fast Fourier transform, characteristic function, moment- and cumulant-generating functions, correlation/convolution theorem, auto/cross-covariance and correlation, Parseval/Plancherel theorem, shift theorem, pricing in Fourier space.
• Model calibration: Implied volatility, Newton-Raphson method, J盲ckel's equivalent form, J盲ckel's modification, complex initial guess, attraction basin, fractals.
• Exotic options: Fourier transform methods for the numerical pricing of discretely and continuously monitored path-dependent options (barrier, hindsight, etc.).

Requisites:

To be eligible to select this module as optional or elective, a student must also select COMP0048 Financial Engineering or an equivalent module like MATH0088 Quantitative and Computational Finance or STAT0013 Stochastic Methods in Finance.

Depending on their previous formation, students should consider taking also the non-examined support course 鈥淚ntroduction to Mathematics and Programming for Finance鈥� available on Moodle.