Postgraduate Course: Numerical Probability and Monte Carlo (MATH11202)
|School||School of Mathematics
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 11 (Postgraduate)
||Availability||Not available to visiting students
|Summary||The course deals with a rigorous introduction to Monte Carlo methods, and numerical methods to find solutions to stochastic differential equations. These methods are immensely important to understanding financial options price sensitivities (Greeks), and so applications to the techniques discussed will be to finance. Students will be expected to understand both the theoretical content, but also to be able to implement numerical techniques in a programming language such as Matlab.
Topics covered in the course include: Random number generation, pseudorandom numbers, inversion method, acceptance/rejection method, Box-Muller method, basic Monte Carlo, quasi Monte Carlo. Variance reduction techniques such as: importance sampling, control variates and antithetic random variable, Option price sensitivities (Greeks): pathwise, likelihood and finite difference approaches. Burkholder-Davis-Gundy inequality and Gronwall' s lemma. Strong and weak approximations of solutions to SDEs. Euler's approximations and Milstein's scheme. Order of accuracy of numerical approximations. Higher order schemes, accelerated convergence. Weak approximations of SDEs via numerical solutions of PDEs.
Entry Requirements (not applicable to Visiting Students)
||Co-requisites|| Students MUST also take:
Stochastic Analysis in Finance (MATH11154) OR
Probability, Measure & Finance (MATH10024)
||Other requirements|| Students not on a mathematics MSc programme MUST have passed (Probability MATH08066 or Probability with Applications MATH08067) AND Several Variable Calculus and Differential Equations (MATH08063) AND Fundamentals of Pure Mathematics (MATH08064)
Additionally, such students MUST also take: Probability, Measure & Finance (MATH10024)
Course Delivery Information
|Academic year 2023/24, Not available to visiting students (SS1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Supervised Practical/Workshop/Studio Hours 5,
Summative Assessment Hours 2,
Programme Level Learning and Teaching Hours 2,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Numerical Probability and Monte Carlo (MATH11202)||2:00|
On completion of this course, the student will be able to:
- Be able to simulate random numbers from standard distributions.
- Be able to use Monte-Carlo techniques to analyse stochastic differential equations.
- Be able to numerically price basic financial options.
- Be able to use various numerical schemes to simulate solutions to stochastic differential equations.
- Be able to use variance-reduction techniques, and to be able to explain their importance.
|Ross, S. M. (2002). Simulation (3rd ed.). Academic Press. |
Boyle P, Broadie M, and Glasserman P (1997). Monte Carlo methods for security pricing, Journal of Economic Dynamics and Control, 4, 1267-1321. . Hull, J. C. (2002). Options, Futures and Other Derivatives, 5th edition. Prentice Hall.
Glasserman, P. (2004). Monte Carlo methods in Financial Engineering. Springer.
Asmussen, S., Glynn, P. W., (2007) Stochastic Simulation: Algorithms and Analysis, Springer.
Kloeden, P. E., and Platen, E. (1999) Numerical Solution of Stochastic Differential Equations, Springer.
|Graduate Attributes and Skills
|Course organiser||Dr Goncalo Dos Reis
Tel: (0131 6)51 7677
|Course secretary||Miss Gemma Aitchison
Tel: (0131 6)50 9268