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Page 1 of 8
Assessment Information for Exam in 24-hour
timed window
Module name: MSIN0107
Module code: Advanced Quantitative Finance
Module leader names: Dennis Kristensen & Ming Yang
Academic year: 2020/21
Term 1, 2 or 3: 2
Type of assessment: 24-hour timed Online Exam
Nature of assessment – individual or group: Individual
Content of this Assessment Brief
Section Content
A Core information
B Requirements
C Module learning outcomes covered in this
assessment
D Assessment criteria
E Groupwork instructions (if applicable)
F Additional information from module leader (if
applicable)
Page 2 of 8
Section A: Core information
This assessment is
marked out of:
100 marks
% weighting of this
assessment within total
module mark
80%
Time allowed for
completion of this
assessment
This assessment should take approximately two hours to
complete. You may take longer to complete it if you wish to.
You have a window of 24 hours from release to submission to
complete it.
In addition to answering/responding to the
questions/requirements, this 24-hour period provides
enough time for you to prepare your document for
submission (including, as appropriate, copying, pasting,
saving electronically) and loading to Moodle.
If you have a SORA which allows for additional writing time
for examinations/tests, this has been factored into the 24-
hour window and no additional time in addition to the 24-
hour period is available.
Word count/number of
pages - maximum
2,000
Determining word count
impacted by Turnitin
After submission to Turnitin, the Turnitin recorded word
count is usually higher than the word count in a Word
document.
Where the assessment brief specifies a maximum word
count, on the front cover of your submission record the
number of words as recorded in your Word document.
It is the Word document word count which will be taken
account of in marking, NOT the Turnitin word count.
Footnotes, appendices,
tables, figures, diagrams,
charts included
in/excluded from word
count/page length?
Any footnotes, appendices are not included in page limit.
Bibliographies, reference
lists included
in/excluded from word
count?
Title page, table of contents, any bibliography are excluded from the
page limit.
Penalty for exceeding
specified word
count/page length?
Where there is a specified word count/page length and this is
exceeded, yes there is a penalty: 10 percentage points
deduction, capped at 40% for Levels 4,5, 6, and 50% for Level
7. Refer to Academic Manual Section 3: Module Assessment -
3.13 Word Counts.
Where there is no specified word count/page length no
penalty applies.
Requirements for/use of
references
This assessment is an ‘open book’ exam/test which you
attempt at home, at UCL, or indeed in any other location. It is
not invigilated. In principle it should take no longer than the
Page 3 of 8
time specified above to complete. However, you have a 24-
hour timed window in which to download the assessment, to
complete it, and to submit it to Moodle.
In responding to the demands of this assessment, you may
draw upon course materials – lecture slides, notes, handouts,
readings, textbook(s) - you engaged with in your studying of
this module.
You are not expected or required to find and use new
materials. In a formal ‘sit-down’ invigilated exam/test you
would not be able to find and draw upon new materials – you
would draw upon what you learned from your studying of the
module.
You may refer to such course materials but you should not be
copying word for word from lecture slides, notes, handouts,
readings, textbook(s) you engaged with in your studying of
this module.
You should capture, articulate and communicate your views,
thoughts and learning in your own words.
If you do provide quotes from any lecture slides, notes,
handouts, readings, textbook(s) you should cite them and
provide references in the usual way.
Be aware that a number of academic misconduct checks,
including the use of Turnitin, are available to your module
leader.
If required/where appropriate UCL Academic Misconduct
penalties may be applied (see immediately below).
Academic misconduct
(including plagiarism)
Academic integrity is paramount.
It is expected that your submission and content will be your
own work with no academic misconduct.
Academic Misconduct is defined as any action or attempted
action, including collusion with other students, that may
result in a student obtaining an unfair academic
advantage. There are severe penalties for Academic
Misconduct, including, where appropriate and required,
exclusion from UCL.
Refer to Academic Manual Section 9: Student Academic
Misconduct Procedure - 9.2 Definitions.
