The Mathematics of the Black and Scholes Theory

**This information is for the 2019/20 session.**

**Teacher responsible**

Prof Johannes Ruf

**Availability**

This course is compulsory on the MSc in Financial Mathematics. This course is available on the MSc in Statistics (Financial Statistics), MSc in Statistics (Financial Statistics) (LSE and Fudan) 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 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.

**Teaching**

20 hours of lectures and 20 hours of seminars in the MT.

The MA415 course has 40 hours of teaching.

**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 summer exam period.

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Key facts
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Department: Mathematics

Total students 2018/19: 31

Average class size 2018/19: 31

Controlled access 2018/19: No

Value: Half Unit