CURRENCY ARBITRAGE
5.7 OPTIONS HEDGING OF SHORT-TERM TRANSACTION EXPOSURE
offset by identical contracts, and there is no scope for the advantages of clearing houses and settlement by the payment of the difference. Through their rules and standardisation, futures provide liquidity and eliminate counterparty risk. Penings and Leuthold (2000) argue that futures contracts can provide jointly preferred contracting arrangement, enhancing relation
ships between firms. What motivates the use of futures contracts is then contract preference, level of power and conflicts in contractual relationships of firms.
CIP and the relative effectiveness of forward and money market hedging
We have seen that money market hedging entails the conversion of payables and receivables at the interest parity forward rate, whereas forward hedging amounts to conversion at the actual forward rate. Under covered interest parity these rates are equal, which means that money market hedging and forward hedging will produce similar results. Al-Loughani and Moosa (2000) use this idea to devise an indirect test of CIP. The idea is that if money market hedging and forward hedging produce similar results then CIP must be valid.
They tested this hypothesis using five exchange rates and found that CIP actu
ally holds.
5.7 OPTIONS HEDGING OF SHORT-TERM TRANSACTION
1. If it assumes the value St+1,1, such that St+1,1 < St, then the option will not be exercised, and the foreign currency will be bought on the spot market.
The total cost of covering the payables is the base currency value, KSt+1,1 and the option premium lost, which is equal to Km, where m is the premium per unit of the underlying currency. The total cost will thus be K(St+1,1 + m).
2. If it assumes the value St+1,2, such that St+1,2 > St, then the option will be exercised, and the foreign currency will be bought at the exercise exchange rate, St. The total cost of covering the payables is the base currency value, KSt, and the option premium paid up front, which is equal to Km. The total cost will thus be K(St + m).
Hedging receivables with a put option
We now consider the case of hedging receivables worth K units of currency y via a put option. In this case the put option used as a hedging instrument gives the hedger the right to sell Kunits of currency yat the exercise exchange rate.
Making the same assumptions as before, we have the following possibilities:
1. If St+1,1 < St, then the option will be exercised, and the underlying amount of currency ywill be sold to the option writer. The value obtained will be equal to the domestic currency value, KSt, minus the option premium paid up front, which is equal to Km. Thus, the net amount is equal to K(St – m).
2. If St+1,2 > St, then the option will not be exercised, and the foreign currency will be sold on the spot market at St+1,2. The net amount received is there
fore K(St+1,2 – m).
A summary of the decision rules
If the exchange rates St+1,1 and St+1,2 are expected with probabilities of p1 and p2 respectively, then the decision to hedge payables and receivables will depend on the expected domestic currency values under the hedge and no- hedge decisions. Table 5.7 lists all of the possibilities.
TABLE 5.7 The options hedge/no-hedge decision.
Condition Payables Receivables
) t ) ,
(St+1 1, + m p1+ (S + m p2 < (S t+1 1 1, p + St+1 2p 2) Hedge
) t ) ,
(St+1 1, + m p1+ (S + m p2 > (S t+1 1 1, p + St+1 2p 2) No hedge
) t ) ,
(St+1 1, + m p1+ (S + m p2 = (S t+1 1 1, p + St+1 2p 2) Indifference
(S t -m p) 1+ (St+1 2, -m p) 2 > (S t+1 1, p1+ St+1 2, p2) No hedge (S t -m p) 1+ (St+1 2, -m p) 2 < (S t+1 1, p1+ St+1 2, p2) Hedge
t ) , ) t+1 11+ St+1 2 2) Indifference
(S -m p1+ (St+1 2-m p2 = (S , p , p
_ _
Payables Receivables
+ +
Unhedged position
t 0 S
Unhedged position
St+ 1 S t
St+ 1
_
+ +
Long call _
Long put
0 St+ 1
St+ 1
+ +
Hedged position Hedged position
St+ 1 St+ 1
_ _
FIGURE 5.6 Options hedging of payables and receivables.
Figure 5.6 illustrates options hedging of payables and receivables assuming an exercise exchange rate of St, which implies that the option is at the money.
Consider the hedging of payables, which is shown in the left-hand panel of Figure 5.6. The top part of the diagram shows the payoff on the unhedged position measured as the difference between the value of the payables at St and St+1. The middle part of the diagram shows the payoff on a long call posi
tion at an exercise exchange rate of St. The bottom part of the diagram shows the payoff on the combined position. Unlike the unhedged position, the combined position shows an upper limit on the possible loss arising from
adverse exchange rate movements. To hedge receivables, a long put position is taken as the hedging instrument. The hedged position shows that while there is a lower limit on the loss arising from adverse exchange rate move
ments, there is no limit on the potential gain arising from favourable exchange rate movements.
Hedging a contingent exposure
A contingent exposure means that the exposure arises only if a certain outcome materialises (for example, when a contract is awarded to the under
lying firm). In this case, an option hedge is preferable to a forward hedge because the latter can cover all possible eventualities. Let us assume that a firm bids for a contract valued at K units of currency y. Exposure to currency y would arise only if the contract is awarded to the firm, in which case the firm would have receivables worth the value of the contract. If, in the mean time, currency y depreciates against the base currency, the firm would incur some losses.
Let us see what happens when the firm uses forward contracts and options to hedge this contingent exposure. To use a forward contract, the firm takes a short forward position on currency y by selling K units of the currency forward. In any case (that is, whether or not the contract is awarded) the firm is committed to come up with K units of currency y on the maturity of the contract. There are then two outcomes:
1. The contract is awarded, in which case the amount of the receivables covers the forward contract. Hence there is no problem, as the hedge works in the sense that the firm manages to lock in the base currency value of the receivables.
