# FINANCIAL MARKETS AND DERIVATIVE PRICING - 2025/6

Module code: MATM068

## Module Overview

Mathematics underpinning real-world uncertain events has become indispensable in many applications, including in particular financial markets. This module will begin with the introduction to probability theory and stochastic processes, with an emphasis on the Ito calculus for treating functions of Brownian motion. Such functions are commonly used in financial markets to model asset price dynamics, required for the valuation of financial contracts.

The module then discusses structures of financial markets, with an emphasis on the equity market. Several of the standard and exotic contingent claims will be introduced, and the need for mathematical models for the valuation and risk management of these products will be explained.

The pricing of a standard call option will then be worked out in a single-period binomial model, for which option price will be worked out in two ways: first using the portfolio replication and no arbitrage argument, and second using the risk-neutral expectation argument.

The model is then extended into multi-period binomial tree model, leading to the Cox-Ross-Rubinstein option pricing formula.

Finally, a continuous-time geometric Brownian motion model, originally introduced by Samuelson, will be considered, and used to deduce the famous Black-Scholes option pricing formula. This can be applied for the purpose of both pricing, as well as risk-management purposes, which will be demonstrated by working out the hedging strategy. The meaning of the pricing formula, and how it can be used in practical investment banking context, will be explained.

### Module provider

Mathematics & Physics

BRODY Dorje (Maths & Phys)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 52

Lecture Hours: 33

Guided Learning: 32

Captured Content: 33

Semester 1

N/A

## Module content

Indicative content includes: measurable spaces; probability spaces; random variables; conditional probability; martingales; supermartingales; submartingales; Brownian motion; Ito calculus; binomial tree model; geometric Brownian motion model; Black-Scholes option pricing formula

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test In-Semester Test (1 hour, during the term, invigilated) 20
Examination Examination (2 hours, at the end of the term, invigilated) 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate

• The ability to work out expectations involving Gaussian random variables;

• The ability to manipulate functions of Brownian motion using the Ito formula;

• The ability to work out option pricing formulae using the binomial model and the geometric Brownian motion model;

• The ability to explain the concept of no arbitrage pricing.

Thus, the summative assessment for this module consists of:

• One in-semester test, covering LO1 and LO2.

• One two-hour examination, covering LO2, LO3, LO4, and LO5.

Formative assessment and feedback

There is feedback from coursework assignments. Verbal feedback is provided by the lecturer during the lectures (e.g., when exercises are worked out), and also in office hours.

## Module aims

• To equip students with the understanding of elements of probability theory and stochastic calculus, sufficient to apply these ideas in financial modelling.
• To equip students with the understanding of equity market structures and products, and how these products are priced and hedged.
• To equip students with the basic understanding of the valuation and hedging of financial contracts within the Black-Scholes framework.

## Learning outcomes

 Attributes Developed 001 Demonstrate working knowledge of probability spaces, probability measures, random variables, stochastic processes, and their use in mathematical models for asset prices. K 002 Demonstrate working knowledge of applications of Brownian motion and Ito calculus. K 003 Demonstrate a working knowledge of pricing options using binomial tree model. K 004 Demonstrate understanding the concept of arbitrage pricing of contingent claims in financial markets. CPT 005 Demonstrate understanding the use of pricing formulae in financial markets PT

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Offer an introduction to the theory of stochastic processes and how to use them to model price dynamics in financial markets.

• Introduce structures of equity market and the various contingent claims, and how mathematical models are required to price and hedge these products.

• Introduce how the binomial tree model and the geometric Brownian motion model can be used to both price and hedge stock options.

• Explain how pricing formulae are used in practice.

The learning and teaching methods include:

Weekly lectures during the term. Typeset notes, containing exercises and examples, will be provided along the way.

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: MATM068

## Other information

Resourcefulness and Resilience: The knowledge on manipulating stochastic processes gained through this module will be hugely helpful in modelling a wide range of random phenomena, thus enhancing the resourcefulness of the students.

Global and Cultural Intelligence: The module will also explain the practicalities of how models are used in real-world applications, which in turn are beneficial towards enhancing global and cultural intelligence of the students.

Employability: This module teaches the basic mathematical skills required to model random processes such as financial asset prices, as well as basic structures of financial markets. These are used to price and risk-manage financial contracts. All of these will significantly enhance the employability of the students in financial and related industries.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Mathematics with Statistics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MPhys 1 Optional A weighted aggregate mark of 50% is required to pass the module
Mathematics and Physics MMath 1 Optional A weighted aggregate mark of 50% is required to pass the module
Financial Data Science MSc 1 Compulsory A weighted aggregate mark of 50% is required to pass the module
Mathematics MSc 1 Optional A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2025/6 academic year.