Submission date Friday 26th March 2021
Submission time 16.00 UK Time
Penalty for late
submission?
Yes. Standard UCL penalties apply. Students should refer to
https://www.ucl.ac.uk/academic-manual/chapters/chapter-4-
assessment-framework-taught-programmes/section-3-module-
assessment#3.12
Submitting your
assignment
Late submissions are not permitted
Anonymity of identity.
Normally, all submissions
are anonymous unless
Anonymity is required.
Your name should NOT appear anywhere on your submission.
Page 4 of 8
the nature of the
submission is such that
anonymity is not
appropriate, illustratively
as in presentations or
where minutes of group
meetings are required as
part of a group work
submission
Return and status of
marked assignments
At the latest this will be within 4 weeks from the date
of submission as per UCL guidelines, but we will
endeavour to return it earlier than this.
Assessments are subject to appropriate double
marking/scrutiny, and internal quality inspection by
a nominated School of Management internal
assessor. All results when first published are
provisional until confirmed by the relevant External
Examiner and the Examination Board.
No appeals regarding your published mark are
available until after confirmation by that
Examination Board. UCL regulations specify that
academic judgment applied within the marking
process cannot be challenged.
Academic Support with this Assessment
Given the nature of this assessment, during the 24-hour window no questions should be directed to
the Module Leader/Module Team. If you have doubts about wording or requirements etc., state your
assumptions. If they are appropriate they will be taken into consideration in marking.
Uploading your submission
Unless specifically instructed otherwise in the assessment document, please upload your
work as a single file via the submission link on Moodle.
o Wherever possible you should type/use Excel for (as appropriate) your answers and
follow instructions later in this assessment document.
o If you do have to include any elements that are not typed/computer generated (e.g.
figures, diagrams, equations etc.), or you are unable to type your answers for any
reason, please follow the advice for submitting handwritten answers for any
submission that requires scanning documents (the webpage refers to 24-hour timed
exams but is applicable to all online submissions including this one).
Page 5 of 8
o If for any reason you are not able to use the app recommended by ISD at the link
above, you can consult the following resources for advice about preparing your
submission:
Submitting handwritten assignments to Moodle using mobile or tablet Devices
- Device Camera
Submitting handwritten assignments to Moodle using mobile or tablet devices
- MS One Drive App
Please DOUBLE CHECK that the file you are uploading is the correct one and is complete
(with all pages visible).Resubmission will not be permitted.
Technical Problems
If you encounter difficulties downloading or submitting your assessment via Moodle, then please
immediately notify (by email) your department (Programme Administrators ONLY), explaining the
problem and including a copy of the work you are trying to submit. ONLY use this approach if you
can show that you have tried to download from/upload to Moodle and encountered technical
difficulties.
Advice and other support
Student Support and Wellbeing
Page 6 of 8
Section B: Requirements
See exam paper at the end of the brief
Page 7 of 8
Section C: Module Learning Outcomes covered in this
Assessment
This assignment contributes towards the achievement of the following stated module
Learning Outcomes as below:
1. Dynamic stochastic asset pricing models
2. Hedging
3. Arbitrage-free pricing
4. Risk-neutral pricing
5. Option pricing
5. Numerical methods in asset pricing
Page 8 of 8
Section D: Assessment criteria
Within each section of this coursework you may be assessed on the following aspects, as
applicable and appropriate to this particular assessment, and should thus consider these aspects
when fulfilling the requirements of each section:
The accuracy of any calculations;
The strengths and quality of your overall analysis and evaluation;
Appropriate use of relevant theoretical models, concepts and frameworks;
The rationale and evidence that you provide in support of your arguments;
The credibility and viability of the evidenced conclusions/recommendations/plans of
action you put forward;
Structure and coherence of your considerations and reports;
As and where required, relevant and appropriate, any references should use either
the Harvard OR Vancouver referencing system (see References, Citations and
Avoiding Plagiarism)
Academic judgement regarding the blend of scope, thrust and communication of
ideas, contentions, evidence, knowledge, arguments, conclusions.
