FNCE 4004 Derivatives Chapter 5 Determination of Forward and - - PowerPoint PPT Presentation

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

FNCE 4004 – Derivatives Chapter 5

Determination of Forward and Futures Prices

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Consumption vs. Investment Assets

• Investment assets are assets held by

significant numbers of people purely for investment purposes (Examples: stocks, bonds, gold, silver)

• Consumption assets are held primarily for

consumption (Examples: copper, oil, corn)

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

SHORT SELLING

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• Forward pricing depends on short selling
• Short selling involves selling securities you do

not own

• Reasons for short selling?

– You are hedging an existing exposure – You are speculating that the security will fall in price

Short Selling

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Securities Lending

• You cannot short sell if no one will lend you a
• security. This business is called securities lending.
• Generally seen as low risk business
• Lender can accept cash as collateral in which case

they pay sub-market interest rates on the collateral

– They invest the cash, earn market rates and earn the spread

• Lender can also accept low-risk securities – for

example Treasuries.

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Securities Lending Pitfalls

• AIG set up AIG Securities Lending to centrally manage

securities lending across insurance company subsidiaries.

• Beginning in late 2005 AIG started to use the cash to invest

in RMBS. At its peak AIG had $76bn invested of which 60% was in RMBS. The securities were AAA rated when they were purchased but fell in quality and price.

• Part of the AIG bailout was to buy the MBS so that the

securities that had been lent could be returned to the insurance companies.

• Otherwise the securities lending subsidiary of AIG could

have caused “collateral” damage to all of the insurance companies which were sound.

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Mechanics of Short Sale

(www.interactivebrokers.com)

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Example

• You short 100 shares when the price is $100

and close out the short position three months later when the price is $90

• During the three months a dividend of $3 per

share is paid and there is no cost to borrowing the stock

– What is your profit? – Do you earn interest on the money you’ve received? – Do you have to pay a borrowing cost?

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

SIMPLE FORWARD PRICES

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Forward Price

• Consider an investment asset that does not

pay any income. For example:

– Non-dividend paying stock – Zero coupon bond – Treasury bill, strip

• Assume that you can short the asset at no

cost and that lending the asset does not provide income

• Assume that there are no transaction costs or

bid/offer spreads

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

An Arbitrage Opportunity?

• Suppose that:

– Spot price of a non-dividend-paying stock = $40 – The 3-month forward price is $43 – The continuously compounded risk free interest rate is 5% – The borrowing cost is 0%.

• Is there an arbitrage opportunity?
• Assume that the forward contract is

uncollateralized.

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage Opportunity

Today

• Sell stock forward in 3

months for $43

• Borrow $40 for 3

months

• Buy Stock

3 months

• Deliver stock into

forward contract and receive $43

• Repay borrowing

$40 × 𝑓0.05×0.25 = 40.50

• Make $2.50 risk-free

In these transactions nothing happens between today and 3 months

• time. If something can happen then we need to understand whether

that affects the arbitrage.

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage Opportunity – mistake!

Today

• Buy stock forward in 3

months for $43

• Borrow stock for 3 months
• Sell stock and receive $40
• Invest $40 at risk-free rate

In 3 months

• Receive

$40 × 𝑓0.05×0.25 = 40.50

• Borrow $2.50
• Buy stock for $43
• Deliver stock into short
• Lose $2.50 (!!!)

If you make a mistake like this you simply have to reverse all of the trades

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Another Arbitrage Opportunity?

• Suppose that:

– The spot price of a non-dividend-paying stock is $40 – The 3-month forward price is USD 40.25 per share – The cont. compounded USD interest rate is 5% – All contracts are uncollateralized – Is there an arbitrage opportunity?

