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Forward Volatility Agreement Chapter 11 Currency by ppp12752

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									Chapter 11: Currency Options and Volatility

A currency option contract is an agreement between two parties that gives the purchaser the right, but
not the obligation, to exchange a given amount of one currency for another, at a specified rate, on an
agreed date in the future. Options provide many benefits. Speculators purchase options because they are
an efficient way to express a view on the underlying asset. Options also provide insurance because
effectively they transfer risk from one party to another. Intuitively, currency options insure the purchaser
against adverse exchange rate movements. For example, many international business firms- use options to
hedge the value of their overseas revenues. The up-front, sunk cost one pays to purchase an options
contract is analogous to an insurance premium.

This chapter is organised in three major sections: The first section reviews basic concepts in currency
options including option valuation, limitations on option valuation models, and common option trading
strategies. If you are already familiar with currency options, this material can be skipped without loss of
continuity. Section 3 focuses on establishing relative value in options markets. Given that options are a
derivative instrument - meaning they derive their value from an underlying security - options themselves
have value relative to other options. We discuss how volatility is defined, introduce volatility cones,
establish seasonals in currency volatility, study correlation and volatility, and conclude with a model of the
term structure of volatility. The final major section of this chapter examines the fundamentals of currency
volatility. We develop a mean-reversion model of implied volatility and describe both short-run and
long-run influences on currency volatility.


Elementary Terminology: A call option on a particular currency gives the holder the right to buy that
currency. A put option gives the holder the right to sell the currency. The seller, or writer of the option,
receives a payment, referred to as the option premium, that then obligates him to sell the exchange
currency at the pre-specified price, known as the strike price, if the option purchaser chooses to
exercise his right to buy or sell the currency. The holder will only decide to exchange currencies if the
strike price is a more favourable rate than can be obtained in the spot market at expiration. The majority of
currency options traded over-the-counter (OTC) are European exercise s tyle; that is, they can only be
exercised on the expiration date. The altemative is an American style option, that can be exercised at
any time during its life. The date on which the contract ends is called the expiration date.

The holder of a call option on a currency will only exercise the option if the underlying currency is trading
in the market at a higher price than the strike price of the option. The call option gives the right to buy, so
in exercising it the holder buys currency at the strike price and can then sell it in the market at a higher
price. Similarly, the holder of a put option on a currency will only exercise the option if the spot currency is
trading in the market at a lower price than the strike price. The put option gives the right to sell, so in
exercising it the holder sells currency at the strike price and can then buy it in the market at a lower price.
From the point of view of the option holder, the negative profit and loss represent the premium that is paid
for the option. Thus, the premium is the maximum loss that can result from purchasing an option.
The nearby exhibit contains several important points. The point at which the graph crosses the horizontal
axis is known as the break-even point. Also note that the point at which the graph kinks represents the
strike price of the option. Referring to the figure, the option would not be exercised if the spot currency
were below the strike price. The option has value at expiration once the spot currency moves above strike
and so the option is exercised if the spot price is at or above strike (in this case, exercising the option and
selling the currency back in the market will result in a positive cash flow). The value the option has at
expiration (the difference between spot and strike) is known as the parity value of the option. An option
which has parity is known as an in-the-money-option.

Option Valuation: An option's value is made up of two components: its intrinsic value and its time value.
'Me intrinsic value of an option is the greater of zero or the value that it would have if it were
exercised immediately. From this definition, it is clear that the only options that have non-zero intrinsic
value are in-the-money options. The time value of an option is the difference between the total premium
and the intrinsic value. For at-the-money or out-of-the-money options, the time value equals the total
premium of the option (since the intrinsic value is zero). In this subsection, we describe the fundamental
variables that determine the value of an option and explain how each of them works.

There are six fundamental variables that determine the price of an option on an exchange rate: the current
(spot) price, the strike price, the time to expiration, forward points, the domestic currency interest rate, and
the implied volatility. The most common currency options pricing model, the Garman-Kohlhagen option
model is a variation on the famous Black-Scholes (B-S) model. These inputs establish the value of a
currency option as the combination of its intrinsic value and its time value:

Call = S * N(d1) * eRft – E * N(d2) * eRdt,


d1 = ln (S / E) + (Rd – Rf + ζ2 / 2 * t

d2 = d1 = ζ * √ t,

t = time to expiration (measured in percent of a year),
E = option exercise price,
Rd = risk free domestic interest rate,
Rf = risk free foreign interest rate,
ln = natural log function, .
N = cumulative normal density function,
ζ = variance of the rate of return on the underlying exchange rate,
S = currency spot rate,
and e = exponential function.

The option premium changes as each of these variables change: As the market price of the exchange
rate increases (S), the value of a call on this exchange rate will increase and the value of a put will

The strike price (E) is the price at which the currency is exchanged if the option is exercised. For a call,
the higher the strike is set, the lower the value of the call since it will be increasingly less likely that the call
will end up in the money - or if it does, the payoff is likely to be smaller. For a put, the lower the strike, the
lower the value of the option.