Each part has requirements with allocated marks, maximum word count limits/page
limits and where applicable, templates that are required to be used.
You are advised to refer to the UCL Assessment Criteria Guidelines, located at
https://www.ucl.ac.uk/teaching-learning/sites/teaching-learning/files/migrated-
files/UCL_Assessment_Criteria_Guide.pdf
Section E: Groupwork Instructions
Not applicable as this is an individual assessment
Section F: Additional information from module leaders
N/A
MSIN0107 Advanced Quantitative Finance
Examination paper
2020/21
Examination length: TWENTYFOUR (24) hours
NOTE: Although the window for completion is TWENTY-FOUR (24) hours, this
exam paper is designed to be completed in TWO (2) hours.
There is ONE (1) section to the examination paper. The section consists of FOUR
(4) compulsory questions. It is worth ONE HUNDRED (100) marks.
Module Leaders: Dennis Kristensen and Ming Yang
Internal Assessor: Wei Cui
MSIN0107 2020/21 1 TURN OVER
MSIN0107 Advanced Quantitative Finance
1. Consider a nancial market with two assets over the time interval [0; T ]. Trading
can take place at n+1 discrete time points t0; :::; tn where ti = i, i = 0; 1; :::; n,
and = T=n is the time distance between trading times. The rst asset is risk-
free with interest rate er 0 per time period. The second asset is a stock
whose price in period i (= 1; :::; n) is given by
Si = S(i1) exp (+ vi) ;
where and are coe¢ cients and vi, i = 1; ::; n, are i.i.d. with
P (vi = 1) = p1; P (vi = 0) = p2; P (vi = 1) = p3:
Here, p1 + p2 + p3 = 1.
(a) [5 pts] Suppose you know that p1 = p3. How would you estimate the two
parameters and using observed stock prices, Si, i = 1; :::; n?
(b) [5 pts] State necessary and su¢ cient conditions on the parameters of the
model under which you cannot construct a self- nancing portfolio in period
i that earns a non-negative pay-o¤ with probability one in period i + 1,
and a positive pay-o¤ in at least one of the states of period i+ 1.
(c) [5 pts] Consider a contingent claim that expires in period i+ 1. Can you
always establish a portfolio in period i that replicates the claims pay-o¤s
in period i+ 1? Explain.
(d) [5 pts] Let Q be an alternative measure characterised by
Q (vi = 1) = q1; Q (vi = 0) = q2; Q (vi = 1) = q3;
with q1 +q2 +q3 = 1. When is Q a risk-neutral measure? Is there a unique
risk-neutral measure in this market? Explain.
(e) [5 pts] Suppose that = n = 0:1=n and = n = 0:2=
p
n and T = 1.
Under a given sequence of risk-neutral measures Qn characterised by
Qn (vi = 1) = qn;1; Qn (vi = 0) = qn;2; Qn (vi = 1) = qn;3;
with qn;1 = qn;3, derive the limiting distribution of log (ST =S0) as n!1.
2. Consider a given non-dividend paying stock whose price, St, satis es
dSt = Stdt+ StdWt;
whereWt is a Brownian motion. We here measure time in years and the risk-free
rate is 1.5% per annum.
MSIN0107 2020/21 2 CONTINUED
MSIN0107 Advanced Quantitative Finance
(a) [5 pts] Suppose that you have observed the following weekly prices of the
stock:
30:5; 32:2; 31:1; 30:4; 30:3; 31:9; 32:1; 31:0; 30:1; 30:0:
Estimate the stock price volatility. Use this volatility estimate to answer
the following questions.
(b) [5 pts] The current stock price is £ 30. What is todays price of a European
call that expires in two years with a strike price of £ 40?
(c) [5 pts] Suppose you have purchased an option giving you the right to pay
£ 2 one year from today to buy the call option described in part (b) of the
question. Verify that you would exercise this option if the stock price in
one year is greater than 35:3458.
