APPENDIX 4A: CONTINUOUS COMPOUNDING
6.2 OPTION PAYOFF PROFILES
OTC options are custom-tailored.
Retail clients include corporations and financial institutions that have a need to manage their currency risk exposures. These clients value the right to exercise a currency option and typically do not want the obligation from writing option contracts. International commercial and investment banks are the principal writers (sellers) of currency options. This asymmetry between buyers and sellers is not seen in currency forward and futures markets. International banks also maintain an active wholesale market in which they hedge — or reinsure — the currency risk exposures in their asset/liability portfolios.
in the spot market than at the exercise price of $1.45/£. The call is in-the-money when the spot rate is above the exercise price. Suppose the spot rate at expiration is
$1.50/£ on a £62,500 CME option. The option holder has the right to buy pounds at a price of $1.45/£. The option holder can then sell this £62,500 in the spot market at $1.50/£ for a five-cents-per-pound profit, or (£62,500)($0.05/£)=$3,125.
The right-hand graph plots call value from the perspective of the option writer.
This contract is a zero-sum game, in that any value gained by the option holder is a loss to the option writer. The risk profile — orpayoff profile— of a short call is the mirror image of the long call.
Currency Put Options The payoff profile of a long pound put option at expiration is shown below on the left, with its corresponding short position on the right.
Puts are options to sell.
Payoff profile of a long pound put at expiration
Payoff profile of a short pound put at expiration
PutT$/£ –PutT$/£
In-the- money
In-the- money Out-of-the-
money
Out-of-the- money
$1.45/£
$1.45/£
ST$/£ ST$/£
Put options are options to sell the underlying asset, so currency put options are in-the-money when the exercise price is greater than the underlying exchange rate. If the exercise price on a £62,500 CME put option is $1.45/£, then the option holder will exercise the option at expiration when the underlying exchange rate is below $1.45/£. For example, if the spot rate at expiration is $1.40/£, then the option holder can buy £62,500 in the spot market for $1.40/£ and simultaneously exercise the option to sell £62,500 to the option writer for $1.45/£, for a net profit of (£62,500)($0.05/£)=$3,125. As in the case of a call option, any gain in value to the option holder in the left-hand graph is a loss to the option writer in the right-hand graph.
Profit and Loss on a Currency Option at Expiration
Options to buy or sell currencies are not free; option sellers demand an option premium for writing an option. The premium depends on the writer’s expected losses should the option expire in-the-money. The effect of this premium on the profit or loss of an option is obtained by superimposing the premium on the option’s payoff profile, as described below.
$0.6280/A$ $0.6400/A$ $0.6520/A$ $0.6640/A$ $0.7720/A$
–$0.0120/A$
Profit at expiration ($/A$)
FutTd/f
$0.6280/A$ $0.6400/A$ $0.6520/A$ $0.6640/A$ $0.7720/A$
Premium cost –$1,200 –$1,200 –$1,200 –$1,200 –$1,200
Exercise price $0 –$64,000 –$64,000 –$64,000
A$ sale $0
$0
$0 $65,200 $66,400 $77,200
Net profit or loss –$1,200 –$1,200 $0 $1,200 $12,000
CME option quotation “A$ Dec 6400 call” selling for $0.0120/A$ on an A$100,000 contract
=> Exercise price = ($0.6400/A$)(A$100,000) = $64,000
=> Premium cost = ($0.0120/A$)(A$100,000) = $1,200 Exercise price K$/A$= $0.6400/A$
Option premium = $0.0120/A$
Kd/f
FIGURE 6.3 Profit or Loss on a Call Option at Expiration.
The option premium is the price of the option.
Currency Call Options Figure 6.3 displays the profit or loss at expiration of an Australian dollar call option quoted as ‘‘A$ Dec 6400 call’’ and selling on the CME at an option premium of $0.0120/A$. This option has an exercise price of
$0.6400/A$ and expires on the third Wednesday in December. The deliverable instrument of a CME currency option is the corresponding CME futures contract.
Each Australian dollar option contract on the CME is worth A$100,000, so this option costs $64,000=($0.6400/A$)(A$100,000) to exercise. At a price of
$0.0120/A$, the option costs $1,200=($0.0120/A$)(A$100,000) to purchase.
The value of this option at expiration depends on the difference between the futures price and the exercise price. Profit or loss at expiration is shown in Figure 6.3 at several exchange rates. This graph combines option value at expiration with the initial cost of the option. For example, if the actual futures price is $0.6520/A$ at expiration, then selling A$100,000 in the futures market yields $65,200, which just covers the $64,000 exercise price and the $1,200 option premium.
This is a zero-sum game between the option writer and the option holder, as the writer’s payoff is a mirror image of the seller’s. The option holder gains (and the writer loses) whenever the futures price closes above $0.6520/A$. The option holder loses (and the writer gains) whenever the futures price closes below $0.6520/A$.
Currency Put Options Figure 6.4 shows the profit or loss at expiration on a CME
‘‘A$ Dec 6400 put’’ selling at an option premium of $0.0160/A$. At this price, one A$100,000 contract costs $1,600=($0.0160/A$)(A$100,000). The cost of exercise is again $64,000 at the $0.6400/A$ exercise price. The option writer’s payoff is the mirror image of the option holder’s payoff. The option holder gains when the exchange rate closes at any price below $0.6240/A$. The option writer gains whenever the exchange rate closes above $0.6240/A$. Again, currency options are a zero-sum game; the option holder’s gain equals the option writer’s loss.
