### FINANCIAL MATHEMATICS

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##### SCHOOL OF MATHEMATICAL SCIENCES

##### A LEVEL 4 MODULE, RES IT 2021-

##### FINANCIAL MATHEMATICS

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Time to complete THREE Hours plus THIRTY Minutes upload time
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Paper set: 26/08/2022 - 09:
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Paper due: 26/08/2022 - 12:
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Answer ALL questions
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*Your solutions should be written on white paper using dark ink (not pencil), on a tablet, or typeset. Do not write close to the margins. Your solutions should include complete explanations and all intermediate derivations. Your solutions should be based on the material covered in the module and its prerequisites only. Any notation used should be consistent with that in the Lecture Notes.*

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Submit your answers as a single PDF with each page in the correct orientation, to the appropriate
dropbox on the modules Moodle page. Use the standard naming convention for your document:
[StudentID]_[ModuleCode].pdf.
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A scan of handwritten notes is completely acceptable. Make sure your PDF is easily readable and does not
require magnification. Text which is not in focus or is not legible for any other reason will be ignored. If your
scan is larger than 20Mb, please see if it can easily be reduced in size (e.g. scan in black & white, use a
lower dpi but not so low that readability is compromised).
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Staff are not permitted to answer assessment or teaching queries during the assessment period. If you spot
what you think may be an error on the exam paper, note this in your submission but answer the question as
written. Where necessary, minor clarifications or general guidance may be posted on Moodle for all students
to access.
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If you submit your paper after the deadline, you will receive a mark of zero, unless you have an EC
accepted.
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**Academic Integrity in Alternative Assessments**

Work submitted for assessment should be entirely your own work. You must not collude with others or engage the services of others (paid for or not) to work on your assignment. As with all assessments, you also need to avoid plagiarism. Plagiarism, collusion and false authorship are all examples of academic misconduct. They are defined in the University Academic Misconduct Policy at:https://www.nottingham. ac.uk/qualitymanual/assessment-awards-and-deg-classification/pol-academic-misconduct.aspx

**Plagiarism** : representing another persons work or ideas as your own. You could do this by failing to correctly acknowledge others ideas and work as sources of information in an assignment or neglecting to use quotation marks. This also applies to the use of graphical material, calculations etc. in that plagiarism is not limited to text-based sources. There is further guidance about avoiding plagiarism on the University of Nottingham website.

**False Authorship** : where you are not the author of the work you submit. This may include submitting the work of another student or submitting work that has been produced (in whole or in part) by a third party such as through an essay mill website. As it is the authorship of an assignment that is contested, there is no requirement to prove that the assignment has been purchased for this to be classed as false authorship.

**Collusion** : cooperation in order to gain an unpermitted advantage. This may occur where you have consciously collaborated on a piece of work, in part or whole, and passed it off as your own individual effort or where you authorise another student to use your work, in part or whole, and to submit it as their own. Note that working with one or more other students to plan your assignment would be classed as collusion, even if you go on to complete your assignment independently after this preparatory work. Allowing someone else to copy your work and submit it as their own is also a form of collusion.

**Statement of Academic Integrity**

By submitting a piece of work for assessment you are agreeing to the following statements:

- I confirm that I have read and understood the definitions of plagiarism, false authorship and collusion.
- I confirm that this assessment is my own work and is not copied from any other persons work (published or unpublished).
- I confirm that I have not worked with others to complete this work.
- I understand that plagiarism, false authorship, and collusion are academic offences and I may be referred to the Academic Misconduct Committee if plagiarism, false authorship or collusion is suspected.

#### 1. InthisquestionusethevaluesofandinthetablebelowthatcorrespondtotheSEVENTH(PENULTIMATE)

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digit of your 8-digit Student ID number.
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7th digit 0 1 2 3 4 5 6 7 8 9
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#### 0.02 0.01 0.04 0.05 0.03 0.05 0.04 0.03 0.02 0.