(d) [5 pts] What is todays price of the option described in part (c)?
(e) [5 pts] What is the price of the option giving you the right to sell the the
option described in part (b) in one years time for £ 2?
3. A rm has total asset value of V0 = $100 million. The volatility of the rms
existing asset is = 30% per annum. The rm does not pay any dividend to
equity holders. The rm also has a zero-coupon debt with total face value of
$150million and maturity of T = 5 years. Suppose the rms asset value evolves
as a Geometric Brownian motion (i.e., satis es the assumptions of the Black-
Scholes formula). In all of the following exercises, assume that the continuously
compounded interest rate is r = 8% per annum.
(a) [4 points] What is the present value of the existing debt, B0? What is the
present value of the existing equity, E0?
At year 0, the rm is considering an investment project that costs f = $10million
in present value. If the project is nanced, it will increase the rms asset value
immediately by $11 million, i.e., the rms asset value becomes $111 million.
Suppose the rms asset value follows the same Geometric Brownian motion
(only initial asset value changes) after the new investment. The rm is consid-
ering nancing the investment by issuing new debt with 5 years maturity (i.e.,
matures at the same time as the existing debt).
Financing through junior debt. Suppose that the rm will issue junior
zero-coupon debt to nance the project (This means the newly issued debt will
have lower seniority than the existing debt.).
(b) [2 points] What is the NPV of the project? Should the rm invest in the
project according to the NPV rule?
(c) [8 points] Note that if the rm were to nance the investment through
junior debt with face value of F million, the junior debt holder will be
paid F at the maturity date if the total value of the rms asset exceeds
MSIN0107 2020/21 3 TURN OVER
MSIN0107 Advanced Quantitative Finance
150 + F million. The junior debt holder will recover only part of the face
value of the debt if the total asset value of the rm is between 150 and
150 + F , and will not be paid at all if the total value of the rms asset is
below 150 (in which case, only the senior debt holder gets paid). What is
the lowest face value of the zero-coupon debt that the rm has to promise
to the new debt holders in order to nance the $10 million ? (You may
need to compute this numerically, e.g., using the EXCEL solver add-in).
Note that in order to nance $10 million from the new debt holder, the
present value of the debt has to be at least $10 million.
(d) [5 points] Suppose the rm did issue the debt with the lowest face value
calculated in (c), what will be the present value of the equity after the
issuance of the debt? Does the nance of the debt increase or decrease the
value of the equity? What is the present value of the existing debt (i.e.,
the senior debt) after the issuance of the new debt (i.e., the junior debt)?
(e) [6 points] Suppose you are the manager of the rm and you make decisions
on behalf of the equity holders. Would you nance the project? Does your
decision agree with the NPV rule? Explain the intuition.
4. This question will guide you to solve the Merton problem with T = 1 and
u (c; t) = et ln c. There are two assets in the economy. One is a non-dividend
stock, the price of which follows
dSt
St
= dt+ dZt ,
where Zt is a standard Brownian motion. The other is a riskfree asset, the price
of which follows dXt = rXtdt. An agent starts with wealth W0 and needs to
choose consumption fctg and allocate between the two assets to maximize his
expected utility.
(a) [3 points] Let t denote the fraction of the agents wealth invested in the
risky asset at t. Derive the stochastic di¤erential equation for the agents
wealth Wt.
(b) [4 points] Since T = 1, the problem is stationary and we can solve the
problem under the stationary value function J (W; t) = etV (W ). Write
down the HJB equation for V ().
(c) [4 points] Derive the rst order conditions with respect to c and .
(d) [5 points] Derive the ODE for V (). (Note that the ODE should not
explicitly contain c and . It only consists of , r, , , W , V , VW , and
VWW .)
(e) [5 points] Solve V () from the ODE. (Hint: conjecture V (W ) = a lnW +b
and solve for a and b.)
(f) [4 points] Solve for the optimal policy of c and .
