# MODERN FINANCIAL METHODS - 2025/6

Module code: PHY3065

## Module Overview

The first half of this module covers various applications of statistical physics to model share prices and financial markets. This mathematics is then applied to calculating prices for some examples of financial derivatives.

The second half of the module then focusses on optimization problems with examples including logistics, aerospace, traffic control and finance (which includes pricing, risk managements and portfolio optimizations in financial markets). There is a particular focus on the role of quantum optimization and the use of quantum computer algorithms in finance.

### Module provider

Mathematics & Physics

NOEL Noelia (Maths & Phys)

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

Independent Learning Hours: 81

Lecture Hours: 22

Tutorial Hours: 10

Laboratory Hours: 5

Guided Learning: 10

Captured Content: 22

Semester 1

None

## Module content

Introduction to Financial derivatives: markets, speculation, hedging, investment; overview of derivatives and non-arbitrage pricing; types of interest rates used in the module.

Futures and Forwards: forward contracts: over the counter agreements, underlying asset, future time, delivery price; future contracts: exchange traded, zero worth at inception; formulas for delivery price via non-arbitrage arguments.

Options (call and put): Definition, European vs American; Vanilla vs Exotic; pricing factors: asset price, interest rate, volatility, strike, time to maturity; call-put parity significance; construction of complex strategies using options. Overview of non-vanilla options (Exotics): binary, forward, compound, path-dependent, spread, Bermuda, customizable strategies with exotic options.

Binomial Model: Basis of option pricing using Binomial model; Delta and its role in option values; extension to more periods and scenarios; calculation of option values using discounted expected values backward; sensitivities of option prices (Greeks); continuous time-pricing model.

Black-Scholes-Merton framework: construction from probabilistic and non-arbitrage arguments; convergence of the binomial model to the Black-Scholes-Merton framework; analytical formulae for Greeks and their interpretation.

Volatility: Spot (instantaneous), historical, and implied volatility; implied volatility surface: smile and skew effects, flattening with time to maturity.

Replicating portfolios: using assets and risk-free investments; Delta hedging technique to reorganise portfolio to maintain a quantity of assets equal to Delta; application of Delta hedging to the binomial model and to Black-Scholes-Merton.

Introduction to Quantum computing: a brief introduction to quantum computing with quantum circuits as well as discussing quantum annealing, both being relevant strategies for quantum optimization.

Case studies of quantum optimization problems: Application of Quantum Approximation Optimization Algorithm and Quantum annealers to optimization problems.

Latest research. The latest research will be used to update content to take account of hardware capabilities and algorithms

## Assessment pattern

Assessment type Unit of assessment Weighting
Coursework Coursework on financial derivatives 50
Coursework Coursework on optimisation problems 50

None

## Assessment Strategy

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

• Individual knowledge, skill, and problem-solving abilities

• Understanding of the concepts of derivative pricing, and underlying mathematics.

• The ability to use quantum optimizers for given simple optimization problems

• The ability to analyse the suitability of a given optimization problem for quantum optimization and devise a strategy to choose the correct algorithm

Thus, the summative assessment for this module consists of two coursework assignments:

• Coursework assignment 1, involving real data analysis and modelling of financial instruments.

• Coursework assignment 2, covering a quantum optimization project

Formative assessment:

• Problem sheets are issued during the course, and feedback will be given during tutorial sessions.

Feedback:

• Students will receive immediate verbal feedback during computational laboratory hours where they will be working on problems, with help from staff.

• Verbal feedback is provided by the lecturer during the tutorials (e.g., when exercises are worked out)

## Module aims

• To expose the students to the fundamentals of financial derivatives, introducing theoretical concepts and practical applications using Python.
• To develop skills in pricing and analyzing various derivatives instruments
• To enable students to construct and analyse complex strategies using derivatives.
• To equip students with an understanding of the key difference between classical and quantum computers.
• To equip students with the understanding of the types of optimization challenges in industry and finance which can be handled by classical and quantum optimization algorithms.
• To equip students with the basic understanding of the different approaches and quantum algorithms and how real-life problems could be translated and solved by quantum processors and quantum annealers.

## Learning outcomes

 Attributes Developed 001 Understand the mathematics and models that underpin the analysis of financial data, including the properties of random variables, probability distributions and share price models and be able to assess their validity and remit. CKT 002 Know about a range of common financial derivatives, be able to explain financial terminology and produce pay-off and profit diagrams for forward contracts, put and call options. CKP 003 Understand basic portfolio optimization theory and types of trading and traders CKP 004 To understand optimization algorithms and to be able to implement them with an understanding of errors and the advantages associated with quantum computers CKPT 005 Demonstrate an understanding of the principles of quantum circuits and quantum annealing. CKT 006 Demonstrate working knowledge of applying quantum computing to optimization problems. CKP

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:

• Help students develop an understanding of how the ideas of stochastic processes can be applied to financial derivatives.

• Develop computational skills with a focus on financial data

• Introduce the structure and flow of quantum algorithms for optimization problems in finance and elsewhere and investigate how they achieve the desired speedup compared to classical algorithms

• Give students the skills to translate real-world optimization problems into a form that may be suitable for processing on quantum processors

• Introduce students to the challenges and limitations of current hardware and how to decide on suitability of using quantum optimization

Thus, the learning and teaching methods include:

• traditional lecture-based sessions to cover background theory.

• hands-on sessions in a computer laboratory for computer-based problem solving.

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: PHY3065

## Other information

The School of Mathematics and Physics is committed to developing graduates with strengths in Employability, Digital Capabilities, Global and Cultural Capabilities, Sustainability, and Resourcefulness and Resilience. This module is designed to allow students to develop knowledge, skills, and capabilities in the following areas:

Sustainability. Sustainable investing aiming at generating long-term financial returns while advancing sustainable solutions and outcomes are a central aspect throughout the module; we explore environmental, social, and governance investing as well as ethical and green investing throughout the module. Optimization of resources is a central topic of discussion in quantum optimization, and this has a wider set of applications in resource management, crucial for sustainability.

Digital capabilities. Students will apply computational skills in Python to engage with and analyse complex financial datasets (¿big data¿). Bespoke software packages will also be used to present complex data clearly and understandable.

Resourcefulness and Resilience: The knowledge of how to optimize (as applied here to various logistics, traffic, communications, finance, and science) will be beneficial in a range of applications, thus enhancing the resourcefulness of the students.

Global and Cultural Capabilities: The first part of the module introduces students to real-life Quant professionals working in the city. In the second half of this module, students will be introduced to the idea of quantum computers and how they compare to their classical counterparts, along with the areas in which they may transform the world. These ideas, in turn, are beneficial towards enhancing global and cultural intelligence of the students.

Employability: During the first part of the module, the leaders in the financial sector will visit either in person or online to share their experience and provide advice regarding graduate jobs and interviews. This module introduces students to the structures of financial markets, along with the concepts of financial derivatives, portfolio management, valuation, and risk management. All of these will greatly enhance student employability in financial and related industries.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Physics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Astronomy BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Nuclear Astrophysics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Statistics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics with Music BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Quantum Computing BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Nuclear Astrophysics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Astronomy MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics with Quantum Computing MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Physics MSc 1 Optional A weighted aggregate mark of 40% is required to pass the module
Financial Mathematics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics MMath 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics BSc (Hons) 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MPhys 1 Optional A weighted aggregate mark of 40% is required to pass the module
Mathematics and Physics MMath 1 Optional A weighted aggregate mark of 40% 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.