# DEGREE REGULATIONS & PROGRAMMES OF STUDY 2019/2020

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# 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 SCQF Credits 10 ECTS Credits 5 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. Course description 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.
 Pre-requisites Co-requisites Students MUST also take: Stochastic Analysis in Finance (MATH11154) OR Probability, Measure & Finance (MATH10024) Prohibited Combinations 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)
 Academic year 2019/20, Not available to visiting students (SS1) Quota:  None Course Start Semester 2 Timetable Timetable Learning and Teaching activities (Further Info) Total Hours: 100 ( 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 69 ) Assessment (Further Info) Written Exam 80 %, Coursework 20 %, Practical Exam 0 % Additional Information (Assessment) Coursework :20% Examination :80% Feedback Not entered Exam Information Exam Diet Paper Name 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 Not entered Keywords NP_MC,Stochastic,Finance,Monte-Carlo
 Course organiser Dr Goncalo Dos Reis Tel: (0131 6)51 7677 Email: g.dosreis@ed.ac.uk Course secretary Miss Gemma Aitchison Tel: (0131 6)50 9268 Email: Gemma.Aitchison@ed.ac.uk
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