APPENDIX 4A: CONTINUOUS COMPOUNDING
6.4 HEDGING WITH CURRENCY OPTIONS
The same principle holds for out-of-the-money call options, as shown below. At expiration, only that portion of the distribution that expires in-the-money has value.
The out-of-the-money call option on the left has little value because there is little likelihood of the forex (FX) rate climbing above the exercise price. As the variability of end-of-period FX rates increases in the graph on the right, there is an increasing probability that the spot rate will close above the exercise price.
Exchange rate volatility and out-of-the-money call option value
CallTd/f CallTd/f
STd/f STd/f
STd/f STd/f
The same general principle holds for the in-the-money call options shown below.
If an underlying exchange rate is below the exercise price at expiration, the option has zero value regardless of how far the closing price falls below the exercise price.
On the other hand, the call option continues to increase in value as the spot rate increases. Thus, in-the-money call options also benefit from higher volatility in the underlying asset.
Option values gain from volatility.
Exchange rate volatility and in-the-money call option value
CallTd/f CallTd/f
STd/f’ STd/f
STd/f STd/f
Similarly, currency puts gain more in value from exchange rate decreases than they lose in value from increases of the same magnitude. The general rule is that currency options gain from increasing variability in the distribution of end-of-period exchange rates regardless of whether the option is in-the-money, at-the-money, or out-of-the-money. In turn, variability in the distribution of end-of-period exchange rates depends on exchange rate volatility and on the time to expiration.
Static Hedging Strategies That Match on Expiration
Suppose the Japanese firm Toyota anticipates a £1 million cash inflow from a U.K. customer on December 13, which happens to be a Friday on which CME currency options expire. If left unhedged, the yen value of this cash flow will depend on the spot rate at expiration. Toyota can use a currency option as a form of insurance or ‘‘disaster hedge’’ against an unfavorable change in the value of the pound. Suppose interest rates in yen and pounds are equal, and the spot rate is equal to the forward rate at S¥/£0 =F¥/£T =¥200/£. The CME yen-per-pound cross-rate futures contract has a contract size of £62,500, so it will take 16 CME futures to hedge the £1 million cash inflow. The CME pound option with a striking price of K¥/£=¥200/£ is at-the-money, and the price of both call and put options are Call¥/£=Put¥/£=¥40/£.
Toyota’s long pound exposure is shown on the left in the following graph.
Toyota needs to offset the downside risk of this exposure, and so needs a hedge with a negative exposure below the ¥200/£ exercise price. The long pound put option in the middle graph does the trick. When combined with the underlying long pound exposure, the long pound put transforms the payoffs to Toyota’s net position as shown on the right.
¥200/£
¥200/£
Long £ exposure + Long £ put option hedge = Net (hedged) position
A long pound exposure hedged with a long pound put
V¥/£ V¥/£ V¥/£
ST¥/£ ST¥/£ ST¥/£
¥200/£ –¥40/£
¥160/£
–¥40/£
¥160/£
The net position is found by adding the y-axis values of the underlying position and its hedge at each point along the x-axis. For example, at a closing spot price of S¥T/£=¥0/£, the underlying position is worthless while the pound put pays ¥160/£.
Between S¥/£T =¥0/£ and S¥/£T =¥200/£, for every increase in value on the underlying exposure there is a corresponding decrease in value from the long put, so the payoff on the combined position remains ¥160/£. Above a closing price of S¥/£T =¥200/£, the option is out-of-the-money while the long exposure continues to gain in value.
In essence, Toyota has paid an option premium (in this case, an insurance premium) of ¥40/£ to ensure that it’ll receive at least ¥160/£ on its net position.
As with other forms of insurance, Toyota would prefer that it not have to exercise its option. Toyota benefits when the pound rises above ¥200/£. This is the preferred outcome. Just like auto insurance that is exercised only when there is an accident, the option insurance is exercised only when bad events unfold; that is, when the pound falls in value and erodes the value of the underlying position.
Suppose Toyota has a short euro exposure of 10 million due on December 13, as shown on the left in the figure below.
Net (hedged) position A short euro exposure hedged with a long euro call
V¥/€ V¥/€ V¥/€
ST¥/€ ST¥/€
¥160/€ ¥200/€
ST¥/€
¥200/€ –¥20/€
–¥220/€ Short € exposure Long € call option hedge
The CME trades a 125,000 cross-rate contract against the yen, so ( 10 million)
÷( 125,000/contract)=80 contracts will offset Toyota’s underlying position. The spot and forward rates are S¥0/ =F¥T/ =¥160/ . Toyota decides to hedge the short euro exposure with a long euro call at an exercise price of K¥/ =¥200/ . The option premium on this contract is Call¥/ =¥20/ , as shown in the middle graph.
The combined or net position is shown on the right. If the spot rate closes at
¥160/ , Toyota will owe ( 10 million)(¥160/ )=¥1,600 million on its underlying exposure. It’ll lose even more on its underlying exposure if the euro appreciates above ¥160/ . The long euro call at K¥/ =¥200/ protects Toyota against a euro increase of more than ¥200/ , but the cost of this hedge is the option premium of
¥20/ . The net result is that Toyota has hedged against an increase in the value of the euro above ¥200/ at a cost of ¥20/ . At worst, Toyota will pay ( 10 million)(¥220/ )=¥2,200 million to fulfill its obligation. At best, the yen-per-euro spot rate will fall and Toyota’s obligation will correspondingly decrease in value.
