1. Currency Options and Options Markets
Chapter 6
Currency Options and Options Markets
Learning objectives
 Option payoffs
– Payoff profiles and profit/loss diagrams
– Option value determinants
 Hedging with options
– Combinations of options & with underlying positions
 Exchange rate volatility
– Implied volatility

2. 6.1 What Is an Option?
An option is an agreement giving the option holder (buyer or owner) the right to buy or sell an underlying asset at a specified price, on (or perhaps before) a specified date.
European: exercisable only at expiration
American: exercisable anytime until maturity
The option writer (seller) holds the obligation to fulfill the other side of the contract.
Just for fun (not in the text )
- Options typically have more value if left unexercised because the time value of the option is related to the time to expiration.
- As a consequence, the value of a European option typically is close to the value of a similar American option.
(Exceptions can occur for deep-out-of-the-money options when interest rates are high, although these circumstances are rare.)

3. 6.1 What Is an Option?
A currency option is a contract giving the option holder the right to buy or sell an underlying currency at a specified price by a specified date.
Call option = an option to buy
Put option = an option to sell
Note that when you buy one currency you are selling another currency.

4. 6.1 What Is an Option? CME pound Dec 1500 call (American)
Type of option: a call option to buy pounds
Underlying asset: CME December pound sterling futures contract
Contract size: £62,500
Expiration date: 3rd week of December
Exercise price: $1.50/£
Rule for exercise: an American option exercisable anytime until expiration

5. 6.1 What Is an Option? CME currency option quotations
British pound (CME contract £62,500; prices in ¢/£)
Strike Calls-Settle Puts-Settle
Price Oct Nov Dec Oct Nov Dec
The CME pound Dec 1500 call sells for $0.0500/£

6. 6.2 Option Payoff Profiles
A forward obligation
A £1 million obligation due in four months
Currency exposure
Underlying transaction
v$/£
s$/£
-£1,000,000

7. 6.2 Option Payoff Profiles
A forward hedge
Buy £1 million in the forward market at the forward price F1$/£ = $1.50/£.
Exposure of forward contract
Long pound forward
v$/£
+£1,000,000
s$/£
-$1,500,000

8. 6.2 Option Payoff Profiles
A currency option hedge
A currency option is like one side of a currency forward contract:
The option holder gains if pound sterling rises.
The option holder does not lose if pound sterling falls.
v$/£
Long pound call
(option to buy
pound sterling)
s$/£

9. 6.2 Option Payoff Profiles
Payoff profile of a pound call at expiration

10. 6.2 Option Payoff Profiles
Profit (loss) on a long call at expiration
FX rate at expiration $1.50/£ $1.55/£ $1.60/£
(Premium)*(contract size) -$3,125 –$3,125 –$3,125
(Exercise price)*(contract size) $0 –$93,750 –$93,750
(Spot £ sale)*(contract size) $0 $96,875 $100,000
Net profit –$3,125 +$0 +$3,125

11. 6.2 Option Payoff Profiles
Payoff profile of a call option at expiration
Long call
Short call
CallT$/£
-CallT$/£
KT$/£
ST$/£
ST$/£
KT$/£
Out-of-the-
money
Out-of-the-
money
In-the-
money
In-the-
money

12. 6.2 Option Payoff Profiles
Payoff profile of a put option at expiration
PutT$/£
-PutT$/£
Long put
Short put
KT$/£
ST$/£
ST$/£
KT$/£
In-the-
money
Out-of-the-
money
In-the-
money
Out-of-the-
money

13. 6.2 Option Payoff Profiles
Puts and calls
Call option to buy
pounds at KT$/£
An option to sell
dollars at KT£/$
DCallT$/£
DPutT£/$ =
DST$/£
DST£/$

14. 6.2 Option Payoff Profiles
Forwards, puts, and calls
A combination of a long call and a short put at the same exercise price and with the same expiration date results in a long forward position at that forward price.
Long call
Short put
Long forward
DCallT$/£ +
–DPutT$/£
DFT$/£
DST$/£
DST$/£ =
DST$/£

15. 6.2 Option Payoff Profiles
Put-call parity at expiration:
CallTd/f - PutTd/f + Kd/f = FTd/f
CallT$/£
-PutT$/£
KT$/£ + +
ST$/£
ST$/£
ST$/£
KT$/£
Exercise price
Long call
Short put
FT$/£ =
Long forward
ST$/£

16. 6.3 Currency Option Values Prior to Expiration
Option value = (Intrinsic value) + (Time value)
Intrinsic value = Value if exercised immediately
Time value = Additional value prior to exercise
The time value of a currency option is a function of the following six determinants
Exchange rate underlying the option
Exercise price or strike price
Riskless rate of interest id in currency d
Riskless rate of interest if in currency f
Volatility in the underlying exchange rate
Time to expiration

17. 6.3 Currency Option Values Prior to Expiration
Time value and volatility

18. 6.3 Currency Option Values Prior to Expiration
The interaction of time and volatility
If instantaneous changes are a random walk, then T-period variance is T times the one-period variance
sT2 = T s  sT = (√T) s
where s2 = 1-period variance
sT2 = T-period variance
Estimation of exchange rate volatility
Historical volatility
Implied volatility

19. 6.4 Hedging with Currency Options
Currency option hedges
Advantages
Disaster hedge insures against unfavorable currency movements.
Disadvantages
Option premiums reflect option values and so are high when:
The option hedge is in-the-money.
Exchange rate volatility is high.
There is a long time until expiration.

