MTH208: 2025 Coursework
• Total marks for the coursework are 15.
• You just need to choose one of the following two questions to write a report.
• There is no specific word limitations but you are expected to address all the questions clearly and thoroughly with 8000 words.
• You may complete the work by using the provided solution sheet in word or tex.
• Please submit the completed solution via a submission link provided on the LMO.
• All the learning materials on the LMO can be referenced including lecture notes, lab codes, extra reading materials. Seeking help from AI is allowed, but you must complete the coursework inde- pendently and explain how the AI is used and the percent of contributions by AI. Any form of plagiarism, collusion and fabrication of data is not allowed.
• Please convert your file into pdf before submission.
• Please name your submission in the form MTH208Final+ID+ZhangSan.pdf
• The coursework will be available at 9:00 AM on May 15th, and the deadline for submission is 9:00 AM on May 23rd. Submissions will still be accepted after the deadline, but a penalty of 5% of the total marks will be deducted for less than 5 days of delay. Work received more than five working days after the submission deadline will receive a mark of zero.
Question I: Application of Numerical Methods in Financial Mathematics
1 Introduction
Options are financial instruments that give the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price (strike price) on or before a specified date (maturity). The price of an option depends on several factors, including the current price of the underlying asset, the strike price, the time to maturity, the risk-free interest rate, and the volatility of the underlying asset.
1.1 Implied Volatility
Implied volatility is a crucial parameter in option pricing. It represents the market’s expectation of the future volatility of the underlying asset. The Black-Scholes model is a widely used formula for pricing European options. However, the Black-Scholes model does not provide a direct formula for implied volatility. Instead, it must be found by solving the Black-Scholes equation for the volatility parameter, given the market price of the option.
1.2 Black-Scholes Formula
The Black-Schoies formula for the price of a European cail option is given by:
where
and N (·) is the cumulative distribution function of the standard normal distribution. To find the implied volatility σ, you need to solve the equation C(σ) = Cmarket for σ .
2 Problem Statement
In the context of implied volatility, we need to solve the equation C(σ) = Cmarket , where C(σ) is the Black-Scholes price of the option as a function of volatility σ, and Cmarket is the observed market price of the option. Suppose you are given the following information about a European call option:
- Current stock price S = 100;
- Strike price K = 100;
- Time to maturity T = 1 year;
- Risk-free interest rate r = 0.05
- Market price of the option Cmarket = 10.
Using the Black-Scholes formula, you need to find the implied volatility σ by solving the equation
C(σ) = Cmarket .
Change the problem into a root finding problem. Then compare the performance of different root finding methods including the bisection method (Use an initial interval for volatility, such as [0.1,0.5] (10% to 50%) and initial guess, e.g., σ = 0.2); Newton’s method, Secant method etc.
In the report, you should recall the background of the problem, then present the idea and detailed im- plementation of each numerical methods; Evaluate the performance of each method in terms of
-Convergence speed (number of iterations to reach a solution).
-Computational time (time taken to reach a solution).
-Accuracy (how close the solution is to the true implied volatility).
You may also have a discussion of the advantages and disadvantages of each method. Provide a recom- mendation on which method might be most suitable for calculating implied volatility in different scenarios. Also put your MATLAB (or other programming language) code for each method as an appendix. If you seek help from AI in completing project, please also mention how the work is carried out and the percent of contribution by AI.
Project assessment: The assessment of the report will mainly consider two parts:
• the content of the report (10 marks) including research contribution, methodology and analysis, results and discussions;
• format of the report (5 marks) including clear structure and format, clear figures and tables with captions, proper citations and references, precise presentation of results
3 Suggested Structure of the Report
- Title: xxxxxxxxxxxx
- Author: [Your Name]
- ID: [Current Date]
- Module Code: [Course Name] Table of Contents
1. Introduction
2. Numerical Methods
3. Performance Evaluation and Discussion
4. References
5. Appendix-Codes
Question II: Wave Packet
A Gaussian wave packet refers to a wave packet whose profile follows a Gaussian distribution, and it has important applications in both optics and quantum mechanics. In optics, it can be used as the electric field function of a laser pulse. In quantum mechanics, it is often employed as a wave function. The Gaussian wave packet is represented by a complex function as (where A is a complex number).
As an specific example, we take A = 1, a = 1/20 and k = 5. The plot is shown in Figure 1. In this
Figure 1: Gaussian wave packet. Blue curve for real part; red curve for imaginary part.
project, we just consider
and its approximation in [-10, 10].
Project goal: Explore and compare various numerical methods (interpolation with various basis or fitting in discrete and continuous cases) for approximating the absolute value function. You may provide basic ideas of each numerical method, give the implementation details and then compare their performance in terms of accuracy, implementation details and computational efficiency (You don't need to include the code in the report)
Project suggestions: You are encouraged to refer to textbooks, academic papers, and online resources on numerical methods for function approximation to enrich your understanding and imple- mentation of the numerical methods. By incorporating these additional details, you are expected to be prompted to delve deeper into the theoretical and practical aspects of numerical methods, enabling a more comprehensive exploration and comparison of the methods for approximating the absolute value function.
Project assessment: The assessment of the report will mainly consider two parts:
• the content of the report (10 marks) including research contribution, methodology and analysis, results and discussions;
• format of the report (5 marks) including clear structure and format, clear figures and tables with captions, proper citations and references, precise presentation of results