2. The contract is not awarded, in which case the amount of y has to be provided at the pre-specified forward rate. If the spot rate at that time happens to be higher than the forward rate, the firm would incur unlimited loss, proportional to the difference between the spot and forward rates.
Now, let us consider what happens if a put option is used to hedge this contingent exposure. In this case there are four possible outcomes, because the option may or may not be exercised. The following outcomes are possible:
1. The contract is awarded and the actual exchange rate turns out to be less than the exercise exchange rate (St+1 < St). The firm exercises the option, converting the proceeds of the contract to the base currency at the exercise exchange rate, obtaining KSt units of x.
2. The contract is awarded but the exchange rate turns out to be greater than the exercise exchange rate (St+1 > St). The firm does not exercise, losing the premium on the option but the proceeds from the contract are converted at the higher rate (KSt+1).
(a) Forward contract
Firm receives KFt units of x Contract awarded
for K units of y
Firm receives KSt + 1units of Contract not awarded
x for K units of y
St +1 < St Æ Firm exercises put to obtain KSt units of x
Contract awarded
Firm does not exercise put, St +1 > St Æ
losing premium (b) Put option
St +1 < St Æ Firm exercises put for profit
Contract not awarded
Firm does not exercise put, St +1 > St Æ losing premium
FIGURE 5.7 Hedging a contingent exposure with forward contracts and options.
3. The contract is not awarded and the actual exchange rate turns out to be less than the exercise exchange rate (St+1 < St). The firm exercises the option, making profit of K(St – St+1).
4. The contract is awarded but the exchange rate turns out to be greater than the exercise exchange rate (St+1 > St). The firm does not exercise, losing the premium on the option.
All of these outcomes are illustrated in Figure 5.7. It is obvious that if a forward contract is used to hedge a contingent exposure, then the loss will be unlimited if the exchange rate rises and the contract is not awarded. If, on the other hand, a put option is used to hedge this exposure, then the maximum loss would be the premium on the option, whether or not the contract is awarded.
Writing an option in itself creates a contingent exposure that can be hedged by taking an opposite option position. Figure 5.8 shows the payoff on a short call, which arises when a firm writes a call, giving the holder the right to buy the currency at an exercise exchange rate of St. As long as the actual exchange rate does not turn out to be higher than the exercise exchange rate (St+1 < St), the option will not be exercised and the firm writing the option will make profit equal to the option premium paid by the holder (m). But if the actual exchange rate turns out to be higher than the exercise exchange rate (St+1 >
St), the option will be exercised and the loss will be unlimited, increasing with the level of St+1. Suppose now that the firm decided to hedge this exposure
+
(exposure) S t
S t
S t
S t
S t
S t
_ m
m +
_ +
_
Short call
1 +
1 +
1 +
Long forward
Hedged position
FIGURE 5.8 Hedging a contingent exposure with a forward contract.
with a forward contract, assuming for simplicity that the forward rate is equal to the exercise exchange rate. The payoff on the forward contract is shown by the middle part of the diagram. Profit will be made on this position as long as the actual exchange rate turns out to be higher than the exercise exchange rate and the forward rate. The bottom part of the diagram shows the payoff on the combined (hedged) position. If the actual exchange rate turns out to be higher than the exercise exchange rate, profit on the forward contract will offset the loss on the option, and the firm will gain the premium on the put option. If, on
+
Short call (exposure) m
St St + 1
_ +
Long call St
St+ 1 m ¢
_ +
Hedged position
m – m ¢ St+ 1
St
_
FIGURE 5.9 Hedginga contingent exposure with an option (sameexerciseexchange rate).
the other hand, the actual exchange rate turns out to be lower than the exer
cise exchange rate, losses on the forward contract will be unlimited.
Consider now Figure 5.9, which shows the situation when the exposure is hedged by a long call position with the same exercise exchange rate but at a lower premium, m¢. If the actual exchange rate is lower than the exercise rate, neither of the two options will be exercised, and the firm will make profit that is equal to the difference between the two premiums (m m- ¢). At higher exchange rates both of the options will be exercised, and the losses on the unhedged
+
m
¢ S t
S t
S t
_ +
_ +
t¢ S
t¢ S
t¢ S
) ¢
Hedged position St + 1
St + 1
¢ m
¢ m – m
St + 1 ( m – m – (S t – St )
_
FIGURE 5.10 Hedging a contingent exposure with an option (different exercise exchange rate).
position will be offset by the gains on the hedge (the long call). Net profit remains at m m- ¢.
Finally, Figure 5.10 shows the situation when the hedging instrument is an option with a lower premium and a higher exercise exchange rate (S¢t). In this
- ¢) -( ¢ -
case, the firm will incur a maximum loss of (m m St St), if the market exchange rate turns out to be higher than S¢t.
Hedging against exchange rate volatility
A trading firm engaged in exporting and importing may find it desirable to work with stable exchange rates. This firm can hedge the risk arising from exchange rate volatility by taking an option position that compensates it if the underlying exchange rate rises above or falls below a certain level. This posi
tion is called a long straddle, which consists of a long call and a long put at the same exercise exchange rate.
A long straddle is shown in Figure 5.11. The top part of the diagram shows the payoff on a long call with a premium of mc. The middle part shows the payoff on a long put with a premium of mp. The total cost of the long straddle,
+
S t
S t
S t
S t
S t
S t
_ +
_ + m c
m p
p c m m +
1 +
1 +
1 +
Long straddle Long put
Long call
_
FIGURE 5.11 Hedging against exchange rate volatility by using a long straddle.
whose payoff is shown in the bottom part of the diagram, is mc+ mp. If the exchange rate at time t+ 1 goes above or below the exercise exchange rate, St, by more than mc+ mp, the firm will be compensated for the difference by exer
cising the call (above) or the put (below).