Chapter 6 Currency Options and Options Markets

1
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
Butler, Multinational Finance, 4e
2
A forward obligation

• A 1 million obligation due in four months

Currency exposure
Underlying transaction
v/
s/
-1,000,000
Option payoffs
3
A forward hedge

• Buy 1 million in the forward market at the
  forward price F1/ 1.45/

Exposure of forward contract
Long pound forward
v/
1,000,000
s/
-1,450,000
Option payoffs
4
An option hedge

• A currency option is like one-half of a 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/
Option payoffs
5
CME pound Dec 1450 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 1.45/
• Rule for exercise an American option exercisable
  anytime until expiration

Option payoffs
6
Currency option quotations

• British pound (CME)
• 62,500 cents per pound
• Strike Calls-Settle Puts-Settle
• Price Oct Nov Dec Oct Nov Dec
• 1430 2.38 . . . . 2.78 0.39 0.61 0.80
• 1440 1.68 1.94 2.15 0.68 0.94 1.16
• 1450 1.12 1.39 1.61 1.12 1.39 1.61
• 1460 0.69 0.95 1.17 1.69 1.94 2.16
• 1470 0.40 0.62 0.82 2.39 . . . . 2.80
• Note S0/ 1.45/

Option payoffs
7
Payoff profile of a pound callat expiration
Option payoffs
8
Profit (loss) on a call optionat expiration
Option premium Callt/ 0.40/
FX rate at expiration 1.45/ 1.69/ 1.93/ Prem
ium (cost) -25,000 -25,000 -25,000 Exercise
price 0 -90,625 -90,625 Spot
sale 0 105,625 120,625 Net profit
-25,000 -10,000 5,000
Option payoffs
9
Payoff profile of a call optionat 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
Option payoffs
10
Payoff profile of a put optionat expiration
Long put
Short put
PutT/
-PutT/
KT/
ST/
ST/
KT/
In-the- money
Out-of-the- money
Out-of-the- money
In-the- money
Option payoffs
11
Puts and calls
Call option to buy pounds at KT/
An option to sell dollars at KT/
DCallT/
DPutT/

DST/
DST/
Option payoffs
12
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/
DFT/
-DPutT/

DST/

DST/
DST/
Option payoffs
13
Put-call parity at expiration CallTd/f -
PutTd/f Kd/f FTd/f
Exercise price
Long call
Short put
CallT/
KT/
-PutT/


ST/
ST/
ST/
KT/
FT/

Long forward
ST/
Option payoffs
14
The time value of an option

• Time value Option value - intrinsic value
• Intrinsic value value if exercised immediately
• 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

Option value determinants
15
Time value and volatility
Option value determinants
16
Time value and volatility
Option value determinants
17
The interaction of time and volatility

• If instantaneous changes are a random walk, then
  T-period variance is T times one-period variance
• sT2 T s2
• where s2 1-period variance
• sT2 T-period variance
• Estimation of exchange rate volatility
• Historical volatility
• Implied volatility

Option value determinants
18
Combinations of options
Combinations of options
19
Advanced Pricing currency options

• Suppose the Australian-per-US dollar spot rate
  A2.4/ bifurcates by a continuously compounded
  4 percent per period for 4 periods
• 4 successive bifurcations result in 24 16 price
  paths
• Value after 1 period is P1 P0e0.04
• (A2.4/)e-0.04 A2.306/
• (A2.4/)e0.04 A2.498/
• each with 50 percent probability

Currency option valuation
20
4 percent for 4 periods
Currency option valuation
21
24 16 possible price paths

• n 1 2 3 4
• 1
• 1
• 1 4
• 1 3
• 1 2 6
• 1 3
• 1 4
• 1
• 1
• 2n 2 4 8 16

Currency option valuation
22
End-of-period distribution for n 4
Probability
0.40
0.30
0.20
0.10
0.00
2.816
2.600
2.400
2.045
2.215
Currency option valuation
23
More frequent compounding

• Suppose we apply this binomial model with 1 per
  period for 16 periods
• This results in 216 65,536 price paths and
  (n1) 17 possible prices

Currency option valuation
24
1 percent for 16 periods
Currency option valuation
25
End-of-period distribution for n 16
0.20
0.15
0.10
0.05
0.00
2.045
2.129
2.215
2.306
2.400
2.498
2.600
2.706
2.816
Currency option valuation
26
The Binomial and B-S OPMs

• As the binomial process generating up and down
  movements bifurcates over shorter and shorter
  intervals
• the binomial distribution approaches the normal
  distribution
• continuous-time pricing methods (e.g., as in the
  Black-Scholes OPM)

Currency option valuation