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage Opportunity

Today

• Buy stock forward in 3

months for $40.25

• Borrow stock for 3

months

• Sell stock and receive

$40

• Invest $40 at risk-free

rate In 3 months

• Receive

$40 × 𝑓0.05×0.25 = 40.50

• Buy stock for $40.25
• Deliver stock into short
• Earn $0.25

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Some Notation

𝑇: Spot price today 𝐺: Futures or forward price today 𝑈: Time until delivery date 𝑠: Risk-free interest rate for maturity T (cont. compounding)

Notation for Valuing Futures and Forward Contracts

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The Forward Price

• Consider an investment asset that has no

yield and does not cost to borrow

• Assume that the market does not require

collateral for any transactions

• If the spot price of an investment asset is 𝑇

and 𝑠 is the T-year risk-free rate of interest then the forward price for a contract deliverable in 𝑈 years is

𝐺 = 𝑇𝑓𝑠𝑈

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

No Arbitrage Argument – part 1

Today

• Agree to sell stock

forward at time T for F

• Borrow S at risk-free

rate

• Buy Stock for S

Time T

• Deliver stock into

forward contract and receive F

• Repay borrowing

S × 𝑓𝑠𝑈

• For there to be no

arbitrage 𝐺 ≤ S × 𝑓𝑠𝑈

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

No Arbitrage Argument – part 2

Today

• Agree to buy stock

forward at time T for 𝐺

• Borrow Stock today
• Sell Stock and receive S
• Invest $S at risk-free

rate Time T

• Redeem investment for

𝑇𝑓𝑠𝑈

• Pay F and receive

stock

• Deliver stock to lender
• For there to be no

arbitrage 𝐺 ≥ S × 𝑓𝑠𝑈

The no-arbitrage argument says that 𝐺 ≥ S × 𝑓𝑠𝑈 and 𝐺 ≤ S × 𝑓𝑠𝑈, which means

𝐺 = 𝑇 × 𝑓𝑠𝑈

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FORWARD CONTRACTS

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Valuing a Forward Contract

• A forward contract is a contract to buy/sell an

asset at some time in the future.

• A forward contract is worth zero (excluding

bid-offer spreads) when it is first negotiated

• Later it may have a positive or negative value
• Suppose:

– K is the delivery price – F is the forward price for a contract that would be negotiated today for settlement at time 𝑈

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Valuing a Forward Contract

By considering the difference between a contract with delivery price 𝐿 and a contract with delivery price 𝐺 we can deduce that:

• the value of a long forward contract, ƒ, is

(𝐺 − 𝐿)𝑓−𝑠𝑈

• the value of a short forward contract is

(𝐿 − 𝐺)𝑓−𝑠𝑈

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COLLATERAL

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Collateral

• Collateral is posted to protect the

counterparties in a contract from non-delivery by the other counterpart.

• We will assume for simplicity that collateral is

in the form of US treasuries.

• If a contract is collateralized then the

discounting should be done at the same rate earned by the collateral.

• What happens if it isn’t?

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Collateralized Lending/Borrowing

• Assume that you can lend/borrow $95 today

and have to return $100 tomorrow.

• Assume that this transaction is collateralized

with US treasuries and that there is a $100 strip with a maturity of 1 day trading at $97.

• Is there arbitrage?
• What if the maturity of all transactions is 1

year?

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage Argument

Today

• Lend $95
• Receive strips with

value $95 or

95 97

= 0.9794 strips as collateral

• Sell Strips for $95
• This requires no cash
• utlay.

Tomorrow

• If repaid then

– Receive $100 – Buy the 0.9794 strips for $97.94 – Return the 0.9794 strips worth $97.94 – Profit $2.06

• If not repaid then no
• bligation to return the

strips, so P&L=0

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage with Collateral

• Suppose that:

– The spot price of a non-dividend-paying stock is $40 – The 3-month forward price is $43 – The continuously compounded USD treasury strip rate is 5% – Assume that all collateral is in the form of a STRIP that matures on the same day as the forward

• Is there an arbitrage opportunity?

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage Opportunity

Today

• Agree to sell stock forward in

3 months for $43. Today this contract is worth zero so no collateral posted.

• Borrow $40
• Buy Stock
• Lend Stock
• Receive $40 US Treasury as

collateral

• Sell Treasuries
• Repay $40

in 3 months

• Borrow

$40 × 𝑓0.05×0.25 = 40.50

• Buy Treasury and return

to borrower of Stock

• Receive Stock
• Deliver stock into forward

contract and receive $43

• Repay borrowing
• Make $2.50 risk-free

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage Opportunity – part 2

• What happens if the stock changes in price?