MSIN0107 2020/21 4 END OF PAPER
学霸联盟
Assessment Information for Exam in 24-hour
timed window
Module name: MSIN0107
Module code: Advanced Quantitative Finance
Module leader names: Dennis Kristensen & Ming Yang
Academic year: 2020/21
Term 1, 2 or 3: 2
Type of assessment: 24-hour timed Online Exam
Nature of assessment – individual or group: Individual
Content of this Assessment Brief
Section Content
A Core information
B Requirements
C Module learning outcomes covered in this
assessment
D Assessment criteria
E Groupwork instructions (if applicable)
F Additional information from module leader (if
applicable)
Page 2 of 8
Section A: Core information
This assessment is
marked out of:
100 marks
% weighting of this
assessment within total
module mark
80%
Time allowed for
completion of this
assessment
This assessment should take approximately two hours to
complete. You may take longer to complete it if you wish to.
You have a window of 24 hours from release to submission to
complete it.
In addition to answering/responding to the
questions/requirements, this 24-hour period provides
enough time for you to prepare your document for
submission (including, as appropriate, copying, pasting,
saving electronically) and loading to Moodle.
If you have a SORA which allows for additional writing time
for examinations/tests, this has been factored into the 24-
hour window and no additional time in addition to the 24-
hour period is available.
Word count/number of
pages - maximum
2,000
Determining word count
impacted by Turnitin
After submission to Turnitin, the Turnitin recorded word
count is usually higher than the word count in a Word
document.
Where the assessment brief specifies a maximum word
count, on the front cover of your submission record the
number of words as recorded in your Word document.
It is the Word document word count which will be taken
account of in marking, NOT the Turnitin word count.
Footnotes, appendices,
tables, figures, diagrams,
charts included
in/excluded from word
count/page length?
Any footnotes, appendices are not included in page limit.
Bibliographies, reference
lists included
in/excluded from word
count?
Title page, table of contents, any bibliography are excluded from the
page limit.
Penalty for exceeding
specified word
count/page length?
Where there is a specified word count/page length and this is
exceeded, yes there is a penalty: 10 percentage points
deduction, capped at 40% for Levels 4,5, 6, and 50% for Level
7. Refer to Academic Manual Section 3: Module Assessment -
3.13 Word Counts.
Where there is no specified word count/page length no
penalty applies.
Requirements for/use of
references
This assessment is an ‘open book’ exam/test which you
attempt at home, at UCL, or indeed in any other location. It is
not invigilated. In principle it should take no longer than the
Page 3 of 8
time specified above to complete. However, you have a 24-
hour timed window in which to download the assessment, to
complete it, and to submit it to Moodle.
In responding to the demands of this assessment, you may
draw upon course materials – lecture slides, notes, handouts,
readings, textbook(s) - you engaged with in your studying of
this module.
You are not expected or required to find and use new
materials. In a formal ‘sit-down’ invigilated exam/test you
would not be able to find and draw upon new materials – you
would draw upon what you learned from your studying of the
module.
You may refer to such course materials but you should not be
copying word for word from lecture slides, notes, handouts,
readings, textbook(s) you engaged with in your studying of
this module.
You should capture, articulate and communicate your views,
thoughts and learning in your own words.
If you do provide quotes from any lecture slides, notes,
handouts, readings, textbook(s) you should cite them and
provide references in the usual way.
Be aware that a number of academic misconduct checks,
including the use of Turnitin, are available to your module
leader.
If required/where appropriate UCL Academic Misconduct
penalties may be applied (see immediately below).
Academic misconduct
(including plagiarism)
Academic integrity is paramount.
It is expected that your submission and content will be your
own work with no academic misconduct.
Academic Misconduct is defined as any action or attempted
action, including collusion with other students, that may
result in a student obtaining an unfair academic
advantage. There are severe penalties for Academic
Misconduct, including, where appropriate and required,
exclusion from UCL.
Refer to Academic Manual Section 9: Student Academic
Misconduct Procedure - 9.2 Definitions.
