APPENDIX 4A: CONTINUOUS COMPOUNDING
6.5 EXCHANGE RATE VOLATILITY REVISITED (ADVANCED)
buying futures on the Nikkei index in the hopes of a recovery that, to Barings’ regret, never occurred.
measured over T periods. This impliesσT=(√
T)σ, so standard deviation increases with the square root of time. Equation 6.3 identifies the manner in which volatility σ and time to expiration T interact to increase the variability of the end-of-period return distributionσTin an i.i.d. normal return series.
Volatility can be estimated in several ways. The two most prominent methods are historical volatility and implied volatility. Historical volatility is a backward- looking measure that captures observed variations over the recent past in the hope that history will repeat itself. Implied volatility is a forward-looking measure that uses current option prices to estimate volatility in the underlying asset. Because it is based on current prices, implied volatility reflects the expectations of participants in the options markets.
Historical Volatility
Historical volatilityis the actual volatility of an exchange rate realized over some historical period. For changes in currency values, historical volatility can be estimated by calculating the observed standard deviation of continuously compounded changes stsampled over T periods.
σ=√
[(1/T)!t(st−µ)2] (6.4)
Historical volatility is realized over some historical period.
As an example, suppose the standard deviation of continuously compounded daily changes in the yen/dollar spot rate is estimated from Equation 6.4 to be σ= 0.00645=0.645 percent per trading day over the 252 business days in a particular calendar year. Assuming zero volatility on nontrading days, such as weekends and holidays, the annual standard deviation of continuously compounded changes in the exchange rate isσ=(√
T)σT=(√
252)(0.00645)=0.1024, or 10.24 percent per year. If instantaneous changes in exchange rates are normally distributed, plus or minus one standard deviation results in plus or minus 10.24 percent per year in continuously compounded returns.
Suppose the spot rate is ¥130/$, as in Figure 6.6. Plus two standard deviations of 10.24 percent in continuously compounded returns is 2σ=(2)(0.1024)=0.2048, or 20.48 percent. The periodic rate of change over the period is s¥1/$ =e(2σ√t)−1= e(+0.2048)−1=22.73 percent. Two standard deviations above the ¥130/$ spot rate is thus S¥1/$=S¥0/$e(+0.2048)=(¥130/$)(1.2273)=¥159.55/$. In periodic returns, this is a 22.73 percent increase in the spot rate.
Similarly, two standard deviations below the spot rate is S¥1/$=S¥0/$e(−0.2048)= (¥130/$)(0.8148)=¥105.93/$. This is equivalent to s¥/$1 =(0.8148−1)= −0.1852, or an 18.52 percent decrease in the spot rate. About 95 percent of the normal distribution falls within two standard deviations of the mean, so there is a 95 percent chance that the actual spot rate in one year will fall between ¥105.93/$ and
¥159.55/$.
As a check, let’s back out continuously compounded changes implied by a change in the spot rate from ¥130/$ to either ¥105.93/$ or ¥159.55/$. If the spot rate moves
¥130/$
¥159.55/$
¥105.93/$
Exchange rate distribution –2σ = –0.2048,
or e(–0.2048) – 1 = –18.52%
+2σ = 0.2048, or e(+0.2048) – 1 = –22.73%
FIGURE 6.6 Exchange Rate Volatility.
from ¥130/$ to ¥159.55/$, the percentage change is s¥/$1 =(¥159.55/$)/(¥130/$)− 1=0.2273, or 22.73 percent. In continuously compounded returns, this is equal to s¥/$=ln(1.2273)=0.2048, or 20.48 percent. Conversely, a move from ¥130/$
to ¥105.93/$ results in a continuously compounded return of s¥1/$=ln(S¥1/$/S¥0/$)= ln((¥105.93/$)/(¥130/$))= −0.2048, or−20.48 percent.
Another useful fact is that volatility measured in continuously compounded returns does not depend on the currency of reference. To verify this, let’s perform the same calculations using dollar-per-yen quotes. The yen-per-dollar exchange rates convert into dollar-per-yen spot rates according to S$/¥ =1/S¥/$.
1/S¥/$=1/(¥159.55/$)=$0.0062676/¥=S$/¥
1/S¥/$=1/(¥130.00/$)=$0.0076923/¥=S$/¥
1/S¥/$=1/(¥105.93/$)=$0.0094402/¥=S$/¥
A 22.73 percent dollar appreciation from ¥130/$ to ¥159.55/$ is equivalent to a 19.52 percent yen depreciation from ($0.0076923/¥) to ($0.0062676/¥). A 19.52 percent dollar depreciation from ¥130.00/$ to ¥105.93/$ is the same as a 22.73 percent yen appreciation from ($0.0076923/¥) to ($0.0094402/¥). Alternatively, in dollars per yen
s$/¥1 =ln(S$/¥1 /S$/¥0 )=ln[($0.0062676/¥)/($0.0076923/¥)]=ln(0.8148)
= −0.2048 and
s$1/¥ =ln(S$1/¥/S$0/¥)=ln[($0.0094402/¥)/($0.0076923/¥)]=ln(1.2273)
= +0.2048
Sure enough, these represent±20.48 percent changes in continuously compounded returns.
A variant of historical volatility calledrealized volatilityis becoming increasingly popular.9Realized volatility is formed by cumulating squared returns measured over short (e.g., 15-minute) intervals throughout the day. Realized volatility estimates are relatively good predictors of future volatility over short forecasting horizons because the average of recent high-frequency squared returns closely approximates true variance at a given point in time. They are less useful in predicting volatilities over longer forecast horizons. For long-horizon forecasts of volatility, the market-based
‘‘implied volatility’’ estimate described in the next section has proven useful.
