PAPER CODE
EXAMINER
DEPARTMENT
TEL MTH319 AM 1729
1st SEMESTER 2022/23 Assignment TWO
Financial Engineering
SUBMISSION DEADLINE: 5:00 PM on Sunday December 17, 2023 INSTRUCTIONS TO CANDIDATES
1. The assignment comprises 15% weight of the final module mark.
2. Write a report about the performance of short-rate models (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.
PAPER CODE: MTH319/23-24/S1/ASSIGNMENT TWO
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 individually.
This project aims to practice your skills in modeling short/forward interest rates, model estimation, and pricing bond options in the context of Monte Carlo simulations.
SCENARIO:
Suppose you are working on a CIR model for short-term rates:
Requirements:
PART I: Short-Rate Modeling (30%)
1) Conduct Monte Carlo simulations for the CIR model i) in a standard approach; ii) by using
variance reduction methods and plot the simulated short rates. The required parameters (r(0), a, b and σ) are estimated from the market data in the excel file of “Government (Zero-Coupon) Bond Yield” available on LMO. The estimation details can be referred to the appendix of “Appendix A - Option prices using Vasicek and CIR (Assignment No2)” on LMO.
PART II: Bond Pricing and Yield Curve (30%)
2) Bydiscretizingtheprocessin(1)withatimeinterval(∆),e.g., 0=t0 <tD <t2D <L<tND =T , work out the solution r(t) to the CIR model with the following conditional distribution:
dr(t)=a(b-r(t))dt+s r(t)dWt
(1)
This model assumes that short rate is normally distributed and has so-called mean-reverting process
(under Q).
where
r(t���∆)|F(���������)∆~���(���, Σ),
μ = b���1 − ���������∆��� + ���������∆r���t(���������)∆���
PAPER CODE: MTH319/23-24/S1/ASSIGNMENT TWO
��������� ������
Σ = 2��� (1 − ���������∆)��� + ��� (���������∆ − ������������∆)r���t(���������)∆���
3) Given the zero-coupon bond prices P(t,T) in the CIR model as follows:
where
T
P(t,T)=EQ[e-òt rsds ´1]=eA(t,T)-B(t,T)r(t),
B(t,T)= 2(eg(T-t) -1) (g+a)(eg(T-t) -1)+2g
2g e( a +g )(T -t ) / 2 A(t,T)=ln((g+a)(eg(T-t) -1)+2g)
2 ab / s 2
g= a2+2s2 Accordingly, the yield to maturity y(t,T) is given by:
y(t,T)=- 1 ln(P(t,T))=-[A(t,T)-B(t,T)r(t)] T-t T-t
Please answer the following questions:
• Estimate the dynamics of bond prices over time with simulated rate paths in 1)
• Estimate the dynamics of yields over time with simulated rate paths in 1)
• Estimate the term structure of yields at time 0<t<T with simulated rate paths in 1), given
T = [0.5, 1, 2, 3, 5, 7, 9, 10, 15, 18, 20, 25, 30, 40, 50]
PART III: Forward Rates in HJM Model (30%)
4) If the short-term rate r(t) follows the CIR process, the forward rate then follows the process:
dF(t,T) = m*(t,T)dt +s(t,T)dW ,0 £ t £ T t
Please answer the following questions:
• Work out the drift and diffusion term of the forward rate F(t,T) process
• Verify that under the CIR model, the no-arbitrage condition in the HJM model is satisfied:
m * ( t , T ) = s ( t , T )s * ( t , T ) = s ( t , T ) ò T s ( t , u ) d u t
• Conduct a Monte Carlo simulation for forward rates implied from the CIR Model
• Plot the dynamics of the drift and diffusion term of the forward rates PAPER CODE: MTH319/23-24/S1/ASSIGNMENT TWO
PART IV (Reporting) (10%)
In this section, please outline all the results in Part I-III and make comments on your results accordingly with the maximum of 600 words. For example, you may comment on the procedure of parameter estimation, the performance of the Monte-Carlo simulation in short-rate/forward rate modeling and bond pricing, and the sensitivity of the term structure of yields towards parameters.
PAPER CODE: MTH319/23-24/S1/ASSIGNMENT TWO
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/excel 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 short- rate/forward rates/yields/bond prices). The deadline of the assignment submission to LMO is at 5pm on December 17, 2023 (Week 14).
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,theprocessofinterestratesand the main goals that this project aims to achieve.
2) Process of short-rate/forward rate modeling and bond pricing and performance analysis
After specifying the setup that you are working on, you are ready to complete the tasks in I)-III) which can be presented in three separate sections. It is suggested that you describe the process of parameter estimation and Monte Carlo simulation in detail and report the performance analysis of your implementation for short-rates/bond prices/yields, followed the analysis on the no-arbitrage condition in the HJM model and the performance analysis of the Monte Carlo simulation for forward rates. Also, you may further investigate the sensitivities of your results towards parameters. Please clearly explain how you obtain the required results, supported by your models in code/excel files.
Note: the lecture and tutorial in Week 7-9 will demonstrate the strategy how to complete these tasks in a simplified case. All other references are available on LMO.
3) Comments on results
In the final section, you need evaluate the implementation process of PART I-III in terms of short-rate/forward rate modeling and bond pricing and their 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).
PAPER CODE: MTH319/23-24/S1/ASSIGNMENT TWO
PAPER CODE: MTH319/23-24/S1/ASSIGNMENT TWO
Assessment Form for MTH319 Assignment
Student’s Name: Student's ID: Project Title:
Marks for Compulsory Questions
Coding
1. Code Quality (Reliability/Testability/Reusability)
2. Accuracy of programing outputs
3. Programing design/structure
4.Use of data structure and algorithms in coding blocks
5. Supportive comments/interpretations in key steps/blocks
Report Presentation*
6. A brief description of the project
7. Illustrations and accessibilities in numerical/mathematical results
8. Presentation of report (including writing style, grammar, use of graphics and tables)
Total Marks for Project
Please add up
Max Points
Examiner
Examiner Name:
Signature: Date: _______________ ____________
PAPER CODE: MTH319/23-24/S1/ASSIGNMENT TWO