– It will change the value of the forward contract and the stock lending

• The forward contract

– If the stock price increases by $1 then approximately $1 will be required to be posted as additional collateral (should be exactly $1)

• The stock lending

– If the stock price increases by $1 then you should receive $1 of additional collateral.

• These net approximately

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Arbitrage Opportunity

• What happens if your counterparty to the

forward contract realizes that they have made a mistake and correct it?

– The loss will appear in the market value of the forward contract and they will be required to post collateral to you.

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Collateralized Forward

• We can follow the same arbitrage argument

and find that the only change in the formula for the forward is that the risk-free rate is replaced by the yield on the collateral:

𝐺 = 𝑇𝑓𝑠𝑑𝑝𝑚𝑚𝑏𝑢×𝑈

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NO SHORT SELLING

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Review the Arbitrage

• On the next two slides we review the

arbitrage which determines the forward price. Which works and which does not?

• Think about who is carrying out the arbitrage.
• Consider two underlyings – oil and a stock.

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

No Arbitrage Argument – part 1

Today

• Agree to sell stock

forward at time T for F

• Borrow S at risk-free

rate

• Buy Stock for S

Time T

• Deliver stock into

forward contract and receive F

• Repay borrowing

S × 𝑓𝑠𝑈

• For there to be no

arbitrage 𝐺 ≤ S × 𝑓𝑠𝑈

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

No Arbitrage Argument – part 2

Today

• Agree to buy stock

forward at time T for 𝐺

• Borrow Stock today
• Sell Stock and receive S
• Invest $S at risk-free

rate Time T

• Redeem investment for

𝑇𝑓𝑠𝑈

• Pay F and receive

stock

• Deliver stock to lender
• For there to be no

arbitrage 𝐺 ≥ S × 𝑓𝑠𝑈

The arbitrage argument says that 𝐺 ≥ S × 𝑓𝑠𝑈 and 𝐺 ≤ S × 𝑓𝑠𝑈, which means that

𝐺 = S × 𝑓𝑠𝑈

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

If Short Sales Are Not Possible..

• The first part of the arbitrage argument works

which means:

𝐺 ≤ S × 𝑓𝑠𝑈

• But the arbitrage argument no longer works

for 𝐺 ≥ S × 𝑓𝑠𝑈

• “Generally” the equality still works for an

investment asset because investors who hold the asset will sell it and buy forward contracts when the forward price is too low.

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MORE COMPLICATED FORWARDS

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General Principle

• A general principle for determining a forward

is that

– If an asset has a cost to hold it then the person selling the forward will incur this cost when they

• hedge. They will charge the buyer of the forward

for this. – If an asset earns an income by holding it then the person selling the forward will earn this income when they hedge. They will deduct this income from the cost of the forward.

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Equity with a Dividend

• Imagine a stock with

– Spot price 𝑇 – A fixed dividend 𝐸 paid at time 𝑢𝐸 – Risk free rate 𝑠

• What is the forward price at time 𝑈
• If 𝑈 < 𝑢𝐸 then the forward is the same as

before

𝐺 = 𝑇𝑓𝑠𝑈

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Equity with a Dividend

• Assume that 𝑈 ≥ 𝑢𝐸 (dividend paid before

maturity)

• The present value of the dividend is

𝑄𝑊 𝑒𝑗𝑤𝑗𝑒𝑓𝑜𝑒 = 𝐽 = 𝐸𝑓−𝑠𝑢𝐸

• Again, if you are long the forward then you

need to borrow S and buy the stock.

• At time 𝑢𝐸 you are paid D from your stock.

