Foreign Currency Derivatives - Lakehead...

Foreign Currency Derivatives

Eiteman et al., Chapter 5

Winter 2004

Outline of the Chapter

• Foreign Currency Futures

• Currency Options

• Option Pricing and Valuation

• Currency Option Pricing Sensitivity

• Prudence in Practice

2

Foreign Currency Futures

A foreign currency futures contractis similar to a forward

contract.

Futures contracts are standardized: Size of contracts and maturity

dates are set by the exchange where the contract is traded.

Futures contracts also require daily settlement of gains and losses

and have maintenance margin requirements.

3


Contract specifications (Chicago):

Size of the Contract: €125,000,U12,500,000, etc.

Method of Stating the Exchange Rate:American terms.

Maturity Date: Third Wednesday of January, March, April,

June, July, September, October or December.

Last Trading Day: Second business day prior to MD.

Initial and Maintenance Margins: Contracts are marked to

market.

4


Contract specifications:

Settlement: 5% of all futures contracts involve physical

delivery. The rest of the time the contract is offset by an

opposite position.

Commissions: Round trip fees.

Clearing House: Ensures liquidity of the contracts.

5


A trader takes ashort positionwhensellinga futures contract,

which corresponds to selling the currency forward.

A trader takes along positionwhenbuyinga futures contract,

which corresponds to buying the currency forward.

6

Anatomy of a Futures Trade

On Tuesday morning, an investor takes a long position in a Swiss

franc futures contract that matures on Thursday afternoon.

The agreed-on price is $0.75/SFr and the contract size is

SFr125,000.

Initial margin requirement is $1,485.

Maintenance margin requirement is $1,100.

7


Tuesday Close

Futures price has risen to $0.755.

Cash profit of125,000× (0.755−0.750) = $625is deposited

into the trader’s account (daily settlement).

Investor has1,485+625= $2,110in his account.

Existing futures contract at $0.75 is canceled and the investor

receives a new futures contract with $0.755 as the prevailing

price.

8


Wednesday Close

Futures price has declined to $0.743.

Investor’s payoff:125,000× (0.743−0.755) =−$1,500.

Investor’s account is debited (daily settlement):

2,110−1,500 = $610 < $1,100.

Investor has less than the maintenance margin requirement.

If keeping his contract, he receives a margin call of

1,100− 610 = $490.

9


Thursday Close

Futures price has declined to $0.74.

Investor’s payoff:125,000× (0.74−0.743) =−$375.

Investor’s net loss on the contract is $1,250 (1,500 + 375 - 625)

before paying commissions.

Investor takes delivery of the SFr125,000.

A contract can be closed by an offsetting trade. A trader with a

long position may offset his trade by taking a short position of

equivalent size.

10

Currency Options

A foreign currency optionis a contract giving the option

purchaser (holder) the right, but not the obigation, to buy or sell a

given amount of foreign exchange at a fixed price per unit for a

specified period of time.

• Call vs put

• Holder vs writer

11

Currency Options

• An American optioncan be exercised at any time before the

maturity date.

• A European optioncan be exercised at the maturity date

only.

• Thepremium, or option price, is the cost of the option.

• An option isin-the-moneyif exercising it profitable,

excluding the premium cost. It can also beat-the-moneyor

out-of-the-money.

12

Currency Options

• Foreign Currency Options Markets

• Over-the-Counter Market

13

Currency Options

Quotations and Prices

Spot Rate: In $/currency.

Exercise Price: In $/currency.

Premium: If premium $0.0050/SFr and the contract size is

SFr125,000, then the cost of the option is

SFr125,000×$0.0050/SFr = $625.00.

14

Currency Options

Let

T ≡ Expiration date of the option

Q ≡ Size of the contract

S0 ≡ Current spot rate

ST ≡ Exchange at the option’s expiration date

X ≡ Strike price

C ≡ Price of a call option

P ≡ Price of a put option

15

Currency Options

Let also

πwc ≡ Profit to the writer of a call

πhc ≡ Profit to the holder of a call

πwp ≡ Profit to the writer of a put

πhp ≡ Profit to the holder of a put

16

Currency Options

ST

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πhc

0X¡

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¡

−CST

6

-

πwc

0X

@@

@@

@@

C

17

Currency Options

ST

6

-

πhp

X−P

0X

@@

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@@−P

ST

6

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πwp

0X

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P−X

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18

Foreign Currency Speculation

Median Joe is a currency speculator. He is willing to risk money

based on his view of currencies and he may do so in the spot,

forward or options market.

Assume Joe has $100,000 and he believes that the six month spot

rate for Swiss francs will be $0.6000/SFr.

The current spot price for Swiss francs is $0.5851/SFr.

19


Speculating in the Spot Market

Joe can use the $100,000 to purchase Swiss francs at the rate of

$0.5851/SFr, which gives SFr170,910.96, and hold the francs

indefinitely.

When target rate ($0.6000/SFr) is reached, sell the

SFr170,910.96 for $.

Profit: 170,910.96× (0.6000−0.5851) = $2,546.57 ignoring

interest and opportunity costs.

20


Speculating in the Forward Market

Suppose the six-month forward quote is $0.5760/SFr.

Joe can buy a contract for $100,000 (no cash outlay initially).

At the contract maturity, Joe expects to sell the100,0000.5760 = SFr173,611.11 received at the rate of $0.6000/SFr, for

an expected profit of

100,0000.5760

×0.6000− 100,000 = $4,166.67 .