Submission date Friday 26th March 2021
Submission time 16.00 UK Time
Penalty for late
submission?
Yes. Standard UCL penalties apply. Students should refer to
https://www.ucl.ac.uk/academic-manual/chapters/chapter-4-
assessment-framework-taught-programmes/section-3-module-
assessment#3.12
Submitting your
assignment
Late submissions are not permitted
Anonymity of identity.
Normally, all submissions
are anonymous unless
Anonymity is required.
Your name should NOT appear anywhere on your submission.
Page 4 of 8
the nature of the
submission is such that
anonymity is not
appropriate, illustratively
as in presentations or
where minutes of group
meetings are required as
part of a group work
submission
Return and status of
marked assignments
At the latest this will be within 4 weeks from the date
of submission as per UCL guidelines, but we will
endeavour to return it earlier than this.
Assessments are subject to appropriate double
marking/scrutiny, and internal quality inspection by
a nominated School of Management internal
assessor. All results when first published are
provisional until confirmed by the relevant External
Examiner and the Examination Board.
No appeals regarding your published mark are
available until after confirmation by that
Examination Board. UCL regulations specify that
academic judgment applied within the marking
process cannot be challenged.
Academic Support with this Assessment
Given the nature of this assessment, during the 24-hour window no questions should be directed to
the Module Leader/Module Team. If you have doubts about wording or requirements etc., state your
assumptions. If they are appropriate they will be taken into consideration in marking.
Uploading your submission
Unless specifically instructed otherwise in the assessment document, please upload your
work as a single file via the submission link on Moodle.
o Wherever possible you should type/use Excel for (as appropriate) your answers and
follow instructions later in this assessment document.
o If you do have to include any elements that are not typed/computer generated (e.g.
figures, diagrams, equations etc.), or you are unable to type your answers for any
reason, please follow the advice for submitting handwritten answers for any
submission that requires scanning documents (the webpage refers to 24-hour timed
exams but is applicable to all online submissions including this one).
Page 5 of 8
o If for any reason you are not able to use the app recommended by ISD at the link
above, you can consult the following resources for advice about preparing your
submission:
Submitting handwritten assignments to Moodle using mobile or tablet Devices
- Device Camera
Submitting handwritten assignments to Moodle using mobile or tablet devices
- MS One Drive App
Please DOUBLE CHECK that the file you are uploading is the correct one and is complete
(with all pages visible).Resubmission will not be permitted.
Technical Problems
If you encounter difficulties downloading or submitting your assessment via Moodle, then please
immediately notify (by email) your department (Programme Administrators ONLY), explaining the
problem and including a copy of the work you are trying to submit. ONLY use this approach if you
can show that you have tried to download from/upload to Moodle and encountered technical
difficulties.
Advice and other support
Student Support and Wellbeing
Page 6 of 8
Section B: Requirements
See exam paper at the end of the brief
Page 7 of 8
Section C: Module Learning Outcomes covered in this
Assessment
This assignment contributes towards the achievement of the following stated module
Learning Outcomes as below:
1. Dynamic stochastic asset pricing models
2. Hedging
3. Arbitrage-free pricing
4. Risk-neutral pricing
5. Option pricing
5. Numerical methods in asset pricing
Page 8 of 8
Section D: Assessment criteria
Within each section of this coursework you may be assessed on the following aspects, as
applicable and appropriate to this particular assessment, and should thus consider these aspects
when fulfilling the requirements of each section:
The accuracy of any calculations;
The strengths and quality of your overall analysis and evaluation;
Appropriate use of relevant theoretical models, concepts and frameworks;
The rationale and evidence that you provide in support of your arguments;
The credibility and viability of the evidenced conclusions/recommendations/plans of
action you put forward;
Structure and coherence of your considerations and reports;
As and where required, relevant and appropriate, any references should use either
the Harvard OR Vancouver referencing system (see References, Citations and
Avoiding Plagiarism)
Academic judgement regarding the blend of scope, thrust and communication of
ideas, contentions, evidence, knowledge, arguments, conclusions.