The remaining borrowing at that time is

𝑇𝑓𝑠𝑢𝐸 − 𝐸

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Equity with a Dividend

• At time T the amount you will need to repay

from your loan to buy the stock, minus the investment of the dividend received, is: 𝑇𝑓𝑠𝑢𝐸 − 𝐸 𝑓𝑠 𝑈−𝑢𝐸 = 𝑇 − 𝐸𝑓−𝑠𝑢𝐸

𝐽

𝑓𝑠𝑈= 𝑇 − 𝐽 𝑓𝑠𝑈

• Through a no-arbitrage argument using the

short position as well we see that

𝐺 = 𝑇 − 𝐽 𝑓𝑠𝑈

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Equity with a Dividend (graphically)

At time 𝑈 repay loan to buy stock (𝑇, at time 0). At time 𝑢𝐸 invest Dividend at the risk free rate 𝑠. That determines the forward price F: 𝑓𝑠 𝑈−𝑢𝐸 𝐺 𝐸 𝑢𝐸 𝑈

today S

𝑓𝑠𝑈 No-arbitrage requires: 𝐺 = 𝑇𝑓𝑠𝑈 − 𝐸𝑓𝑠 𝑈−𝑢𝐸 = 𝑇 − 𝐸𝑓−𝑠𝑢𝐸

𝐽

𝑓𝑠𝑈= 𝑇 − 𝐽 𝑓𝑠𝑈

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Equity Index

• An equity index can be thought of as an asset

which provides a known yield.

• The forward price of an equity index is

F0 = S0𝑓(𝑠−𝑟)𝑈

where 𝑟 is the average dividend yield on the portfolio represented by the index during life

• f the contract

University of Colorado at Boulder – Leeds School of Business – FNCE 4040 Derivatives

Equity Index Example

• Consider the S&P 500 index

– The Spot price for the index is 2,041.51 – The Futures price for the March 2015 contract is 2,025.65 – 6-7 week risk-free interest rates are 0.08%

• What is the implied dividend yield on the S&P 500

between now (4-Feb-2015) and 20-Mar-2015?

𝐺 = 𝑇 ∗ 𝑓 𝑠𝑔𝑠𝑓𝑓−𝑟 𝑒𝑏𝑧𝑡

365

solve for 𝑟:

𝑟 = 𝑠

𝑔𝑠𝑓𝑓 −

𝑚𝑜 𝐺 𝑇 𝑒𝑏𝑧𝑡 365 = 0.08% − 𝑚𝑜 2025.65 2041.51 44 365 = 6.55%

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Borrow Costs

• There may be a cost to short an asset.

– This can be considered income on the asset similar to a dividend yield (if the stock you borrowed and shorted pays a dividend, you are responsible to pay that dividend to the lender)

• If this yield is 𝑐 then the Forward Price is:

F0 = S0𝑓(𝑠𝑑−𝑐)𝑈

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Gold Lease Rate

• If you hold physical gold you can lend it out.

The difference between the income you can earn from lending the gold and the cost of storing the gold is called the lease rate.

• You will generally be required to post the

value of the gold as collateral and will be paid interest on this amount.

• The net interest earned will be the difference
• f the risk-free rate and the lease rate.

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Gold Forward

• Assume you have 1 ounce of gold that you

want to sell forward 1 year.

– Spot gold price is $1250.00 per ounce – The one year cont. comp. gold lease rate* is 0.40%. – The one year cont. comp. risk-free interest rate is 0.60%

• What is the one year forward price?
• Assume all contracts are uncollateralized.

* In reality lease rate is quoted like Libor, the lease owed after 𝑒 days is 𝑚𝑓𝑏𝑡𝑓 = 𝑇0 1 + 𝑚 × 𝑒 360

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Gold Forward Arbitrage – part 1

Today

• Agree to sell gold forward in

1 year for F

• Borrow $1,250
• Buy 1oz of gold for $1,250
• Lend gold at lease rate
• Receive collateral and repay

borrowing

Time T

• Borrow $1250𝑓0.006−0.004 =

$1252.50

• Close Gold lending and

receive gold and return collateral of $1252.50

• Deliver gold into forward

and receive F

• Repay borrowing
• No arbitrage means that

𝐺 ≤ 1252.50($ 𝑝𝑨 )

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Gold Forward Arbitrage – part 2

Today

• Agree to buy gold forward in
• ne year for F
• Borrow 1,250
• Borrow gold today and post

1,250 as collateral

• Sell gold and receive 1,250
• Repay borrowing

Time T

• Borrow F
• Pay F and receive gold
• Return gold and receive

1,252.50

• Repay borrowing
• No arbitrage means that

𝐺 ≥ 1252.50

We have both 𝐺 ≥ 1252.50 and 𝐺 ≤ 1252.50 so

𝐺 = 1252.50

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Gold Forward

• The current forward price of gold for time T is

F0,T = S0𝑓(𝑠𝑑−𝑐)𝑈

Where

• 𝑇0 is the spot price of gold
• 𝑠

𝑑 is the continuously compounded risk-free

rate (or c.c. rate on collateral)

• 𝑐 is the lease rate

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Consumption Assets – Costs

• There may be a cost to store a consumption

asset.

