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MTH319 Financial Engineering
Financial Engineering
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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 2.0% (r = 2.0%) 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.
2) the stock price follows a geometric Brownian motion. Under a risk-neutral probability measure
Q, the process for the stock price is given by:
( )
( )
Q
t
dS t
rdt dW
S t
And the current stock price is $100, e.g., 0 $100S , and the volatility of this stock is 20% per
annum, e.g., 20% .
3) 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 1 year (e.g., TA= 1 year). YOU may decide the type of
Option A: either call/put.
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:
2
0 0
2
( ) ; (1 )
T
rT rT
T S K
c c
prob S K e prob e
K K
%
% % % ,
based on the simulation conducted in a) by setting a range of
1
{ }( 0,1,...,100)
2
K K i i . Plot these estimated probabilities. (15%)
II. PART II (Trading Strategies) (40%)
Suppose that there is another European option (Option B) written on this stock with a different
strike K=$90 but with maturity of 2 years (TB=2 years). YOU may decide the type of Option B:
either call/put.
Assume now that you have taken a short position in the European option with the quantity of
1 (e.g., nA= -1)
d) Develop the delta-neutral strategy to hedge your short position in Option A by using the
portfolio of stocks and bonds. Plot 1) the time series of replication errors of your hedging
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 short 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%)
Summary all the results in Part I and Part II and make comments on your results accordingly
with the maximum of 600 words. For example, you may report your numerical results by
briefly describe the methodology at first, and make 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 20, 2022.
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
overview of the project, including the title of the project, the setup of the securities market
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
Max
Points
Examiner
Coding
1. Code Quality
(Reliability/Testability/Reusability)
20
2. Accuracy of programing outputs 20
3. Programing design/structure 15
4.Use of data structure and algorithms in
coding blocks
15
5. Supportive comments/interpretations in
key steps/blocks
10
6. Illustrations and accessibility in
numerical results
10
Report Presentation*
7. A brief description of the project 5
8. Presentation of report (including writing
style, grammar, use of graphics and tables)
MTH319
Financial EngineeringINSTRUCTIONS 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 2.0% (r = 2.0%) 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.
2) the stock price follows a geometric Brownian motion. Under a risk-neutral probability measure
Q, the process for the stock price is given by:
( )
( )
Q
t
dS t
rdt dW
S t
And the current stock price is $100, e.g., 0 $100S , and the volatility of this stock is 20% per
annum, e.g., 20% .
3) 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 1 year (e.g., TA= 1 year). YOU may decide the type of
Option A: either call/put.
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:
2
0 0
2
( ) ; (1 )
T
rT rT
T S K
c c
prob S K e prob e
K K
%
% % % ,
based on the simulation conducted in a) by setting a range of
1
{ }( 0,1,...,100)
2
K K i i . Plot these estimated probabilities. (15%)
II. PART II (Trading Strategies) (40%)
Suppose that there is another European option (Option B) written on this stock with a different
strike K=$90 but with maturity of 2 years (TB=2 years). YOU may decide the type of Option B:
either call/put.
Assume now that you have taken a short position in the European option with the quantity of
1 (e.g., nA= -1)
d) Develop the delta-neutral strategy to hedge your short position in Option A by using the
portfolio of stocks and bonds. Plot 1) the time series of replication errors of your hedging
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 short 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%)
Summary all the results in Part I and Part II and make comments on your results accordingly
with the maximum of 600 words. For example, you may report your numerical results by
briefly describe the methodology at first, and make 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 20, 2022.
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
overview of the project, including the title of the project, the setup of the securities market
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
Max
Points
Examiner
Coding
1. Code Quality
(Reliability/Testability/Reusability)
20
2. Accuracy of programing outputs 20
3. Programing design/structure 15
4.Use of data structure and algorithms in
coding blocks
15
5. Supportive comments/interpretations in
key steps/blocks
10
6. Illustrations and accessibility in
numerical results
10
Report Presentation*
7. A brief description of the project 5
8. Presentation of report (including writing
style, grammar, use of graphics and tables)
Case category
- 经济Economics
- 金融Finance
- 会计Accounting
- 工商管理Business Management
- 计算机computer science
- 商业分析business analysis
- 法律Law
- 管理Management
- 市场营销Marketing
- 哲学Philosophy
- 心理学Psychology
- 物理Physical
- 统计学statistics
- 电气工程electric engineering
- 化学chemistry
- 生物biology
- 数学mathematics
- 语言学linguistics
- 土木工程civil engineering
- 工程力学engineering mechanics
- 艺术art
Tag cloud
关键词一
关键词二
关键词三
关键词四
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2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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