#### 0.03 0.04 0.02 0.01 0.05 0.01 0.05 0.02 0.04 0.

#### Suppose a savings account with interest rateper annum and loans with interest rateper annum are

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available. Both the savings account and loan have monthly compounding.
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(a) Suppose investor A puts 407217 in the savings account. How much money will be in the savings
account after 10 years?
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(b) Suppose investor B puts 0 in the savings account at time 0 but puts 2000 in the savings account
at the end of every month. How much money will be in the savings account after 10 years?
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(c) Suppose investor C borrows 407127 at time 0 and makes fixed repayments of 2000 at the end of
each month. What will be the outstanding balance on the loan after 10 years?
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(d) How many years will it take investor C to pay off their loan?
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(e) Suppose investor C inherits 407127 at time 0. They could either keep their loan (repaying
each month) and put the inheritance in the savings account OR they could immediately pay off their
loan and put 2000 into the savings account each month. Calculate their financial position after 10
years under both scenar ios to determine which is the best action.
[14 marks]
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#### 2. Inthisquestionusethevaluesof 0 ,and 4 inthetablebelowthatcorrespondtotheSEVENTH(PENULTIMATE)

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digit of your 8-digit Student ID number.
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7th digit 0 1 2 3 4 5 6 7 8 9
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#### 0 0.05 0.03 0.07 0.04 0.06 0.06 0.04 0.03 0.07 0.

#### 4 0.07 0.04 0.05 0.06 0.03 0.04 0.05 0.07 0.03 0.

#### A ten year long forward contract on one kilogram of gold is entered into when the commodity price is $ 40

#### per gram and the continuously compounded interest rate is 0 per annum.

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(a) What is the forward price for the contract at date of entry?
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#### (b) Four years later the price is $ 55 per gram and the continuously compounded interest rate is 4 per

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annum. What is the current value of the original long forward contract?
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#### (c) Suppose there is a ten year forward contract on one kilogram of gold with a forward price of $ 65000

#### when the gold price is $ 40 per gram and the continuously compounded interest rate is 0 per annum.

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How would you exploit the arbitrage opportunity?
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#### (d) Suppose, instead, that the continuously compounded interest rate for lending is= 0 0.01and the

#### continuously compounded interest rate for borrowing is= 0 +0.02. Does the arbitrage opportunity

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in (c) still exist? For what forward prices does no arbitrage opport unity exist?
[14 marks]
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##### MATH4060-E1R

#### 3. In this question use the values ofin the table below that correspond to the SEVENTH (PENULTIMATE)

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digit of your 8-digit Student ID number.
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7th digit 0 1 2 3 4 5 6 7 8 9
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#### 2 4 7 6 5 3 9 1 10 8

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The term structure of a bond is given below:
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#### 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00 5.50 6.

#### (0,) 1.00 0.97 0.95 0.93 0.89 0.83 0.76 0.70 0.69 0.68 0.61 0.57 0.

#### Company A issues a fixed coupon bond with nominal value $ 10000 that pays coupons with% of the

#### nominal value per annum at 6 month intervals. The first coupon is to be paid after 3 years and the maturity

#### date is 6 years.

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Company B issues a floating rate note with the same coupon and maturity dates but a nominal value of
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#### $ 30000.

#### (a) Company A issues the fixed coupon bond at time 0. What would be a fair price for this bond?

#### (b) Company B issues the floating rate note at time 0. What would be a fair price for this note?

#### (c) Suppose companiesAand B purchase each others bond at time 0. For each time point in the interval

#### 0 6, describe and, where possible, calculate the money that is transferred between companies

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A and B.
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#### (d) Neither company can afford to pay cash for the others bond at time 0 so they agree to purchase each

#### others bond for the same price as their own. For what value ofwould this be fair?

#### (e) Although this arrangement is a fair swap at time 0 , at what time point will we know whether company

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A or company B has profited from the arrangement?
[16 marks]
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MATH4060-E1R *Turn Over*

#### 4. Inthisquestionusethevaluesof 1 and 2 inthetablebelowthatcorrespondtotheSEVENTH(PENULTIMATE)

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digit of your 8-digit Student ID number.
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7th digit 0 1 2 3 4 5 6 7 8 9
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#### 1 30 30 40 40 50 50 60 60 70 70

