MA415      Half Unit
The Mathematics of the Black and Scholes Theory

This information is for the 2013/14 session.

Teacher responsible

Dr Arne Lokka and Prof Mihail Zervos


This course is compulsory on the MSc in Financial Mathematics. This course is available on the MSc in Management and Regulation of Risk, MSc in Risk and Stochastics, MSc in Statistics (Financial Statistics) and MSc in Statistics (Financial Statistics) (Research). This course is available with permission as an outside option to students on other programmes where regulations permit.


Students must have completed September Introductory Course (Financial Mathmatics) (MA400).

Course content

This course is concerned with a mathematical development of the risk-neutral valuation theory. In the context of the binomial tree model for a risky asset, the course introduces the concepts of replication and martingale probability measures. The mathematics of the Black & Scholes methodology follow; in particular, the expression of European contingent claims as expectations with respect to the risk-neutral probability measure of the corresponding discounted payoffs, pricing formulae for European put and call options, and the Black & Scholes PDE are derived. A class of exotic options is then considered. In particular, pricing formulas for lookback and barrier options are derived using PDE techniques as well as the reflection property of the standard Brownian motion. The course also introduces a model for foreign exchange markets and various foreign exchange options.


20 hours of lectures and 9 hours of seminars in the MT. 1 hour of seminars in the LT.

Indicative reading

N H Bingham and R Kiesel, Risk-Neutral Valuation, Springer; T Björk, Arbitrage Theory in Continuous Time, Oxford; P J Hunt and J Kennedy, Financial Derivatives in Theory and Practice, Wiley; D Lamberton and J Kennedy, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall; D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall/Crc Financial Mathematics Series, 2nd edition, 2007; S E Shreve, Stochastic Calculus for Finance: Continuous-time Models: vol. 2, Springer


Exam (100%, duration: 2 hours) in the main exam period.

Key facts

Department: Mathematics

Total students 2012/13: 37

Average class size 2012/13: 37

Value: Half Unit

Guidelines for interpreting course guide information