Postgraduate Course: Stochastic Analysis in Finance (MATH11154)
Course Outline
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  20 
ECTS Credits  10 
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. 
Course description 
Continuous time processes: basic ideas, filtration, conditional expectation, stopping times.
Continuoustime martingales, sub and supermartingales, martingale inequalities, optional sampling.
Wiener process and Wiener martingale, stochastic integral, Itô calculus and some applications.
Multidimensional Wiener process, multidimensional Itô's formula.
Stochastic differential equations, OrnsteinUhlenbeck processes, BlackScholes SDE, Bessel processes and CIR equations.
Change of measure, Girsanov's theorem, equivalent martingale measures and arbitrage.
Representation of martingales.
The BlackScholes model, selffinancing 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)
Prerequisites 

Corequisites  
Prohibited Combinations  
Other requirements  None 
Course Delivery Information

Academic year 2022/23, 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:
 model the evolution of random phenomena using continuoustime 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

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: Riskneutral 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. PrenticeHall 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. 
Keywords  SAF 
Contacts
Course organiser  Prof Istvan Gyongy
Tel: (0131 6)50 5945
Email: I.Gyongy@ed.ac.uk 
Course secretary  Miss Gemma Aitchison
Tel: (0131 6)50 9268
Email: Gemma.Aitchison@ed.ac.uk 