For either a call or a put, the longer the time to expiration, the greater the value of the option. Increasing
the time period (t) must necessarily raise the expected range of price movements. The asymmetric
nature of an option (the fact that the holder's downside risk is limited to the loss of premium paid) means
there must be greater value to an option that has a wider possible range of upside outcomes.

At expiration, options are valued around the forward price of the underlying exchange rate. In any
currency option pricing model, a necessary input is the difference between the foreign (Rf) and domestic
(Rd) interest rate (for the tenor of the option) measured in basis points, or the forward points, to account
for the forward exchange rate. Where the carry is positive, the forward price of the exchange rate is
lower than the spot price. Thus, a call purchased at-the-money spot (i.e., where the strike price equals the
spot price) is actually expected to be out-of-the-money by the time the option expires. Similarly, a put
purchased at-the-money spot is expected to be in-the-money at expiration. All else being equal, if the
forward points increase, the positive carry will decrease; hence, the forward price of the exchange rate
will be higher. Thus, higher forward points lead to an increase in the call value and a decrease in put value.

If an investor is long a call option, he has the same upside risk profile as one invested in the underlying
currency, but without the benefits of the domestic return (Rd). This means that in comparing at-the-money
options, a higher domestic interest rate will lead to a lower call price, since the owner of the call is missing
out on a larger payment. The reverse is true of puts: Since owning a put has the risk profile of being short
the underlying currency, the put buyer will pay more for a put on a currency with a high domestic interest
rate, since he can use the put premium to maintain a short position without having to pay the domestic

Volatility (ζ) is one measure of a security's risk. This aspect of risk is defined by the dispersion (or
standard deviation; variance [ζ2] is the square of volatility) of possible prices that the exchange rate
can achieve over a certain time period. The higher the volatility on an exchange rate, the greater the range
of values over which the currency pair is likely to move. An option buyer will be prepared to pay more to
participate in larger expected swings in an exchange rate, while limiting his risk to the premium amount
that he pays. Similarly, an option writer will expect to be compensated more highly (in the form of larger
premium) for selling an option when the currency pair is expected to undergo large price moves.
Limitations on option Valuation Models: Underlying the Black-Scholes options valuation formula are a
number of assumptions. To the extent that the world deviates from these assumptions, the B-S model will
be biased in certain, often predictable ways. This subsection explores the pricing biases introduced by two
major simplifying assumptions of the B-S model: constant volatility and a normal (or lognormal) distribution
to price changes. B-S may not be a perfect model of options valuation, but as long as market participants
understand these and other problems with the model, then it is useful in practice.

The B-S model assumes that an exchange rate's volatility is known and never changes. However, causal
observation reveals exchange rate volatility is not constant. This fact has a major impact on the values of
certain options, especially those options that are away-from-the-money, because the dynamics of the
volatility process rapidly change the probability that a given out-of-the-money option can reach the
exercise price. The B-S model consistently underestimates the value of an option to the extent that
volatility is stochastic rather than constant as assumed.

A second major assumption of the B-S model is that exchange rate returns are normally (or lognormally)
distributed with a variance proportional to the length of time over which the asset trades. However, a
                                                      number of academic studies show that exchange
                                                      rate movements are neither normally nor
                                                      lognormally distributed. Empirically, exchange rates
                                                      experience big moves with a frequency that
                                                      exceeds what one would expect theoretically. This
                                                      could be a result of herd-like behaviour among
                                                      currency speculators or it could be due to the
                                                      intervention of central banks. Once again, this
                                                      problem with the distributional assumption of the
                                                      B-S model means that it generally underestimates
                                                      currency option values because the likelihood of
                                                      having an extreme price movement is greater than
                                                      the model expects.

Common Option Trading Strate gies: Finally in this subsection, we consider the implications of holding
long, short, or a combination of options positions. As a starting point, we use the following conventions: a
bullish position is one that is profitable when the market rallies, and a bearish position makes money
when the market sells off. Implicitly for the purposes of this subsection, we are assuming that the options
holder is long a call on the currency in the numerator and short a put on the currency in the dominator for
a given currency pair. The nearby table illustrates these assumptions:

                                            Long(Purchase)                         Short(Write)
Call                                            Bullish                              Bearish
Put                                            Bearish                               Bullish

Owning a call is similar to owning the underlying currency, since it gives the owner the right to buy that
currency at some predetermined exchange rate. Clearly, as the market rallies, the right to buy at a fixed
price must increase in value. Similarly, if you have the right to sell at a pre-determined exchange rate (as
with a put), then a market sell-off will increase the value of this right.
A straddle is the combination of a call and a put struck at the same level (typically, but not necessarily,
at-the-money), with the same expiration date. Along straddle is the purchase of both options; a short
straddle is the sale of both options. A straddle is neither bullish nor bearish if struck at-the-money. It is
market-neutral in terms of direction. The break-even prices on a straddle are equal to the strike plus or
minus the combined premiums.