20. 6.4 Hedging with Currency Options
Summary of option payoffs
CallTd/f
Long call
PutTd/f
Long put
Kd/f
Kd/f
STd/f
STd/f
-CallTd/f
Short call
-PutTd/f
Short put
STd/f
STd/f
Kd/f
Kd/f

21. 6.4 Hedging with Currency Options
A currency call option
A pound call is an option to buy pounds:
The option holder gains if pound sterling rises.
The option holder does not lose if pound falls.
V$/£
A long pound call
is an option to buy
pounds sterling
at a contractual
exercise price.
Exercise
price
$1.50/£
Option premium
S$/£
-$0.30/£

22. 6.4 Hedging with Currency Options
A call option hedge
Call option hedge
V$/£
$1.50/£
S$/£
-$0.30/£
-$1.50/£
Option
hedged
position
-$1.80/£
Short exposure

23. 6.4 Hedging with Currency Options
A currency put option
A pound put is an option to sell pounds
The option holder gains if pound sterling falls.
The option holder does not lose if pound rises.
V$/£
A long pound put
is an option to sell
pounds sterling
at a contractual
exercise price.
Exercise
price
$1.50/£
Option premium
S$/£
-$0.30/£

24. 6.4 Hedging with Currency Options
A put option hedge
Long exposure
V$/£
Option
hedged position
+$1.50/£
+$1.20/£
S$/£
-$0.30/£
$1.50/£
Put option hedge

25. 6.4 Hedging with Currency Options
Profit (loss) on a short straddle
Short straddle
Short put
Short call

26. 6.5 Exchange Rate Volatility Revisited
Continuously compounded returns
Continuously compounded changes in exchange rates s (in italics) are related to holding period changes s according to
s = ln(1 + s) or es = eln(1+s) = 1 + s Eqn (6.2)
Normally distributed returns
If continuously compounded returns are normally distributed, then return variance increases linearly with time. For one-period variance σ2, that means
σT2 = (T)σ2 or σT = (T)σ Eqn (6.3)

27. 6.5 Exchange Rate Volatility Revisited
Volatility estimates
Historical volatility is estimated from actual returns:
The standard deviation of currency returns over a recent period provides a easy-to-calculate and easy-to-understand estimate of volatility.
Time-varying estimates (see Chapter 3) such as realized volatility and conditional volatility (e.g., GARCH) provide more sophisticated estimates of historical volatility.

28. 6.5 Exchange Rate Volatility Revisited
Volatility estimates
Implied volatility is implied by an option’s value given the other determinants of option value:
Currency option value determinants are: (1) price of the underlying asset, (2) exercise price, (3) domestic risk-free rate, (4) foreign risk-free rate, (5) time to expiration, and (6) volatility of the underlying asset.
If option value and the first five determinants are known, then the underlying volatility that generates the option value can be found by trial-and-error given an option pricing model.

29. Appendix 6-A Currency Option Valuation
Learning objectives
 A primer on continuous compounding
 The Black-Scholes option pricing model
 The Biger-Hull currency option pricing model

30. Appendix 6-A Currency Option Valuation
A primer on continuous compounding
Natural logarithms and exponentials
y = logex = ln(x)  ey = x
r = holding period return (e.g., annual return)
r = continuously compounded return
r = ln(1+r) Û er = (1+r)
where e ≈ 2.718
Price changes
Compounding: Vt+1 = Vt (1+r) = Vt er
Discounting: V0 = Vt / (1+r) = Vt (1+r)-t = Vt e -r t

31. Appendix 6-A Currency Option Valuation
The Black-Scholes option pricing model
If continuous returns are normally distributed with mean μ and standard deviation σ, then the value of a European call on a non-dividend-paying stock is
Call = P N(d1) – e(–iT) K N(d2)
P = the current share price
K = exercise price of the call option
i = risk-free rate (continuously compounded)
σ = instantaneous standard deviation of return
T = time to expiration expressed as a fraction of one period
d1 = [ ln(P/K) + (i + (σ2/2))T ] / (σ√T)
d2 = (d1 – σ√T)
N(•) = the standard normal cumulative distribution function

32. Appendix 6-A Currency Option Valuation
The Black-Scholes option pricing model
Call = P N(d1) – e(–iT) K N(d2)
N(d1) = Probability the option expires in the money
P N(d1) = E[ PT | PT > K ]

33. Appendix 6-A Currency Option Valuation
The Biger-Hull currency option pricing model
Calld/f = e(–i dT) [ FTd/f N(d1) – Kd/f N(d2) ]
= e(–i fT) [ S0d/f N(d1) ] – e(–i dT) [ Kd/f N(d2) ]
S0d/f = spot exchange rate
FTd/f = S0d/f e(+i dT) e(–i fT) = forward exchange rate
Kd/f = exercise price on one unit of foreign currency
id = domestic risk-free rate (continuously compounded)
if = foreign risk-free rate (continuously compounded)
σ = instantaneous standard deviation of the FX rate
T = time to expiration
d1 = [ ln(Sd/f/Kd/f ) + ( (id – if) + (σ2/2) ) T ] / (σ√T)
d2 = (d1 – σ√T)
N(•) = the standard normal cumulative distribution function

34. Appendix 6-A Currency Option Valuation
The Biger-Hull currency option pricing model
Put option values are found from put-call parity
Calld/f – Putd/f = e(–i dT) ( FTd/f – Kd/f )
or Putd/f = Calld/f – e(–i fT) S0d/f + e(–i dT) Kd/f
for FTd/f = S0d/f e(+i dT) e(–i fT)

35. Appendix 6-A Currency Option Valuation
The Biger-Hull currency option pricing model