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# Stochastic Calculus & Black-Scholes Theory - MAT00090M

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• Department: Mathematics
• Module co-ordinator: Dr. Christian Litterer
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2023-24
• See module specification for other years: 2024-25

## Module summary

This module introduces and develops fundamental ideas of the theory of stochastic calculus. Building on this mathematical foundation, the Black-Scholes model for financial markets and the theory of arbitrage-free pricing of financial derivatives are presented in the second part of the module.

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## Module will run

Occurrence Teaching period
A Semester 2 2023-24

## Module aims

This module introduces and develops fundamental ideas of the theory of stochastic calculus. Building on this mathematical foundation, the Black-Scholes model for financial markets and the theory of arbitrage-free pricing of financial derivatives are presented in the second part of the module.

## Module learning outcomes

At the end of the module students should be able to:

1. Use the definition and construction of the Ito integral to derive its structure and basic properties.

2. Demonstrate fluency in the use of the Ito formula in applications and be able to solve linear SDEs arising in mathematical finance.

3. Use the relationship between parabolic PDEs and the solution of SDEs (provided by the Feynman-Kac formula) and change of measure techniques utilising the Girsanov’s theorem.

4. Apply the Black-Scholes approach to pricing financial derivatives, both in its PDE and martingale formulations.

5. Explain the impact of the simplifying assumptions in the Black-Scholes model and the role of the ‘Greek parameters’.

6. Compare European and exotic options, and discuss the differences between their pricing methodologies.

## Module content

Indicative Content:

1. Development of the stochastic Ito integral and its fundamental properties.

2. The Ito formula and applications to evaluating stochastic integrals.

3. Stochastic differential equations (SDEs). Examples and explicit solutions to some linear SDEs arising in mathematical finance.

4. Girsanov’s theorem, the Novikov conditions and absolutely continuous changes of measures; the martingale representation theorem.

5. Derivation of the Feynman-Kac formula that relates parabolic PDEs to solutions of SDEs

6. The Black-Scholes model: its fundamental ideas and assumption; delta-hedging; derivation of the Black-Scholes PDE and its solution via the heat equation; role of the ‘Greeks’ as measures of model parameters; derivation of the Black-Scholes formula.

7. The risk-neutral approach to pricing derivatives using the Girsanov’s theorem and martingale representation; model completeness; robustness of the Black-Scholes model under modelling error

8. Comparison of European and American options.

9. (optional, time allowing) A choice of: barrier or lookback options in the Black-Scholes model or extensions of the model to dividend paying stocks.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed Exam : Stochastic Calculus & Black-Scholes Theory
3 hours 100

None

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed Exam : Stochastic Calculus & Black-Scholes Theory
3 hours 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.

M. Capinski, E. Kopp and J. Traple, Stochastic Calculus for Finance, Cambridge University Press, 2012

M. Capinski, E. Kopp: The Black-Scholes Model, Cambridge University Press, 2012

R.A. Dana and M. Jeanblanc, Financial Markets in Continuous Time, Springer 2003.

T. Björk, Arbitrage theory in continuous time, Oxford University Press, 2. ed. 2004.

Z. Brzezniak and T. Zastawniak, Basic Stochastic Processes, Springer 1999

M. Baxter and A. Rennie, Financial Calculus: An Introduction to Derivative Pricing, Cambridge University Press, 1996.

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

P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering, Wiley 1998.

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

D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapmans & Hall/CRC, Second Edition 2007.

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.