MTH319
FAM
1st SEMESTER 2023/24 Assignment ONE
Financial Engineering
SUBMISSION DEADLINE: 5:00 PM on Sunday November 12, 2023 INSTRUCTIONS TO CANDIDATES
1. The assignment comprises 15% weight of the final module mark.
2. Write a report about the performance of hedging strategies (details and guidelines
attached).
3. The report must be written in English, associated with the supportive excel files.
4. University policy on late submission will be followed.
Introduction
This part of the course assessment is worth 15% of the final mark for the course, and consists of a take-home course assignment that will be worked on and submitted properly.
This project aims to practice your skills in pricing, hedging and trading involving options, stocks and bonds in the context of Monte Carlo simulations.
SCENARIO:
Suppose you are working on a security market consisting of a stock, a Treasury bond and a European (call/put) option:
1)
a Treasury bond delivers the annual yield of 2.50% (r = 2.50%) with continuous compounding, and has the infinity maturity (TBond=∞). Under a risk-neutral probability measure Q, the bond price follows a process as follows:
dB(t) = rdt B(t)
And the current bond price is $1.0, e.g., B(0) =1.0.
the stock price follows a geometric Brownian motion. Under a risk-neutral probability measure Q, the process for the stock price is given by:
S(t)
And the current stock price is $100, e.g., S0 = $100 , and the volatility of this stock is 25% per
annum, e.g., s = 25% .
A European option on this stock (or Option A) is specified with the strike price of $100 (e.g., K=$100) and the time-to-maturity of one year (e.g., TA= 1 year). YOU may decide the type of Option A: either call/put.
2)
dS(t) =rdt+sdWQ
3)
t
Requirements:
I. PART I (Numerical Analysis) (50%)
a) Conduct the Monte Carlo simulation and work out the price of this European option with
the number of simulation paths equal to M= 5000. (20%)
b) Conduct the performance analysis 1) by changing the number of simulation paths (e.g.,
M=1000, 3000, 5000, 8000, 10000 and etc.) 2) by changing the number of time steps in each simulation path (e.g., N= 500, 1000, 3000, 5000, 7000 and etc.). Plot your numerical results. (15%)
c) Estimate the probability functions of strike price at maturity (e.g., TA=1 year) by using the result as follows:
%
prob(ST > K) and prob(1ST =K% )
based on the simulated (call/put) option prices obtained in a) with a specific combination of (M, N) by setting a range of
K Î{K ± 2 i}(i = 0,1,...,100) .
For example, for those simulated call option prices, we have:
%
¶c
prob(S >K)=-erT 0 ;prob(1
¶2c )=erT 0
Suppose that there is another European option (Option B) written on this stock with a different strike K=$95 but with maturity of 3 years (TB=3 years). YOU may decide the type of Option B: either call/put.
Assume now that you have taken a long position in the Option A with the quantity of 1 (e.g., nA = +1)
d) Develop the delta-neutral strategy to hedge your long position in Option A by using the portfolio of stocks and bonds. Plot 1) the time series of replication errors of your hedging
%
T %S=K% 2
%
Plot these estimated probabilities. (15%) II. PART II (Trading Strategies) (40%)
T
¶K ¶K .
portfolio (involving option A, stocks and bonds) and 2) terminal payoff of Option A against the payoff of the portfolio of stocks and bonds across different stock prices at maturity. (20%)
e) Develop the delta-gamma neutral strategy to hedge your long position in Option A by using the portfolio of stocks, bonds and options B. Plot 1) the time series of replication errors of your hedged portfolio (involving option A, stocks, bonds and option B) and 2) the terminal payoff of Option A against the payoff of the portfolio of stocks, bonds and option B across different stock prices at maturity. (20%)
III. PART III (Reporting) (10%)
Summarize all the results in Part I and Part II and make comments on your results accordingly with the maximum of 600 words only in this section (PART III). For example, you may report your numerical results by briefly describing the methodology at first, and making observations/comments on your plots in PART I. Also, you may assess the performance and features (i.e., advantages and disadvantages) of the hedging strategies developed in PART II.
Assignment Guideline
This assignment assesses Learning Outcome A-D.
Note that you need:
1) Show all the results with comprehensive interpretations in a report (with the maximum 15 pages);
2) Show other relevant and supportive results in Python programme code or other program language code.
As the outcome from this project, you are expected to submit a report, associated with the code files. Please disclose the detailed process/results as much as possible (e.g., the methodology and implementation of Monte Carlo simulation and the development of delta-neutral and delta-gamma-neutral hedging strategies). The deadline of the assignment submission to LMO is at 5pm on November 12, 2023.
Your analysis has to be your own, demonstrating your own ideas and independent and critical thinking (and not those of somebody else).
However, your report must include: 1) A brief description of the project
In the first section, your work should contain a formal introductory section that provides an overviewof theproject,includingthetitleoftheproject,thesetupofthesecuritiesmarket and the process of asset prices and the main goals that this project aims to achieve.
2) Process of the optimal hedging strategies and performance analysis
After specifying the security market that you are working on, you are ready to complete the tasks in I) and II) which can be presented in two separate sub-sections. It is suggested that you describe the process of Monte Carlo simulation in the market in detail and report the performance analysis of your implementation and the estimated probability plots as well, and then provide the details about the development of the delta-neutral and delta-gamma- neutral hedging strategies, followed the performance analysis of hedging portfolios. Also, please clearly explain how you obtain the required results, supported by your models in code files.
Note: the lecture and tutorial in Week 5&6 will demonstrate the strategy how to complete these tasks in a simplified case.
3) Comments on results
In the final section, you need evaluate the implementation process of PART I and II in terms of the pricing and hedging performance in the context of Monte Carlo simulation. It is very important for you to understand those theoretical results discussed in the lectures by conducting practical investigations in a specified market, which will help to improve your skills in financial modelling and investment management in practice. The section is subject to the limit of 600 words.
Submission: each of you submits only ONE report and ONE code file. In the first page, please list your name(s) and student ID(s).
Assessment Form for MTH319 Assignment
Student’s Name:
Student's ID:
Project Title:
Marks for Compulsory Questions
1. Code Quality (Reliability/Testability/Reusability)
2. Accuracy of programing outputs
3. Programing design/structure
Coding
4.Use of data structure and algorithms in coding blocks
5. Supportive comments/interpretations in key steps/blocks
6. Illustrations and accessibility in numerical results
Report Presentation*
7. A brief description of the project
8. Presentation of report (including writing style, grammar, use of graphics and tables)
Max Points
Examiner
Total Marks for Project
Please add up
Examiner Name:
Signature: _______________
Date: ____________