Each part has requirements with allocated marks, maximum word count limits/page
limits and where applicable, templates that are required to be used.
You are advised to refer to the UCL Assessment Criteria Guidelines, located at
https://www.ucl.ac.uk/teaching-learning/sites/teaching-learning/files/migrated-
files/UCL_Assessment_Criteria_Guide.pdf
Section E: Groupwork Instructions
Not applicable as this is an individual assessment
Section F: Additional information from module leaders
N/A
MSIN0107 Advanced Quantitative Finance
Examination paper
2020/21
Examination length: TWENTYFOUR (24) hours
NOTE: Although the window for completion is TWENTY-FOUR (24) hours, this
exam paper is designed to be completed in TWO (2) hours.
There is ONE (1) section to the examination paper. The section consists of FOUR
(4) compulsory questions. It is worth ONE HUNDRED (100) marks.
Module Leaders: Dennis Kristensen and Ming Yang
Internal Assessor: Wei Cui
MSIN0107 2020/21 1 TURN OVER
MSIN0107 Advanced Quantitative Finance
1. Consider a nancial market with two assets over the time interval [0; T ]. Trading
can take place at n+1 discrete time points t0; :::; tn where ti = i, i = 0; 1; :::; n,
and = T=n is the time distance between trading times. The rst asset is risk-
free with interest rate er 0 per time period. The second asset is a stock
whose price in period i (= 1; :::; n) is given by
Si = S(i1) exp (+ vi) ;
where and are coe¢ cients and vi, i = 1; ::; n, are i.i.d. with
P (vi = 1) = p1; P (vi = 0) = p2; P (vi = 1) = p3:
Here, p1 + p2 + p3 = 1.
(a) [5 pts] Suppose you know that p1 = p3. How would you estimate the two
parameters and using observed stock prices, Si, i = 1; :::; n?
(b) [5 pts] State necessary and su¢ cient conditions on the parameters of the
model under which you cannot construct a self- nancing portfolio in period
i that earns a non-negative pay-o¤ with probability one in period i + 1,
and a positive pay-o¤ in at least one of the states of period i+ 1.
(c) [5 pts] Consider a contingent claim that expires in period i+ 1. Can you
always establish a portfolio in period i that replicates the claims pay-o¤s
in period i+ 1? Explain.
(d) [5 pts] Let Q be an alternative measure characterised by
Q (vi = 1) = q1; Q (vi = 0) = q2; Q (vi = 1) = q3;
with q1 +q2 +q3 = 1. When is Q a risk-neutral measure? Is there a unique
risk-neutral measure in this market? Explain.
(e) [5 pts] Suppose that = n = 0:1=n and = n = 0:2=
p
n and T = 1.
Under a given sequence of risk-neutral measures Qn characterised by
Qn (vi = 1) = qn;1; Qn (vi = 0) = qn;2; Qn (vi = 1) = qn;3;
with qn;1 = qn;3, derive the limiting distribution of log (ST =S0) as n!1.
2. Consider a given non-dividend paying stock whose price, St, satis es
dSt = Stdt+ StdWt;
whereWt is a Brownian motion. We here measure time in years and the risk-free
rate is 1.5% per annum.
MSIN0107 2020/21 2 CONTINUED
MSIN0107 Advanced Quantitative Finance
(a) [5 pts] Suppose that you have observed the following weekly prices of the
stock:
30:5; 32:2; 31:1; 30:4; 30:3; 31:9; 32:1; 31:0; 30:1; 30:0:
Estimate the stock price volatility. Use this volatility estimate to answer
the following questions.
(b) [5 pts] The current stock price is £ 30. What is todays price of a European
call that expires in two years with a strike price of £ 40?
(c) [5 pts] Suppose you have purchased an option giving you the right to pay
£ 2 one year from today to buy the call option described in part (b) of the
question. Verify that you would exercise this option if the stock price in
one year is greater than 35:3458.
