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MTH3025: Financial Mathematics Project

Project: Arbitrage

Arbitrage is a key concept in financial mathematics. In this project, you are expected

to consider some financial trading opportunities and identify, from the given data set, a

mispricing that can lead in turn to an arbitrage opportunity. Then you will have to report a

corresponding arbitrage strategy for returning a risk-free profit and estimate the magnitude

for the expected profit, both in written and in oral form.

1 Report (20% of module mark)

In the report, you will have to show the completion of three main tasks:

Financial instruments. You will have to explain in your own words what is meant

with the term arbitrage to someone with no prior knowledge of financial mathematics.

Then, you are also expected to explain what is meant with the following types of

financial instruments:

– Foreign-exchange swap.

– A rainbow option.

– A lookback option with floating strike price.

These financial instruments are not explained in the lectures. You are nevertheless

expected to explain these instruments in your own words following independent con-

sultation of external sources (to be duely acknowledged in your bibliography).

Misprice identification. For all the four given trading opportunities, you will have

to perform the necessary calculations in order to identify a mispricing in one of them.

Arbitrage strategy. You will have to describe the strategy that you can pursue to

make a risk-free profit. You will have to explain why you have chosen a particular

strategy, the asset(s) and derivative(s) to trade in, the investment that needs to be

made at present time, and what will happen to the portfolios at the maturity date

(final time).

The report should be typed and checked for originality through Turnitin. The submitted

file should be in PDF format. The expected length is no more than 5 pages (references

excluded), using a reasonable font-size and reasonable line spacing (e.g., as the present

document). Longer reports will be penalised.

The marking scheme adopted is the following:

Financial instruments: 30%.

Misprice identification: 30%.

Arbitrage strategy: 30%.

quality of the report (grammar, layout, clarity of exposition): 10%.

MTH3025: Financial Mathematics Project

2 Presentation (10% of module mark)

You will also be expected to explain your trading strategy to the lecturer via a presentation

that you have to upload on Canvas (via a separate upload with respect to the report, see

corresponding Assignment). The presentation should highlight the arbitrage strategy that

you would adopt to make a profit. It should explain what instruments you would prefer to

trade in, and what profit you expect to make. You do not need to include trade opportunities

that do not lead to a profit. The presentation should last between 5 and 8 minutes, and

be aimed at a third-year student in mathematics with no knowledge of financial mathematics.

The presentation will take the form of a Spoken PowerPoint or a pre-recorded video.

Your uploaded files should be readable with the tools available within the platform Of-

fice365, available via your QUB account, otherwise penalisations will be applied.

When you upload your presentation please add a comment that indicate the total duration

of your presentation.

Specific instructions about the presentation can be found on Canvas (in due time). The

marking scheme adopted is the following:

Quality of the slides (fonts, formulas, tables,...): 30%.

Quality of the presentation (timing, narrative, visual e?ects, voice, ...): 30%.

Quality of the technical explanations (assets to trade in, arbitrage strategy, ...): 40%.

MTH3025: Financial Mathematics Project

3 Data

A trader has access to the following four investment opportunities. One of them features a

significant misprice that needs to be identified. Once identified, an arbitrage strategy can

be devised to make a risk-free profit (see below for further guidance).

The trader also has access to bonds at the UK market interest rate at 0.67%.

Opportunity 1: Currency trading

On a currency market, the following currency exchange rates are listed.

GBP USD EUR CHF

1 GBP = 1.0000 1.2724 1.1491 1.3296

1 USD = 0.7859 1.0000 0.9031 1.0450

1 EUR = 0.8702 1.1072 1.0000 1.1638

1 CHF = 0.7521 0.9569 0.8592 1.0000

Opportunity 2: Futures for stocks with dividend

Futures are similar to forward contracts. However, futures are traded on an exchange. As-

sume that fair futures prices are equal to forward prices (see lecture notes Secs 4.8 and 4.9).

The following financial data for the supermarket sector was available on 1 February 2021.

The fair strike price for futures contracts is given for delivery of shares on 1 September 2021.

All prices are given in GBX (pence).

Share Value Dividend Payment date Futures

Tesco 217.58 1.75 1 May 216.68

Sainsbury’s 280.75 0.75 1 March 281.10

Morrisons 245.76 1.25 1 June 245.47

Opportunity 3: Futures for commodities with storage costs

The following financial data for trading in a variety of commodities was available on 1 January

2021. The fair strike price for the futures contracts is given for delivery of the commodity

on 1 September 2021. All prices are given in GBP (pounds) per ounce. Storage costs are

given per ounce for four months of storage and should be paid in advance.

