This module runs in Term 1 and is core for students on the MSc in Mathematical Finance.

PhD students interested in taking the module should consult the module leader.

This module is not available to undergraduate students.

This module provides a thorough introduction into discrete-time martingale theory, Brownian motion, and stochastic calculus, illustrated by examples from Mathematical Finance.

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

1 Conditional expectations

(a) Elementary conditional expectations

(b) Measure-theoretic conditional expectations

(c) Properties of conditional expectations

2 Martingale Theory

(a) Stochastic processes and filtrations

(b) Martingales, submartingales, and supermartingales

(c) Discrete stochastic integral

(d) Stopping times and stopping theorem

(e) Martingale convergence theorems

(1) Applications to Finance (option pricing in complete markets)

3 Markov Processes

(a) Markov processes and Markov property

(b) Strong Markov property

4) Brownian motion and continuous local martingales

(a) Definition and fundamental properties of Brownian

(b) Quadratic variation

(c) Continuous local martingales and semimartingales

5) Stochastic calculus

(a) Integration with respect to local martingales

(b) Finite variation processes and Lebesgue-Stieljes integration

(c) Integration with respect to semimartingales

(d) Ito's formula

(e) Levy's characterisation of Brownian motion

(f) Stochastic exponentials and Novikov's condition

(g) Girsanov's theorem

(h) Ito representation theorem

(i) Feynman-Kac formula

(j) Applications to Finance (Black Scholes model)

6) Stochastic differential equations

(a) Strong solutions and Lipschitz-theory

(b) Examples (0U-processes, CIR processes, etc.)

By the end of the module, students should be able to:

- Explain and apply the concept of measure-theoretic conditional expectations
- Demonstrate an understanding of discrete time martingale theory and apply the theory to option pricing
- Understand the basic properties of Brownian motions
- Explain the main steps in the construction of the stochastic integral
- Be proficient in applying Ito's formula and Girsanov's theorem in problems arising in Mathematical Finance
- Solve standard SDEs appearing in Mathematical Finance

View reading list on Talis Aspire

-Explain and apply the concept of measure theoretic conditional expectations

Show an understanding of martingales and the connection with gains from trade

-Understand the Markov property and the strong Markov property and apply it to examples

-Demonstrate the ability to perform calculations involving martingales and stochastic integrals

-Be proficient in applying Ito's formula and Girsanov's theorem to problems in Mathematical finance

-Demonstrate the ability to translate problems from mathematics to finance and vice-versa

-Demonstrate problem solving skills involving concepts from the module

Type | Required |
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Lectures | 30 sessions of 1 hour (20%) |

Tutorials | 10 sessions of 1 hour (7%) |

Private study | 110 hours (73%) |

Total | 150 hours |

Weekly revising of lecture notes and materials, solving of problem sheets, and preparing for class tests and the final exam.

No further costs have been identified for this module.

You do not need to pass all assessment components to pass the module.

Students can register for this module without taking any assessment.

Weighting | Study time | |
---|---|---|

Class Test 1 (20-minute synchronous online assessment) | 10% | |

This class test takes place in the middle of the term during a lecture. |
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Class Test 2 (20-minute synchronous online assessment) | 10% | |

This class test takes place in the middle of the term during a lecture. |
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Locally Timetabled Examination | 80% | |

The examination paper will contain four questions, of which the best marks of THREE questions will be used to calculate your grade. |

Solutions and written cohort level feedback will be provided for the final exam. Oral cohort level feedback will be provided for the class tests.

Scripts are retained for external examiners and will not be returned to you.

This module is Core for:

- Year 1 of TIBS-N3G1 Postgraduate Taught Financial Mathematics