COMPG008 - Stochastic Processes 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).

CodeCOMPG008
YearMSc
Prerequisites

Calculus, Basic Probability

Term1
Taught By

Guido Germano (100%)

Aims

Rehearse/survey probability theory and give a systematic introduction to stochastic processes and their applications without stressing too much the measure-theoretical aspects and other mathematical formalisms. The main target are students with an undergraduate degree in engineering, physics, computer science and the like, who have a good basis in calculus and have already come into contact with aspects of probability and statistics for ad hoc applications like transport equations, laboratory data treatment, and quantum mechanics, but have not attended yet a dedicated course on stochastic processes. The course material will unfold with references to its historical development and early applications in physics/engineering the students may already heard of, ending with current-day applications in finance.

Learning Outcomes

Familiarity with probability theory, stochastic processes in discrete and continuous time, stochastic calculus, and basic applications in physics/engineering and, mainly, in finance.

Content

  1. Probability Theory - a Survey (3 or 4 weeks)
    • Definitions, Kolmogorov's axioms
    • Joint and conditional probabilities, Bayes' theorem
    • Probability distribution and density functions
    • Multivariate and marginal distributions
      Mean, variance, covariance, correlation, moments, cumulants
    • Moment/cumulant-generating functions, characteristic function, entropy
    • Sum and transformation of random variables
    • Gaussian and Poisson distributions
    • Law of large numbers and central limit theorem
  2. Stochastic Processes (6 or 7 weeks)
    • Definitions and classification
    • Markov and semi-Markov processes, Chapman-Kolmogorov equation
    • Jump processes: the master equation
    • Diffusion processes: the Fokker-Planck equation
    • Deterministic processes: the Liouville equation
    • General processes; martingales and semimartingales
    • Evolution equations, forward and backward equations
    • Stationary and homogeneous Markov processes, ergodicity
    • Auto- and cross-covariance/correlation
    • Random walk, Brownian motion, Wiener process
    • Poisson process, compound Poisson process
    • Renewal process, continuous-time random walk
    • Stochastic integration: Ito and Stratonovich; Ito's formula
    • Langevin equation, stochastic differential equations, Feynman-Kac theorem
    • Euler-Maruyama and Milstein algorithms
    • Geometric Brownian motion and Black-Scholes equation
    • Ornstein-Uhlenbeck process and Vasicek model
    • Rayleigh and Bessel processes, Cox-Ingersoll-Ross model

Method of Instruction

30 hours of lectures plus homework and assignments.

Assessment

The course has the following assessment components:

  • Written exam (2.5 hours, 100%)

To pass this course, students must:

  • Obtain a pass mark of 50%.

Resources

Reading list available via the UCL Library catalogue.