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Mathematical Finance in Continuous Time - MAT00097H

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  • Department: Mathematics
  • Module co-ordinator: Prof. Zdzislaw Brzezniak
  • Credit value: 20 credits
  • Credit level: H
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

Module gives an introduction to classical methods of asset pricing in continuous time.

Professional requirements

Counts towards IFoA exemption.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Module will run

Occurrence Teaching period
A Semester 2 2024-25

Module aims

Module gives an introduction to classical methods of asset pricing in continuous time.

Module learning outcomes

At the end of this module, students will be able to:

  • Use the main notions of Stochastic Calculus in continuous time.

  • Apply the Black-Scholes formula for option pricing.

  • Use basic models of interest rates.

  • Use basic credit risk models.

Module content

Syllabus

  • Stochastic Processes in continuous time. Ito Lemma.

  • Stochastic differential equations and martingales.

  • Equivalent measures and Girsanov Theorem.

  • Black-Scholes model of a stock market.

  • Self-financing trading strategies. No-arbitrage principle.

  • European derivative securities. Pricing by replication.

  • Black-Scholes partial differential equation.

  • Black-Scholes pricing formula for European call options.

  • Equivalent martingale measure and pricing in the Black-Scholes model.

  • The Greek parameters.

  • Merton model as an example of a structural model of credit risk

  • Further examples of stochastic differential equations and the Ornstein Uhlenbeck process

  • Modelling credit risk: reduced form models and intensity based models.

  • The two-state model for credit ratings and the Jarrow-Lando-Turnbull model

  • Models of the term structure of interest rates, including one-factor general diffusion model, and the Vasicek, Cox-Ingersoll-Ross and Hull-White models

  • Pricing some standard interest rate derivatives in the above models

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Mathematical Finance in Continuous Time		 3 hours 80
Essay/coursework
Excel coursework		 N/A 20

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Mathematical Finance in Continuous Time		 3 hours 80
Essay/coursework
Excel coursework		 N/A 20

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.

Indicative reading

Z. Brzezniak, T. Zastawniak, Basic stochastic processes : a course through exercises. Springer

M. Capinski and T. Zastawniak, Mathematics for Finance. An Introduction to Financial Engineering, Springer

M. Capinski and T. Zastawniak, Credit Risk (elect. resource)

D. McInerney and T. Zastawniak, Stochastic Interest rates (elect. resource)

M. Musiela, M. Rutkowski, Martingale Methods in Financial Modelling, Springer.

M. Steele, Stochastic calculus and financial applications, Springer


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.