$0.6080/A$ $0.6240/A$ $0.6400/A$ $0.6560/A$
–$0.0160/A$
Profit at expiration ($/A$)
$0.6080/A$ $0.6240/A$ $0.6400/A$ $0.6560/A$
Premium cost –$1,600 –$1,600 –$1,600 –$1,600
A$ purchase –$60,800 –$62,400 $0 $0
Exercise price +$64,000 +$64,000 $0 $0
Net profit or loss +$1,600 $0 –$1,600 –$1,600
CME option quotation “A$ Dec 6400 put” selling for $0.0160/A$ on an A$100,000 contract
=> Exercise price = ($0.6400/A$)(A$100,000) = $64,000
=> Premium cost = ($0.0160/A$)(A$100,000) = $1,600 Exercise price K$/A$= $0.6400/A$
Option premium = $0.0160/A$
FutTd/f Kd/f
FIGURE 6.4 Profit or Loss on a Put Option at Expiration.
At-the-Money Options and Asset Pricing Relations
Suppose a currency option is at-the-money, with an exercise price equal to the current exchange rate. If exchange rates are a random walk, then the current spot rate and the exercise price equal the expected future spot rate at expiration. Centering the origin of a payoff profile on the exercise price provides a graph of changes in option values against changes in exchange rates, as shown below for a call option on pounds sterling. The deliverable instrument is the pound, so it is convenient to use dollar-per-pound prices.
Payoff profile of a long pound call at expiration
Payoff profile of a short pound call at expiration
∆CallT$/£
∆ST$/£
−∆CallT$/£
∆ST$/£
Out-of-the-money In-the-money Out-of-the-money In-the-money
≡
A Call Option by Any Other Name Buying pounds at S$/£ means that you are simulta- neously selling dollars at S£/$. For this reason, an option to buy pounds at a price of K$/£is the same contract as an option to sell dollars at K£/$. That is, a call option to buy pounds sterling is equivalent to a put option to sell dollars. The payoff profiles of a pound call and its counterpart, the dollar put, are shown here.
A call option on a currency is a put option on another currency.
Payoff profile of a long pound call at expiration
Payoff profile of a long dollar put at expiration
Out-of-the-money In-the-money In-the-money Out-of-the-money
≡
∆CallT$/£
∆ST$/£
∆PutT$/£
∆ST$/£
Prices in these figures are related according to P$/£=(P£/$)−1. This option is in-the-money when the spot rate S$/£is above the exercise price K$/£or, equivalently, when the spot rate S£/$ is below the exercise price K£/$. Since a call option to buy pounds with dollars is equivalent to a put option to sell dollars for pounds, these payoff profiles are equivalent. In this sense, a currency option is simultaneously both a put and a call.
On the other side of the contract, the option writer has an obligation to sell pounds and buy dollars. From the option writer’s perspective, an obligation to sell pounds for dollars is equivalent to an obligation to buy dollars with pounds. These equivalent payoffs are shown next.
Payoff profile of a short pound call at expiration
Payoff profile of a short dollar put at expiration
–∆CallT$/£ –∆PutT$/£
∆ST$/£ ∆ST$/£
Out-of-the-money In-the-money In-the-money Out-of-the-money
≡
Shakespeare wrote, ‘‘A rose by any other name would smell as sweet.’’ This is true for currency options as well. An in-the-money pound call is just as sweet to the option holder as the corresponding in-the-money dollar put.
A Forward by Any Other Name Suppose you purchase an at-the-money pound call and simultaneously sell an at-the-money pound put with the same expiration date.
The payoff profiles of these two option positions at expiration can be combined into a single payoff profile, as shown here.
A forward is the same as a long call and a short put.
Long pound call Short pound put
+ =
Long pound forward
∆CallT$/£ −∆PutT$/£ ∆FT$/£
∆ST$/£ ∆ST$/£ ∆ST$/£
Does the graph on the right look familiar? It should. A combination of a long pound call and a short pound put with the same exercise price and expiration date creates the same payoff at expiration as a long forward position on pounds sterling.
Conversely, a short pound call and a long pound put with the same exercise price and expiration date is equivalent to a short forward position in pounds sterling at expiration.
Short pound call Long pound put
+ =
Short pound forward
−∆CallT$/£ ∆PutT$/£ −∆FT$/£
∆ST$/£ ∆ST$/£ ∆ST$/£
If the value of the pound is below the exercise price at expiration, the long put allows you to sell pounds at the above-market exercise price. If the value of the pound is above the exercise price at expiration, the short call forces you to buy pounds at the below-market exercise price. The resulting payoff profile exactly matches that of a short pound forward position with the same contract price and expiration date.
Put-Call Parity The previous section showed that the exposure of a long forward position can be replicated with a long call and a short put on the underlying asset.
Conversely, the exposure of a short forward position can be replicated with a short call and a long put. Thus, the no-arbitrage condition ensures that the values of puts and calls at a particular exercise price must be related to the value of a forward con- tract on the underlying asset. The general case of this relation is calledput-call parity.
Put and call values are related to forward rates.
Suppose a call and a put option are written on currency f with a single exercise price Kd/f and an expiration date in T periods. Put-call parity relates the option values Calld/fand Putd/fto the discounted present values of the exercise price and the forward price
Calld/f−Putd/f=(Fd/fT −Kd/f)/(1+id)T (6.1) where idis the risk-free rate of interest in the domestic currency. Arbitrage between markets in these currency derivatives ensures that the put-call parity relation holds within the bounds of transaction costs.