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MA50296: Continuous time finance

[Page last updated: 09 August 2024]

Academic Year: 2024/25
Owning Department/School: Department of Mathematical Sciences
Credits: 6 [equivalent to 12 CATS credits]
Notional Study Hours: 120
Level: Masters UG & PG (FHEQ level 7)
Period:
Semester 2
Assessment Summary: EXCB 100%
Assessment Detail:
  • Examination (EXCB 100%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites:
Learning Outcomes: Apply the Black-Scholes formula to find the price of European options.
Define a Brownian motion, and determine basic properties of Brownian motion.
Discuss different financial contexts in which continuous models can be applied.
Evaluate the strengths and weaknesses of discrete or continuous models in finance.
Explain how "Greeks" are used in financial applications to manage risk.
Use Ito's Lemma to manipulate functions of Stochastic processes.


Synopsis: Complementing the discrete time models introduced the previous semester, this module will introduce you to continuous-time models for random processes. The unit will look at applying continuous-time models in a range of financial contexts, primarily derivative pricing (Black-Scholes-Merton) in stock markets and for fixed income products. Other potential topics could include insurance modelling using Levy processes, optimal investment and modelling of other financial contracts, such as in currency, energy or commodity markets.

Aims: This module will introduce students to continuous-time models for random processes, complementing discrete-time models which have been introduced in the previous semester. The unit will look at applications of continuous-time models in a range of financial contexts, primarily derivative pricing (Black-Scholes-Merton) in stock markets and for fixed income products. Other potential topics could include insurance modelling using Levy processes, optimal investment and modelling of other financial contracts, e.g. in currency, energy or commodity markets.

Skills: Problem Solving (T,F,A) Stochastic Modelling (T,F,A)

Content: Continuous-time stochastic models: Brownian motion: definition, basic properties, reflection principle. Sketch introduction to Stochastic Integration and stochastic differential equations. Ito's Lemma, Girsanov's Theorem. Black-Scholes model: Geometric Brownian motion as a model for asset prices, risk-neutral measure, European call price formula (Black-Scholes), Fundamental Theorem of Asset pricing. Role of Greeks in risk management. American options. Stochastic volatility models and the volatility smile/surface. Fixed income products and yield curve modelling: short rate models and the term-structure equation. Complete vs Incomplete Markets. Other topics may be chosen from: Levy processes and risk models for insurance. Stochastic optimal control and optimal investment. Models for commodity prices and/or foreign exchange. Parameter fitting for stochastic differential equations. Simple mean-field games.

Course availability:

MA50296 is Compulsory on the following courses:

Department of Mathematical Sciences

Notes:

  • This unit catalogue is applicable for the 2024/25 academic year only. Students continuing their studies into 2025/26 and beyond should not assume that this unit will be available in future years in the format displayed here for 2024/25.
  • Courses and units are subject to change in accordance with normal University procedures.
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