Implied Volatility
There are six determinants of a currency option value: (1) the spot rate Sd/f, (2) the exercise price Kd/f, (3) the domestic risk-free rate id, (4) the foreign risk-free rate if, (5) time to expiration T, and (6) the volatility of the underlying assetσ. For publicly traded options, the values of five of the six determinants, as well as the option value itself, are published in the financial press. The only unobservable determinant is the volatility of the underlying asset.
Implied volatility is implied by an option price.
Suppose you know the equation specifying how option values are related to these six variables. Then, given five of the six inputs and the option price, the value of the single unknown determinant (exchange rate volatility) can be found by trial and error. Volatility estimated in this way is called implied volatility, because it is implied by the option price and the other option value determinants.
As an example, consider a ‘‘December A$ 73 call’’ trading on the PSE. Suppose the following values are known:
Value of call option Call$/A$ = $0.0102/A$
Price of underlying asset S$/A$ = $0.7020/A$
Exercise price K$/A$ = $0.7300/A$
Domestic risk-free rate i$ = 4% per year (continuously compounded) Foreign risk-free rate iA$ = 0% per year (continuously compounded) Time to expiration T = 21/2months
Volatility of the spot rate σ = ?
Solving the currency option pricing model (OPM) from the appendix to the chapter for the standard deviation of the spot rate yields an implied volatility of 0.148, or 14.8 percent per year. When combined with the five other inputs, this is the only standard deviation that results in an option value of $0.0102/A$.
A Cautionary Note on Implied Volatilities
Beware of prices in thinly traded markets.Let’s look at another quote, a ‘‘December A$ 63 call’’ on the PSE. Suppose the following prices are quoted inThe Wall Street Journal:
Value of call option Call$/A$ = $0.0710/A$
Price of underlying asset S$/A$ = $0.7020/A$
Exercise price K$/A$ = $0.6300/A$
Domestic risk-free rate i$ = 4% per year (continuously compounded) Foreign risk-free rate iA$ = 0% per year (continuously compounded) Time to expiration T = 21/2months
Volatility of the spot rate σ = ?
Both options are based on the December U.S.-per-Australian spot rate, so the implied volatility of this option should be the same as that of the previous option. However, trying to find an implied volatility for the $0.63/A$ call based on these prices is futile. There is no value for volatility that yields a call price of $0.0710/A$. What’s wrong?
The Wall Street Journal reports prices from the last trade of the previous day.
The call option’s time of last trade may or may not correspond to the time of last trade of the exchange rate underlying the option. Suppose the last time this option traded on the PSE was at noon, at which time the spot rate was $0.6900/A$. The implied volatility of the ‘‘December A$ 63 call’’ at that instant is determined from the following:
Value of call option Call$/A$ = $0.0666/A$
Price of underlying asset S$/A$ = $0.6900/A$
Exercise price K$/A$ = $0.6300/A$
Domestic risk-free rate i$ = 4% per year (continuously compounded) Foreign risk-free rate iA$ = 0% per year (continuously compounded) Time to expiration T = 21/2months
Volatility of the spot rate σ = ?
The implied volatility for this ‘‘December A$ 63 call’’ is 14.8 percent per year, the same as in the ‘‘December A$ 73 call.’’ There was no solution to the previous example because the end-of-day exchange rate was used to price an option that
last traded at noon. This example suggests a general result: Beware of prices in thinly traded markets. In this example, we were comparing apples and oranges. Or, more precisely, we were comparing apples (or options) at two different times of the growing season.
Volatility and Probability of Exercise
Let’s go back to the example of a December A$ 73 call on the PSE.
Value of call option Call$/A$ = $0.0102/A$
Price of underlying asset S$/A$ = $0.7020/A$
Exercise price K$/A$ = $0.7300/A$
Domestic risk-free rate i$ = 4% per year (continuously compounded) Foreign risk-free rate iA$ = 0% per year (continuously compounded) Time to expiration T = 21/2months
Volatility of the spot rate σ = 14.8%
What is the probability of this option being in-the-money on the expiration date in December? The spot rate would have to go from S$/A$0 =$0.7020/A$
to S$/A$T =$0.73/A$ for a continuously compounded change of s$/A$T =ln[($0.73/ A$)/($0.7020/A$)]=0.039, or 3.9 percent. The standard deviation over 2.5 months is σT =(√
T)σ=(2.5/12)(1/2)(0.148)=0.0676, or 6.76 percent per 2.5 months. The continuously compounded change in the spot rate must bes$T/A$/σT= (0.039)/(0.067)=0.58, or 58 percent of one standard deviation above the current spot rate. The probability mass of the normal distribution above 0.58σ is about 0.40. Thus, there is about a 40 percent chance of this option expiring in-the-money.
Time-Varying Volatility
Recall that empirical investigations of exchange rate behavior reject the simple random walk model. Instead, researchers have found that exchange rates can be described as having generalized autoregressive conditional heteroskedasticity (GARCH).10
■ At each point in time, instantaneous returns are normally distributed.
■ The instantaneous variance at each point in time depends on whether exchange rate changes in the recent past have been large or small.
The fact that foreign exchange volatility is not a constant means that OPMs that assume stationary price changes (such as the binomial and Black-Scholes models) are misspecified. Implied volatility is actually a time-weighted average of the instantaneous variances prevailing over the life of the option. For this reason, implied volatilities option values may not represent the instantaneous volatility at any point in time during the life of the option.
As an example, implied volatilities can be as large as 20 percent per year during turbulent periods in the foreign exchange markets. A 20 percent implied volatility on a 5-month currency option might represent a 40 percent standard deviation over the first month and a 10 percent standard deviation over the remaining four months.11Because of time-varying volatility, foreign exchange volatilities estimated from OPMs are at best imprecise estimates of current and expected future exchange rate volatility.