– This can be considered as additional interest you need to pay for having purchased the asset. – Interest and storage add up to the full carry cost

• If this cost is 𝑣 then the Forward Price is:

F0,T ≤ S0𝑓(𝑠𝑑+𝑣)𝑈

We’ll see why the inequality in a moment

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Consumption assets – e.g. Wheat

• Assume you want to sell 1,000 bushels of

wheat, forward 1 year.

– Spot wheat price is 7.30 ($/bu) – The one year cont. comp. storage cost is 0.20%. – The one year cont. comp. risk-free interest rates is 0.60%

• What is the one year forward price?
• Assume all contracts are uncollateralized.

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Wheat Forward Arbitrage – part 1

Today

• Agree to sell wheat forward

in 1 year for F

• Borrow $7,300
• Buy 1,000 bushels of wheat

for $7,300

• Store the wheat and agree
• n the storage cost today

Time T

• Borrow $7,300(𝑓20𝑐𝑞−1) =

$14.61 to pay for storage

• Pay $14.61 for wheat

storage and get the wheat

• Deliver wheat into forward

and receive F

• Repay borrowing =

$7,300(𝑓60𝑐𝑞) = $7,343.93

• Repay loan to pay storage
• No arbitrage means that

𝐺 ≤ 7,343.93 + 14.61 = 7,358.54($ 𝑐𝑣 ) This is the max forward price you can charge

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Wheat Forward Arbitrage – part 2

• You cannot easily borrow wheat like you did

for gold – there may be shortage currently, but farmers are expected to flood the market with wheat in one year.

• Therefore “borrowing” wheat today for one

year may be very expensive (“I want wheat now, there will be plenty in one year”)

• Therefore you do not have the other

inequality, so you only have an upper bound for wheat forward price.

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Wheat Forward

• The current forward price of wheat for

delivery at time T must not exceed

F0,T ≤ S0𝑓(𝑠𝑑+𝑣)𝑈

• The right hand side is the full carry cost

Where

• 𝑇0 is the current price of wheat
• 𝑠

𝑑 is the continuously compounded risk-free

rate (or c.c. rate on collateral)

• 𝑣 is the storage cost (cont. compounded)

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A Note on Convenience Yield

• Your book talks about “Convenience Yield”
• In practice, you will not hear the word

“Convenience Yield” on a commodities trading floor.

• Traders talk about forward curve, contango or
• backwardation. The shape of the forward

curve is driven by supply demand for that commodity delivered at various points in time.

• The forward curve contango is constrained by

the full carry cost. If steeper, you can arb it.

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The Cost of Carry

• The cost of carry, 𝑑, is equal to the

storage cost, plus the interest costs, less the income earned.

• For an investment asset 𝐺

0,𝑈 = 𝑇0𝑓𝑑𝑈

• For a consumption asset 𝐺

0,𝑈 ≤ 𝑇0𝑓𝑑𝑈

• The convenience yield on the

consumption asset, 𝑧, is defined so that

𝐺0,𝑈 = 𝑇0𝑓(𝑑−𝑧)𝑈

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Futures and Forwards on Currencies

• A foreign currency is analogous to a security

providing a yield

• We leave this derivation as an exercise to the

students but give you the following picture to help you think about it.

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Relationship Between Spot and Forward explained by no arbitrage condition

1000 units of foreign currency (time zero) 1000𝑓𝑠𝑔𝑈 units of foreign currency (time T) 1000 𝐺

0 𝑓𝑠𝑔𝑈

US Dollars (time T) 1000 𝑇0 US Dollars (time zero) 1000 𝑇0 𝑓𝑠𝑈 US Dollars (time T)

A: earn interest and then convert B: convert and then earn interest