Stochastic Calculus & Black-Scholes Theory (Online Version) - MAT00029M

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Module will run

Occurrence	Teaching period
A1	Semester 1 2025-26
A2	Semester 1 2025-26 to Semester 2 2025-26
B1	Semester 2 2025-26
B2	Semester 2 2025-26 to Semester 1 2026-27

Module aims

The module enables students to acquire in-depth knowledge of the main features of Ito stochastic calculus as applied in mathematical finance, including:

• The role of the Ito integral and Ito formula in solving stochastic differential equations (SDEs);
• Martingale properties of the Ito integral and the structure of Brownian martingales;
• The mathematical relationships between wealth processes, investment strategies and option prices;
• Change of measure techniques and Girsanov theorem.
• Partial differential equation (PDE) approach, and in particular the Black-Scholes equation.
• Feynman-Kac representation of option prices.

The emphasis is on fundamental concepts which underlie the main continuous-time models of option pricing, principally the Black-Scholes model. Both plain vanilla (European) and exotic options (for example, barrier options) are dealt with, and the relationship between the approaches based on martingale theory and partial differential equations is explored. The module aims to equip students with a thorough understanding of the sophisticated mathematical results and techniques encountered in financial market modelling.

Module learning outcomes

By the end of this module students should
a) Be competent in calculations involving the precise mathematical details of the definition and construction of the Ito integral, and understand its structure and properties;
b) Demonstrate fluency in the use of the Ito formula in applications;
c) Be able to solve linear SDEs;
d) Have a thorough grasp of Black-Scholes methodology, both in its PDE and martingale formulations, and its application in deriving option prices in continuous-time models;
e) Have a clear understanding of the impact of the simplifying assumptions in the Black-Scholes model, and understand the role of the 'Greek parameters';
f) Be able to use measure transformations to price European options via expectations of martingale measures, and apply martingale calculus in pricing options and finding optimal trading strategies in complete models;
g) Be familiar with the Feynman-Kac formula and its use in representing the price of an option;
h) Be able to compare European and exotic options, and to discuss the differences between their pricing methodologies;
i) Have a working knowledge of the mathematical analysis of the American put option (time allowing).

Indicative assessment

Task	% of module mark
Coursework - extensions not feasible/practicable	100
Oral presentation/seminar/exam	0

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Coursework - extensions not feasible/practicable	100
Oral presentation/seminar/exam	0

Module feedback

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Indicative reading

1. R.A. Dana and M. Jeanblanc, Financial Markets in Continuous Time, Springer 2001.

2. T. Bjork, Arbitrage theory in continuous time, Oxford University Press 1999.

3. M. Baxter and A. Rennie, Financial Calculus, Cambridge University Press 1996.

4. R.J. Elliott and P.E. Kopp, Mathematics of Financial Markets, Springer 1999.

5. P. Wilmott, Derivatives, Wiley 1997.

6. R. Korn and E. Korn, Option Pricing and Portfolio Optimization, Graduate Studies in Mathematics, vol. 31, American Mathematical Society, 2001.

7. D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapmans & Hall/CRC, 2000.