#### 2 50 60 50 60 70 80 70 80 80 90

#### (a) An investor constructs a portfolio with payoff function at timegiven by

#### ()={

#### 0 if< 1

#### 1 if 1 << 2

#### 2 1 if> 2

#### i) Sketch this payoff function for20<<100.

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ii) Explain when an investor might choose to purchase this portfolio.
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iii) Show how this payoff function can be recreated using a portfolio of European call and put options.
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#### (b) The current stock price is 0 = 60. Assume a bank account exists and has an interest rate fixed at

#### 2%. European put options with maturity of =10years are available at the following premiums.

#### Strike price, 30 40 50 60 70 80 90

#### Put price, 0.836 2.454 5.109 8.749 13.249 18.466 24.

#### i) Calculate the European call option price for strike prices 1 and 2.

#### ii) Let 0 be the total cost (premium) for the portfolio of options considered in part (a). Use part (b)(i)

#### to calculate 0.

#### iii) Prove that0< 0 < 2 1 for any 1 and 2 such that0< 1 < 2.

#### iv) Deriveaformulafor, whereisthevalueofsuchthatif>thentheholderoftheportfolio

#### gets a greater return than simply investing the portfolio premium in a bank account. Calculate.

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[24 marks]
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##### MATH4060-E1R

#### 5. In this question use the option type and the value ofin the table below that correspond to the SEVENTH

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(PENULTIMATE) digit of your 8-digit Student ID number.
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7th digit 0 1 2 3 4 5 6 7 8 9
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#### 3 4 5 6 7 4 5 6 7 8

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Type Call Call Call Call Call Put Put Put Put Put
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Let a three-step binary price model have the structure:
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#### =0 =1 =2 =

#### Suppose the continuously compounded interest rate per annum,, and the time interval between steps,

#### , is such that=

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6
5
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##### .

#### Consider a European call or put option (as indicated in the table) with strike priceand maturity at time

#### step=3.

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(a) Demonstrate that there are no arbitrage opportunities in this market model.
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(b) Calculate the risk-neutral probability of an upward move at each node.
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(c) Calculate the option premium.
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#### (d) What portfolio should the writer hold at time step=2if the stock price is4.

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[16 marks]
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MATH4060-E1R *Turn Over*

#### 6. In this question use the value ofin the table below that correspond to the SEVENTH (PENULTIMATE)

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digit of your 8-digit Student ID number.
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7th digit 0 1 2 3 4 5 6 7 8 9
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#### 6 7 8 4 5 6 7 4 8 5

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Let a three-step binary price model have the structure:
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#### =0 =1 =2 =

#### Suppose the continuously compounded interest rate per annum,, and the time interval between steps,

#### , is such that=^6

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5
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##### .

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NOTE: The binary tree structure and interest rate are the same as the previous question. You may use
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#### your answers to the previous question, but be aware that the value ofis different in this question.

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An American-style discrete lookback put option has the payoff
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#### =( min

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0
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#### )

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+
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#### , =0,1,2,3.

#### which expires at step=3.

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(a) Calculate the payoff value for this option on every node and path.
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(b) Find the price of this option.
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(c) Find the optimal stopping time for this option on each path.
[16 marks]
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##### MATH4060-E1R

#### 7. Consider a stock price process(), where(0) = 0 is assume known. Let()denote a standard

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Wiener process.
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(a) Suppose we model the stock price using
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#### =

#### whereis some constant.

#### i) Derive an expression for().

#### ii) Briefly describe its behaviour for the cases >0, =0and <0.

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(b) Suppose we model the stock price using
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#### =

#### whereis some constant.

#### i) Derive an expression for().

#### ii) Briefly describe the behaviour of()compared to a standard Wiener process.

#### iii) What is the distribution of()?

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(c) Suppose we model the stock price using
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#### =+

#### whereandare constants.

#### i) Derive an expression for().

#### ii) Briefly describe the behaviour of()compared to a standard Wiener process.

#### iii) What is the distribution of()?

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(d) Suppose we model the stock price using
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#### =()+

#### whereandare constants.

#### i) Derive an expression for().

#### ii) Derive an expression[()].

#### iii) Briefly describe the behaviour of().

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[20 marks]
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