Spot price Storage cost per ounce Futures

Gold 1093.22 8 1114.17

Iridium 1179.09 6 1196.41

Palladium 1032.11 5 1046.76

Opportunity 4: Option portfolio trading

The following financial data for FTSE-traded companies was available on 1 February 2021.

All prices are given in GBX (pence). The option price listed is for European options that

can be exercised exclusively on 1 July 2021 for the stated strike price. You may assume that

no dividend is payable between 1 February and 1 July 2021.

Share Value Call option Put option Strike price

AstraZeneca 5725 352.64 411.47 5800

Diageo 3025 197.87 264.22 3100

Unilever 4260 260.23 288.25 4300

MTH3025: Financial Mathematics Project

4 Guidance about the report

The report should contain the following elements:

Your project number in the title.

A short introduction.

A first part (financial instruments) with an explanation of the term arbitrage and of

the financial instruments listed on the first page.

A second part (misprice identification) with an overview of the calculations carried

out for all the four opportunities to identify the significant misprice in one of them

(significant in this case means beyond rounding errors and leading to a risk-free profit

via arbitrage above 2, see below).

A third part (arbitrage strategy) in which you explain a possible arbitrage strategy,

including an estimation of the expected risk-free profit.

A short conclusion.

A references section if deemed necessary.

There is no need to include all data provided in this document in Sec. 3.

The following points should be taken into account in your report:

In opportunity 3, the trader can only buy or sell multiples of ounces.

Regarding the first part (financial instruments) you should report the payo? (if rele-

vant) of each financial instrument, the cost (if any) associated to the corresponding

contract, and any additional information that you deem relevant and interesting to

show your depth of understanding.

Regarding the second part (misprice identification), you should present an overview

of your calculations to prove that you have checked all the 4 opportunities, regardless

where the arbitrage opportunity arises. For each opportunity, you do not need to repeat

all the details of the calculations for all the cases; however, at least one representative

case per opportunity should be analysed in full details.

The arbitrage strategy needs to be self-financed, meaning that the trader does not

need to invest their own money or assets. The trader can self-finance their arbitrage

strategy constructing a portfolio by making as many financial transactions they desire

at initial time. The transactions can consist of borrowing, buying, selling, exchanging

and returning all the following: money (including di?erent currencies), stocks, futures,

options, bonds, or any instrument you think is relevant. However each initial transac-

tion should not exceed the value of 10000 and only one transaction at most of each

type can be made. For example, only 10000 can be borrowed from a bank and only

once; or only stocks up to the value of 10000 can be borrowed and they can be bor-

rowed only once. Initial transactions can be assumed to be made instantaneously and

without fees. With the constraints above, the significant misprice that you have

identified in the second part should allow you to devise an arbitrage strategy to make

a profit greater than 2.

MTH3025: Financial Mathematics Project

You can conclude the third part (arbitrage strategy) with observations showing your

critical assessment of the strategy that you laid out, for example describing the short-

comings due to the assumptions made when compared to a real-world scenario.

There is no need to write numerical codes to complete this project but you can certainly

do that and include them in your report. However, if you wish to do so, please ensure

that your codes are properly commented.

The report should be uploaded via Canvas in pdf format, with automatically embedded

Turnitin check for plagiarism. You do not need to worry as long as your Turnitin

percentage is below 30%. Reports above this threshold will be considered on a case-

by-case basis but, in general, in the past also reports above that threshold did not

present genuine plagiarism issues.

The lecturer will be happy to answer possible questions that you might have either via email

or during the tutorials. However, notice that for what concerns the third part of the report

(arbitrage strategy) no specific indication will be given. It is expected that the student

should be able to devise an arbitrage strategy using the material available (notes, tutorials

and homework problems, as well as additional resources).

Due to the substantial nature of this assessment and the large number of students (approxi-

mately 1 hour is needed to mark and give feedback to each student’s work, and approximately

80 studentes are enrolled), marks and feedback for the full project (thus including both re-

port and presentation) is expected to be provided in 4 semester weeks from the deadline of

the presentation — therefore by week 1 of the second semester.

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