Dynamic Hedging Strategies with Rebalancing
Individual transactions can be hedged against currency risk. However, it is more cost-effective to first offset transactions within the firm and then hedge the firm’s net exposures to currency risks. Thus, exposures evolve over time and dynamic hedging strategies need to adapt to these changing circumstances. This section presents several measures that are useful in dynamically managing the firm’s evolving exposures to currency risk.
Delta Hedges The sensitivity of option value to change in the value of the underlying asset is calledoption delta.Call option deltas are positive, as indicated by the slope of option value in the call option payoff profiles. The delta of a call option increases as the underlying asset increases in price. For deep-in-the-money calls, the slope of option value approaches a delta of one (i.e., a 45-degree line). The delta of a put option is negative and approaches zero from below as the price of the underlying asset increases.
Option delta is the sensitivity of option value to the underlying asset.
Option delta also is called the hedge ratiobecause it indicates the number of options required to offset one unit of the underlying asset and minimize the variance of the hedged position.6 This measure is useful when hedging an underlying spot, forward, or futures position. Suppose the delta of a currency call option on the yen/dollar futures price is +0.50. For a given (small) change in the futures price, option value increases by exactly 50 percent of that amount.
To form a delta-neutral hedgeof a forward position with an option position, an offsetting position is taken according to the hedge ratio. For example, a Japanese firm can hedge a future dollar obligation of $1 million with a long dollar call option.
For a long call with a delta of +0.50, the firm should take a $2 million option position to offset the underlying $1 million obligation. A small increase in the yen value of the dollar will result in a loss in value on the underlying forward obligation.
This loss is offset by a gain in value on the long call position. Note that the expiration date of this option does not need to match that of the underlying forward obligation.
The $1 million forward obligation also could be offset by writing a $4 million put option with a delta of −0.25. An increase in the value of the dollar futures price then increases the yen value of the forward obligation at the same time that it decreases the yen obligation on the short put option.
Conversely, a future cash inflow of $1 million can be delta-hedged with (1) a short position of $1.25 million on a dollar call option with a delta of+0.80, or (2) a long position of $3 million on a dollar put option with a delta of−0.33.
More Funny Greek Letters Option delta is a measure of the rate of change or sensitivity of option value to change in the underlying asset value. A delta hedge uses this measure to offset an underlying exchange rate exposure with a currency option position that has the same sensitivity to an exchange rate change. However, option delta changes as the underlying price changes. As delta changes, so does the hedge ratio that matches the sensitivities of the option and the underlying positions.
When the delta of an option hedge changes at a different rate than that of the underlying position, even small changes in an underlying exchange rate can quickly throw a delta hedge out of balance. The option pricing methods in the appendix to this chapter assume continuous rebalancing. In practice, option hedges must be closely monitored to make sure they do not become too unbalanced.
Option gammais the rate of change of delta with a change in underlying asset price; that is, the curvature of option value in the option payoff profiles.7 Many option hedges are designed to be gamma-neutral as well as delta-neutral. Matching on gamma usually means forming a hedge with payoffs that match those of the underlying position. Hedges that are both delta-neutral and gamma-neutral are far less likely to become unbalanced with changes in underlying asset values.
Another useful measure of option sensitivity isvega, which is the sensitivity of option value to changes in the volatility (or standard deviation) of the underlying asset. Option vega is greatest for long-term options, all else being constant. As time to expiration decreases, so too does option vega. Vega also is larger for near-the-money options than for deep in-the-money or deep out-of-the-money options.
Finally,thetais the sensitivity of option value to change in the time to expiration.
All else being constant, theta increases in absolute value as the time to expiration decreases, so that currency options lose most of their value just prior to expiration.
Theta is also greater in absolute value for near-the-money than for deep in-the-money or deep out-of-the-money options.
Combinations of Options
Two or more option positions can be combined by snapping together the correspond- ing option payoff profiles. This is a simple yet powerful technique for understanding the risks and potential payoffs of even the most arcane option positions.
Here’s an example. In early 1995, a rogue trader named Nick Leeson drove the United Kingdom’s Barings Bank into bankruptcy through unauthorized speculation in Nikkei stock index futures on the Singapore and Osaka stock exchanges. Leeson sold option straddleson the Nikkei index at a time when volatility on the index was low. A long option straddle is a combination of a long call and a long put on the same underlying asset and with the same exercise price, as shown below.
Long call at Kd/f + Long put at Kd/f = Long straddle
CallTd/f PutTd/f
STd/f STd/f
Kd/f Kd/f
Vd/f
STd/f Kd/f
Leeson formed a short straddle by selling calls and puts. After including the proceeds from these option sales, the profit/loss diagram on the short straddle position at expiration is as follows:
Profit (loss) on a short straddle VTNikkei
KTNikkei
STNikkei
STNikkei
Leeson placed a bet on the volatility of the Nikkei index. In option parlance, Leeson ‘‘sold volatility.’’ So long as the Nikkei index did not vary too much, Leeson would have won his bet. As seen in the diagram, Leeson wins if the end-of-period Nikkei index falls between the two points at which the profit/loss pyramid crosses the x-axis. Leeson loses if the Nikkei index rises too high or falls too low. Volatility on the Nikkei index was low at the time Leeson sold his position, so the proceeds from the sale were small (and Leeson’s gamble was large) relative to what would have been received on this position in a high-volatility market. As it turned out, the Nikkei index fell below the profitable range. Leeson incurred further losses by
buying futures on the Nikkei index in the hopes of a recovery that, to Barings’ regret, never occurred.