(d) [5 pts] What is todays price of the option described in part (c)?
(e) [5 pts] What is the price of the option giving you the right to sell the the
option described in part (b) in one years time for £ 2?
3. A rm has total asset value of V0 = $100 million. The volatility of the rms
existing asset is = 30% per annum. The rm does not pay any dividend to
equity holders. The rm also has a zero-coupon debt with total face value of
$150million and maturity of T = 5 years. Suppose the rms asset value evolves
as a Geometric Brownian motion (i.e., satis es the assumptions of the Black-
Scholes formula). In all of the following exercises, assume that the continuously
compounded interest rate is r = 8% per annum.
(a) [4 points] What is the present value of the existing debt, B0? What is the
present value of the existing equity, E0?
At year 0, the rm is considering an investment project that costs f = $10million
in present value. If the project is nanced, it will increase the rms asset value
immediately by $11 million, i.e., the rms asset value becomes $111 million.
Suppose the rms asset value follows the same Geometric Brownian motion
(only initial asset value changes) after the new investment. The rm is consid-
ering nancing the investment by issuing new debt with 5 years maturity (i.e.,
matures at the same time as the existing debt).
Financing through junior debt. Suppose that the rm will issue junior
zero-coupon debt to nance the project (This means the newly issued debt will
have lower seniority than the existing debt.).
(b) [2 points] What is the NPV of the project? Should the rm invest in the
project according to the NPV rule?
(c) [8 points] Note that if the rm were to nance the investment through
junior debt with face value of F million, the junior debt holder will be
paid F at the maturity date if the total value of the rms asset exceeds
MSIN0107 2020/21 3 TURN OVER
MSIN0107 Advanced Quantitative Finance
150 + F million. The junior debt holder will recover only part of the face
value of the debt if the total asset value of the rm is between 150 and
150 + F , and will not be paid at all if the total value of the rms asset is
below 150 (in which case, only the senior debt holder gets paid). What is
the lowest face value of the zero-coupon debt that the rm has to promise
to the new debt holders in order to nance the $10 million ? (You may
need to compute this numerically, e.g., using the EXCEL solver add-in).
Note that in order to nance $10 million from the new debt holder, the
present value of the debt has to be at least $10 million.
(d) [5 points] Suppose the rm did issue the debt with the lowest face value
calculated in (c), what will be the present value of the equity after the
issuance of the debt? Does the nance of the debt increase or decrease the
value of the equity? What is the present value of the existing debt (i.e.,
the senior debt) after the issuance of the new debt (i.e., the junior debt)?
(e) [6 points] Suppose you are the manager of the rm and you make decisions
on behalf of the equity holders. Would you nance the project? Does your
decision agree with the NPV rule? Explain the intuition.
4. This question will guide you to solve the Merton problem with T = 1 and
u (c; t) = et ln c. There are two assets in the economy. One is a non-dividend
stock, the price of which follows
dSt
St
= dt+ dZt ,
where Zt is a standard Brownian motion. The other is a riskfree asset, the price
of which follows dXt = rXtdt. An agent starts with wealth W0 and needs to
choose consumption fctg and allocate between the two assets to maximize his
expected utility.
(a) [3 points] Let t denote the fraction of the agents wealth invested in the
risky asset at t. Derive the stochastic di¤erential equation for the agents
wealth Wt.
(b) [4 points] Since T = 1, the problem is stationary and we can solve the
problem under the stationary value function J (W; t) = etV (W ). Write
down the HJB equation for V ().
(c) [4 points] Derive the rst order conditions with respect to c and .
(d) [5 points] Derive the ODE for V (). (Note that the ODE should not
explicitly contain c and . It only consists of , r, , , W , V , VW , and
VWW .)
(e) [5 points] Solve V () from the ODE. (Hint: conjecture V (W ) = a lnW +b
and solve for a and b.)
(f) [4 points] Solve for the optimal policy of c and .
MSIN0107 2020/21 4 END OF PAPER
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