APPENDIX 4A: CONTINUOUS COMPOUNDING
6.3 CURRENCY OPTION VALUES PRIOR TO EXPIRATION
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.
∆ Option value determinant ∆Calld/f ∆Putd/f
↑ Underlying exchange rate (Sd/f or Futd/f) ⇒ ↑ ↓
↑ Exercise price (Kd/f) ⇒ ↓ ↑
↑ Risk-free rate of interest in currency d (id) ⇒ ↑ ↓
↑ Risk-free rate of interest in currency f (if) ⇒ ↓ ↑
↑ Volatility in the underlying exchange rate (σ) ⇒ ↑ ↑
↑ Time to expiration (T) ⇒ ↑ ↑
FIGURE 6.5 The Determinants of Currency Option Values.
Currency option values are a function of the six variables shown in Figure 6.5.
The price of an American currency call or put option will respond as indicated when each of these determinants is increased while holding the other determinants of option value constant.2With the exception of volatility, each of these determinants of option value is readily observable for currency options quoted on major exchanges.
The exercise price and expiration date are stated in the option contract, and the underlying exchange rate and the foreign and domestic interest rates are quoted in the financial press. The volatility of the underlying asset is not directly observable, which makes it an extremely important ingredient in option valuation.Volatilityrefers to the standard deviation of continuously compounded returns to the underlying asset or exchange rate. (Section 6.5 discusses volatility in more detail.)
Volatility must be estimated.
Options have two sources of value prior to expiration: the intrinsic value of immediate exercise and thetime valuereflecting the value of waiting until expiration before exercise.
APPLICATION An Application of Put-Call Parity
The CME trades a call option on U.K. pounds sterling with an exercise price of K$/£ =$1.7500/£ and an expiration date in six months. The risk-free rates of interest are i£=4.08 percent and i$=0.50 percent per annum. The spot exchange rate is S$/£0 =$1.7600/£. The call option sells for an option premium of $0.0717/£.
Put-call parity allows us to calculate the value of a pound put with the same exercise price and expiration date as the pound call. The forward rate from interest rate parity is F$/£T =S$/£0 [(1+i$)/(1+i£)]T=($1.7600/£)[(1.0408)/ (1.0050)]1/2=$1.7911/£. Solving Equation 6.1 for the value of the put leads to Put$/£=Call$/£−(F$/£T −K$/£)/(1+i$)1/2=$0.0717/£−($1.7911/
£−$1.7500/£)/(1.0408)1/2 =$0.0314/£.
The Intrinsic Value of an Option
The intrinsic valueof an option is the value of the option if it is exercised today.
Consider the currency call and put options presented here.
Intrinsic value is the value of immediate exercise.
Call option value when exercised Put option value when exercised
CallTd/f PutTd/f
STd/f Kd/f
STd/f Kd/f
If an option is out-of-the-money, its intrinsic value is zero. If an option is in-the- money, its intrinsic value is equal to the difference between the exercise price and the value of the underlying asset. Option values at exercise on the spot exchange rate are determined as follows:
Call option value when exercised=Max[(Sd/ft −Kd/f),0]
Put option value when exercised=Max[(Kd/f−Sd/ft ),0]
These are the intrinsic values of the call and put options, respectively. Every graph that has appeared up to this point in the chapter has been a graph of intrinsic value.
As the underlying asset value moves away from the exercise price, option values follow a one-way path. Currency call option holders gain when the underlying exchange rate rises above the exercise price, but cannot lose more than the option premium as the underlying exchange rate falls below the exercise price. Put option holders gain as the underlying exchange rate falls below the exercise price, but lose, at most, the option premium as the exchange rate rises. It is this asymmetry that gives options their unique role as a disaster hedge.
The Time Value of an Option
Thetime valueof an option is the option’s market value minus its intrinsic value. Two important variables in determining the time value of an option are the volatility in the underlying exchange rate and the time to expiration. Volatility in the underlying (spot or futures) exchange rate determines how far in- or out-of-the-money an option is likely to expire. Time to expiration has an effect that is similar to volatility, in that more time until expiration results in more variable outcomes at expiration. Here’s the general rule for American options.
Time value is market value less intrinsic value.
That is, American option values are greater if volatility in the underlying asset (e.g., the exchange rate for a currency option) increases or if the time to expiration is longer.
As time to expiration or volatility increases, the values of American call and put options increase.
Consider the payoffs to a dollar call and a dollar put option, each with an exercise price of K¥/$=¥100/$. Suppose the spot rate at expiration will be either ¥90.484/$ or ¥110.517/$ with equal probability.3Payoffs to these options are as follows:
Closing spot exchange rate S¥/$T
¥90.484/$ ¥110.517/$
Value of a call at expiration ¥0/$ ¥10.517/$
Value of a put at expiration ¥9.516/$ ¥0/$
Suppose the volatility of the spot rate increases such that the spot rate at expiration can be as low as ¥81.873/$ or as high as ¥122.140/$.4The values of a dollar call and a dollar put at these spot rates and with an exercise price of ¥100/$ are as follows:
Closing spot exchange rate S¥T/$
¥81.873/$ ¥122.140/$
Value of a call at expiration ¥0/$ ¥22.140/$
Value of a put at expiration ¥18.127/$ ¥0/$
Because option holders continue to gain on one side of the exercise price but do not suffer continued losses on the other side, options become more valuable as the end- of-period exchange rate distribution becomes more dispersed. For this reason, prior to expiration there are more good things than bad that can happen to option value.5 The figure below illustrates how at-the-money call options gain from an increase in volatility. An at-the-money call gains if the spot rate closes above the exercise price, but does not lose if the spot rate closes farther below the exercise price. As volatility increases as in the distribution at the right, more good things can happen for the call as it can close even farther in-the-money.
Exchange rate volatility and at-the-money call option value
CallTd/f CallTd/f
STd/f STd/f
STd/f STd/f
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.