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时间：2021-03-25

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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