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# Stochastic Calculus & Black-Scholes Theory - MAT00028M

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• Department: Mathematics
• Module co-ordinator: Dr. Christian Litterer
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2021-22

• None

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## Module will run

Occurrence Teaching cycle
A Spring Term 2021-22
B Summer Term 2021-22

## Module aims

This 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

At the end of the module you should:

1. Be competent in calculations involving the precise mathematical details of the definition and construction of the Ito integral, and understand its structure and properties;
2. Demonstrate fluency in the use of the Ito formula in applications;
3. Be able to solve linear SDEs;
4. 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;
5. Have a clear understanding of the impact of the simplifying assumptions in the Black-Scholes model, and understand the role of the ‘Greek parameters’;
6. 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;
7. Be familiar with the Feynman-Kac formula and its use in representing the price of an option;
8. Be able to compare European and exotic options, and to discuss the differences between their pricing methodologies;
9. Have a working knowledge of the mathematical analysis of the American put option (time allowing).

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## Module content

Indicative Content:

1. Brownian motion and its properties.
2. Development of the Ito integral in  framework and its extension to wider classes of integrands; isometry and martingale properties of the integral.
3. Ito calculus, Ito formula and its application to evaluating stochastic integrals.
4. Stochastic differential equations.
5. Risk-neutral pricing: Girsanov’s theorem and equivalent measure change in a martingale setting; representation of Brownian martingales.
6. Feynman-Kac formula.
7. The Black-Scholes model: assumptions and scope; delta-hedging; derivations of the Black-Scholes PDE and its solution via the heat equation and Brownian motion; role of the ‘Greeks’ as measures of model parameters; the Black-Scholes formula and simple extensions of the model.
8. Application of Girsanov’s theorem to Black-Scholes dynamics; self-financing strategies and model completeness; derivation of the Black-Scholes formula via expectations; wealth processes and minimal hedges.
9. A choice of: barrier or lookback options in the Black-Scholes model; reflection principle for Brownian motion;
10. (optional, time allowing) Comparison of European and American options; dividend-paying stocks; path-dependence.

## Assessment

Task Length % of module mark
Online Exam
Stochastic Calculus & Black-Scholes Theory
N/A 100

None

### Reassessment

Task Length % of module mark
Online Exam
Stochastic Calculus & Black-Scholes Theory
N/A 100

## Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

1. M. Capinski, E. Kopp and J. Traple, Stochastic Calculus for Finance
2. M. Capinski, E. Kopp: The Black-Scholes Model
3. R.A. Dana and M. Jeanblanc,  Financial Markets in Continuous Time, Springer 2001.
4. T. Björk, Arbitrage theory in continuous time, Oxford University Press 1999.
5. Z. Brzezniak and T. Zastawniak, Basic Stochastic Processes, Springer 1999
6. M. Baxter and A. Rennie, Financial Calculus, Cambridge University Press 1996.
7. R.J. Elliott and P.E. Kopp, Mathematics of Financial Markets, Springer 1999.
8. P. Wilmott, Derivatives, Wiley 1997.
9. R. Korn and E. Korn, Option Pricing and Portfolio Optimization, Graduate Studies in Mathematics, vol. 31, American Mathematical Society, 2001.
10. D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapmans & Hall/CRC, 2000.

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The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.