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DRPS : Course Catalogue : School of Mathematics : Mathematics

Postgraduate Course: Stochastic Analysis in Finance (MATH11154)

Course Outline
SchoolSchool of Mathematics CollegeCollege of Science and Engineering
Credit level (Normal year taken)SCQF Level 11 (Postgraduate) AvailabilityNot available to visiting students
SCQF Credits20 ECTS Credits10
SummaryThis 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.
Course description 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)
Pre-requisites Co-requisites
Prohibited Combinations Other requirements None
Course Delivery Information
Academic year 2023/24, Not available to visiting students (SS1) Quota:  None
Course Start Semester 1
Timetable Timetable
Learning and Teaching activities (Further Info) Total Hours: 200 ( Lecture Hours 36, Seminar/Tutorial Hours 8, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 152 )
Assessment (Further Info) Written Exam 90 %, Coursework 10 %, Practical Exam 0 %
Additional Information (Assessment) Two assignments will be marked out of 50 each, and the best two will be counted towards the final result.

Examination 90%
Coursework 10%
Feedback Not entered
Exam Information
Exam Diet Paper Name Hours & Minutes
Main Exam Diet S2 (April/May)Stochastic Analysis in Finance (MATH11154)3:00
Learning Outcomes
On completion of this course, the student will be able to:
  1. model the evolution of random phenomena using continuous-time stochastic processes
  2. understand the Weiner process, stochastic calculus, Itô integral, and Itô's formula, and apply Ito calculus
  3. apply stochastic calculus to option pricing problems
  4. 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
  5. understand stochastic differential equations and be able to use them in modelling generally and in finance in particular
Reading List
(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
Additional Information
Graduate Attributes and Skills Not entered
Special Arrangements MSc Financial Mathematics, MSc Financial Modelling and Optimization and MSc Computational Mathematical Finance students only.
Course organiserProf Istvan Gyongy
Tel: (0131 6)50 5945
Course secretaryMiss Gemma Aitchison
Tel: (0131 6)50 9268
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