21


Speculating in the Options Market

Joe could buy the August call on francs at a strike price of5812

($0.5850/SFr) at a premium of 0.50 or $0.0050/SFr. We’re

currently in February. Suppose the contract size is SFr125,000.

If spot rate is below strike price, Joe won’t exercise his options

and he will lose125,000×0.005= $625per contract.

If spot rate is above5812, the options will be exercised and Joe’s

profit per contract will be

125,000× (Spot rate−0.5850−0.0050) = 125,000× (Spot rate−0.5900).

22


Joe could alsowrite a put option , hoping that it won’t be

exercised.

LettingS, X andP denote the spot price at maturity date, the

strike price and the option premium, respectively, his profit at

maturity would be

πw =

P − (X − S) if S≤ X,

P if S> X.

23



If he were expecting the value of SFr to decrease, Joe could

write a call option, hoping that it won’t be exercised.

LettingS, X andC denote the spot price at maturity date, the


maturity would be

πw =

C if S≤ X,

C − (S−X) if S> X.

24



If he were expecting the value of SFr to decrease, Joe couldbuya put option.

LettingS, X andP denote the spot price at maturity date, the


maturity would be

πw =

X − S− P if S≤ X,

−P if S> X.

25

Option Pricing and Valuation

The value of an option can be divided in two components

Total Value (Premium)= Intrinsic Value+ Time Value.

26


Consider a call option with a premium of $0.033/£ and a strikeprice of $1.70/£. The premium is calculated from the following:

• Present spot rate: $1.70/£.

• Time to maturity: 90 days.

• Forward rate on 90-day contracts: $1.70/£.

• USD interest rate: 8.00% per annum.

• British pound interest rate: 8:00% per annum.

• Standard deviation of daily spot price movement: 10.00% per annum.

27


Theintrinsic valueof an option is its value if exercised

immediately, i.e. the spot exchange rate minus the strike price

when the option is in-the-money. When out-the-money, the

option price is zero.

Thetime valueof an option arises from the fact that the spot rate

can potentially rise above the spot price.

Note that the time value of an option is symmetric, as it is based

on an expected distribution of possible outcomes around the

forward rate that is also symmetric.

28


Intrinsic, time and total value of the 90-Day Call Option on

British pounds (in Cents per Pounds, Except for the Spot Rate)

Spot ($/£) 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74

Intrinsic value 0.00 0.00 0.00 0.00 0.00 1.00 2.00 3.00 4.00

Time value 1.67 2.01 2.39 2.82 3.30 2.82 2.39 2.01 1.67

Total value 1.67 2.01 2.39 2.82 3.30 3.82 4.39 5.01 5.67

29

Currency Option Pricing Sensitivity

The value of an option depends on:

• The forward rate

• The spot rate

• The time to maturity

• The volatility of the spot rate

• The interest rate differential

• The strike price

30


Forward Rate Sensitivity

Foreign currency options are priced around the forward rate.

Let F90 denote the 90-day forward rate in $/£, letSdenote the

current spot rate in $/£, and leti$ andi£ denote the annual

interest rate in dollars and pounds, respectively.

The absence of (risk-free) arbitrage opportunities means that

saving in $ yields the same return as saving in £.

31


Forward Rate Sensitivity

That is,

1+ i$×90360

=1S×

(1+ i£× 90

360

)×F90.

and thus

F90 = S× 1+ i$/41+ i£/4

.

32


Spot Rate Sensitivity (delta)

delta =∆Premium∆Spot Rate

Strike ($/£) Spot ($/£) Premium Intrinsic Time Delta

1.70 1.75 6.37 5.00 1.37 .71

1.70 1.70 3.30 0.00 3.30 .50

1.70 1.65 1.37 0.00 1.37 .28

33



Delta measures the sensitivity of the premium to small changes

in the spot rate. That is, it is the slope of the curve in Exhibit 5.8.

At S= $1.73/£, delta is approximately equal to

.0567− .04391.74−1.72

= .64

34



The delta of a call option varies between 0 and 1. The greater the

spot rate, the greater the option’s delta.

The delta of a put option varies between−1 and 0. The greater

the spot rate, the smaller the option’s delta.

35



Why is delta always between−1 and 1?

36


Time to Maturity and Value Deterioration (theta)

Option values increase with the length of time until maturity. The

expected change in the option premium given a small change in

the time to maturity is calledtheta.

theta =∆Premium

∆Time

37



If, all else being equal, the $1.70/£ call option is worth

• 3.28 cents/£ 90 days before maturity


then the option’s theta at that point in time is

theta =3.30−3.28

89−90= 0.02

38



Option premiums deteriorate at an increasing rate as they

approach expiration. All else being equal, the $1.70/£ call option

is worth



the option’s theta at that point in time is then

theta =1.32−1.37

14−15= 0.05

39


Sensitivity to Volatility (lambda)

Option volatility is the standard deviation of daily percentage

changes in the underlying exchange rate.

lambda=∆Premium∆Volatility

Premiums increase with volatility. Why?

40


Sensitivity to Interest Rates (rho and phi)

rho =∆Premium

∆Domestic Interest Rate

phi =∆Premium

∆Foreign Interest Rate

Do we expect rho to be positive or negative? What about phi?

41

Prudence in Practice

Read the mini-case “Rogue Trader, Nicholas Leeson”.

42

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