## MA50089: Stochastic processes & finance

[Page last updated: 03 August 2022]

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: CW 25%, EX 75%
Assessment Detail:
• Coursework (CW 25%)
• Examination (EX 75%)
Supplementary Assessment:
Like-for-like reassessment (where allowed by programme regulations)
Requisites: Before taking this module you must take MA50125
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
* Demonstrate critical thinking and an in-depth understanding of some aspects of stochastic processes.

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.

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

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:

#### MA50089 is Optional on the following programmes:

Department of Mathematical Sciences

 Notes: This unit catalogue is applicable for the 2022/23 academic year only. Students continuing their studies into 2023/24 and beyond should not assume that this unit will be available in future years in the format displayed here for 2022/23. Programmes and units are subject to change 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. Find out more about these and other important University terms and conditions here.