# COMPG008 - Stochastic Processes for Finance

**This database contains 2016-17 versions of the syllabuses.** For current versions please see here.

Code | COMPG008 |
---|---|

Year | MSc |

Prerequisites | Calculus, Basic Probability |

Term | 1 |

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

**Recommended book**

Crispin W. Gardiner, Stochastic Methods - A Handbook for the Natural and Social Sciences, 4th ed., Springer 2009, Chapters 1-4, 10, 15, plus a few other selected parts, for a total of 150-200 out of 420 pages. See table of contents on http://www.springer.com/physics/complexity/book/978-3-540-70712-7 and a review of the 3rd edition (2004) on http://www.ejwagenmakers.com/2006/GardinerBookReview2006.pdf.

**Further books**

- J. L. McCauley, Stochastic Calculus and Differential Equations for Physics and Finance, Cambridge University Press, 2013.
- N. van Kampen, Stochastic Processes, 3rd edition, Elsevier, 2007.
- W. Paul, J. Baschnagel, Stochastic Processes, Springer, 1999.
- Sheldon N. Ross, Stochastic Processes, 2nd edition, Wiley, 1996.
- S. Karlin, H. M. Taylor, An Introduction to Stochastic Modeling, 3rd edition, Academic Press, 1998.
- S. Karlin, H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, 1981.
- S. Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd edition, Academic Press, 1975.