Postgraduate Course: Stochastic Analysis in Finance (MATH11154)
|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||This course aims to provide a good and rigorous understanding of the mathematics used in derivative pricing and to enable students to understand where the assumptions in the models break down.
Continuous time processes: basic ideas, filtration, conditional expectation, stopping times.
Continuous-time martingales, sub- and super-martingales, martingale inequalities, optional sampling.
Wiener process and Wiener martingale, stochastic integral, Itô calculus and some applications.
Multi-dimensional Wiener process, multi-dimensional Itô's formula.
Stochastic differential equations, Ornstein-Uhlenbeck processes, Black-Scholes SDE, Bessel processes and CIR equations.
Change of measure, Girsanov's theorem, equivalent martingale measures and arbitrage.
Representation of martingales.
The Black-Scholes model, self-financing strategies, pricing and hedging options, European and American options.
Option pricing and partial differential equations; Kolmogorov equations.
Further topics: dividends, reflection principle, exotic options, options involving more than one risky asset.
Entry Requirements (not applicable to Visiting Students)
||Other requirements|| None
Course Delivery Information
|Academic year 2021/22, Not available to visiting students (SS1)
|Learning and Teaching activities (Further Info)
Lecture Hours 36,
Seminar/Tutorial Hours 8,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Two assignments will be marked out of 50 each, and the best two will be counted towards the final result.
||Hours & Minutes
|Main Exam Diet S2 (April/May)||Stochastic Analysis in Finance (MATH11154)||3:00|
On completion of this course, the student will be able to:
- model the evolution of random phenomena using continuous-time stochastic processes
- understand the Weiner process, stochastic calculus, Itô integral, and Itô's formula, and apply Ito calculus
- apply stochastic calculus to option pricing problems
- understand the martingale representation theorem, its role in financial applications, and the role of martingales in fincancial mathematics generally and the theory of derivative pricing in particular
- understand stochastic differential equations and be able to use them in modelling generally and in finance in particular
|(1) M. Baxter, A. Rennie: Financial Calculus: an introduction to derivative pricing, ,Cambridge University Press, 1997. |
( Lib. Number: HG6024.A3 Bax.)
(2) N. H. Bingham: Risk-neutral valuation : pricing and hedging of financial derivatives,Springer, 1998. ( Lib. Number: HG4515.2 Bin.)
(3) D. Lamberton and B. Lapeyre: Introduction to stochastic calculus applied to finance,Chapman & Hall, 1996.
(4) T. Bj ¿ork: Interest rate theory. Financial mathematics (Bressanone, 1996), 53¿122,Lecture Notes in Math., 1656, Springer, Berlin, 1997.
(5) J. C. Hull: Options, futures, and other derivatives, 4th ed. Prentice-Hall International,2000.
( Lib. Number: HG6024.A3 Hul.)
(6) T. Bj ¿ork: A geometric view of interest rate theory. Option pricing, interest ratesand risk management, 241¿277, Handb. Math. Finance, Cambridge Univ. Press,Cambridge, 2001.
(7) N. V. Krylov: Introduction to the theory of random processes. Graduate studies inmathematics ; v. 43, American Mathematical Society, Providence, RI, 2002.
(8) B. Oksendal: Stochastic differential equations : an introduction with applications, 5thed. Springer, 1998.
( Lib. Number: QA274.23 Oks.)
(9) T. Mikosch: Elementary stochastic calculus with finance in view. Advanced series on statistical science & applied probability ; vol. 6, World Scientific, Singapore, London,1998.
(10) R. J. Williams: Introduction to the Mathematics of Finance, Graduate Studies in Mathematics V. 72, American Mathematical Society, Providence, RI, 2006.
(11) S. E. Shreve: Stochastic Calculus for Finance I: The Binomial Asset Pricing Model:Binomial Asset Pricing Model v. 1, Springer 2004.
(12) S. E. Shreve: Stochastic C
|Graduate Attributes and Skills
||MSc Financial Mathematics, MSc Financial Modelling and Optimization and MSc Computational Mathematical Finance students only.
|Course organiser||Prof Istvan Gyongy
Tel: (0131 6)50 5945
|Course secretary||Miss Gemma Aitchison
Tel: (0131 6)50 9268