##
MA415 Half Unit

The Mathematics of the Black and Scholes Theory

**This information is for the 2024/25 session.**

**Teacher responsible**

Dr Arne Lokka

**Availability**

This course is compulsory on the MSc in Financial Mathematics. This course is available on the 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.

**Pre-requisites**

Students must have completed September Introductory Course (Financial Mathematics and Quantitative Methods for Risk Management) (MA400).

**Course content**

This course develops the mathematical theory of risk-neutral valuation. 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. The course expands on PDE techniques for the pricing and hedging of several options. Implied volatilities as well as stochastic volatility models are then considered. The course also introduces the Black & Scholes model for foreign exchange markets and various foreign exchange options.

**Teaching**

20 hours of lectures and 10 hours of seminars in the AT.

**Formative coursework**

Weekly homework

**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

**Assessment**

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

** Key facts **

Department: Mathematics

Total students 2023/24: 25

Average class size 2023/24: 13

Controlled access 2023/24: No

Value: Half Unit

**Course selection videos**

Some departments have produced short videos to introduce their courses. Please refer to the course selection videos index page for further information.