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Programme & Unit Catalogues

## MA30089: Stochastic processes & finance

Owning Department/School: Department of Mathematical Sciences
Credits: 6
Level: Honours (FHEQ level 6)
Period: Semester 2
Assessment: EX 100%
Supplementary Assessment: MA30089 Mandatory Extra Work (where allowed by programme regulations)
Requisites: Before taking this unit you must take MA30125
Description: Aims:
To present the Black-Scholes-Merton approach to pricing financial derivatives, and the mathematical results which underpin this theory. To perform simple calculations to compute certain quantities relating to Brownian motion, and to understand how these quantities can be important in pricing financial derivatives.

Learning Outcomes:
On completing the course, students should be able to:
* Compute the prices of options in the one-period Binomial model;
* Explain how the principle of arbitrage can be used in determining the prices of derivative contracts;
* Define a Brownian motion, and determine basic properties of Brownian motion;
* Use the martingale property to find important quantities relating to Brownian motion;
* Apply the Black-Scholes formula to find the price of a European Call option.

Skills:
Numeracy T/F A
Problem Solving T/F A
Written and Spoken Communication F (in tutorials).

Content:
Discrete time: trading portfolio, Binomial model, arbitrage, derivative pricing using arbitrage. Radon-Nikodym derivative, change of measure, Fundamental Theorem of Asset pricing.
Brownian motion: definition, basic properties, reflection principle. Using related martingales, and computing quantitative properties of Brownian motion.
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, Fundamental Theorem of Asset pricing.
Programme availability:

#### MA30089 is Optional on the following programmes:

Department of Mathematical Sciences
• USMA-AFB15 : BSc (hons) Mathematical Sciences (Full-time) - Year 3
• USMA-AKB16 : BSc (hons) Mathematical Sciences (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB16 : BSc (hons) Mathematical Sciences with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB13 : BSc (hons) Mathematics (Full-time) - Year 3
• USMA-AKB14 : BSc (hons) Mathematics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AFB01 : BSc (hons) Mathematics and Statistics (Full-time) - Year 3
• USMA-AKB02 : BSc (hons) Mathematics and Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB02 : BSc (hons) Mathematics and Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AAB14 : BSc (hons) Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFB05 : BSc (hons) Statistics (Full-time) - Year 3
• USMA-AKB06 : BSc (hons) Statistics (Full-time with Thick Sandwich Placement) - Year 4
• USMA-AAB06 : BSc (hons) Statistics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 3
• USMA-AFM14 : MMath Mathematics (Full-time) - Year 4
• USMA-AAM15 : MMath Mathematics with Study Year Abroad (Full-time with Study Year Abroad) - Year 4
• TSMA-AFM09 : MSc Mathematical Sciences (Full-time)
• TSMA-APM09 : MSc Mathematical Sciences (Part-time)
• TSMA-AFM08 : MSc Modern Applications of Mathematics (Full-time)
• TSMA-AWM14 : MSc Modern Applications of Mathematics (Full-time incorporating placement)
• TSMA-AFL02 : PG Dip Modern Applications of Mathematics (Full-time)

Notes:
* This unit catalogue is applicable for the 2013/4 academic year only. Students continuing their studies into 2014/15 and beyond should not assume that this unit will be available in future years in the format displayed here for 2013/14.
* Programmes and units are subject to change at any time, in accordance with normal University procedures.
* Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.