In the short-term, a straddle is a pure volatility position. A straddle buyer believes that the market is
undervaluing volatility - i.e., that the price of the underlying security will tend to move, either upward or
downward, further than is priced into the premium of the options. A straddle seller believes the reverse:
that the underlying security's price will tend to move less than the market expectation as priced into the

A strangle is similar to a straddle in that it is
the purchase (or sale) of both a call and a put,
with the same expiration date. However, the call
and put have different strikes, which are
typically out-of-the-money by the same amount.
For example, a 1-unit out-of-the-money
strangle has the strike on the call I -unit higher
than spot, and the strike on the put 1-unit lower
than spot. The net effect of this 'difference is
that the overall premium paid will be lower
(since out-of-the-money options are always
cheaper than at-the- money options), but there
will be a broader range of prices on the
underlying currency pair for which the value of the position does not change.

Along call spread is the combination of a long call at one strike and a short call with the same expiration
at a higher strike. This is a conservatively bullish strategy, with limited upside. If the spot price rallies
above the strike of the long call, the strategy continues to increase in value until the spot price is higher
than the strike on the short call. A long put spread is a conservatively bearish strategy. Like the call
spread, the upside is limited to the difference between the two strikes, less the net premium paid.

The motivation for a spread strategy versus a single option strategy is that the spread is cheaper. Some of
the cost of the option purchased is defrayed by the (lower) cost of the option sold. In return for the smaller
net premium, the investor effectively caps his maximum potential profit. It is worth noting that the payoff
diagrams on a long call spread and a short put spread are virtually identical. Similarly, the payoff diagrams
on a short call spread and a long put spread are virtually identical.

A covered call write is also known as a buy-write. If an investor is long the underlying asset and wants
to earn some income on the position, he can sell calls against the currency. If the currency appreciates and
the call is exercised, the investor already has the currency to sell to the call buyer. If the currency drops in
value, the investor has reduced his losses on the underlying currency by taking in call premium. A covered
call has the same risk profile as a short put; it is an income-generating bearish strategy with a limited
upside and unlimited downside.
An investor may know in advance that he needs to buy a currency at a specified price (below the present
level of the market). If he sells a put struck at his target price, he will receive income (the put premium),
and if the currency trades down through the strike, he will be obligated to purchase the currency from the
put buyer at the strike price.

The purchase of a call and the sale of a put, to the same expiration date with the same strike, has a payoff
equivalent to that of the synthetic purchase of the underlying currency at the strike price of the option.

That is to say, the risk exposure is identical to the risk exposure of buying the currency at the strike price.
Similarly, the risk exposure of being short a security at a specific price can be replicated by buying a put
and selling a call at the same strike price.

If the risk exposure on a synthetic long is identical to the risk exposure of buying the currency outright,
why not just buy the underlying? There are many reasons why an investor may not wish to buy the
currency outright. For instance, he may have balance sheet constraints that limit his holdings in certain
currencies (remember that options are off-balance-sheet items and do not affect capital asset ratios).


An option is a derivative instrument in that it derives its value from a movement in the underlying security.
When comparing two options, one option's value can be deduced from or defined relative to another
option's value. This section introduces several techniques that investors can use to establish relative value
between options.

Deflning Volatility: Volatility is a central element in the valuation of options because options securitise
this risk. Thus, how volatility is measured or estimated is an important issue. Three common methods for
measuring volatility are historical, implied, and time series volatility.

Historical volatility is usually calculated by computing the standard deviation of daily exchange rate
returns. The disadvantage of this approach is that it is necessarily backward looking when what investors
really need is an estimate of future volatility.

Implied volatility is computed by using an option price and solving the B-S model for the unknown
volatility parameter. Like any other market, implied volatility is a function of the underlying supply and
demand for options with a given set of characteristics. Intuitively, it represents the market's best estimate
of exchange rate volatility over the term of the option.

Some market participants also estimate and forecast the underlying volatility of an exchange rate using
time series econometric techniques. The critical assumption here is that volatility is a stationary time series
drawn from a stable distribution so that its history provides some insight into its future. However, the
distribution of exchange rates returns has large, unpredictable jumps that diminish the usefulness of this
class of models.

Volatility Cones: Comparing historical and implied volatility is one simple technique that many market
participants use to gauge an option's relative value.
To compute a volatility cone, one calculates the
average, maximum, and minimum historical volatility
over a given horizon and graphs the information. As
the sample period lengthens, we observe that the
distance between the maximum and minimum
historical volatilities declines. Hence, this pattern is
called a "cone."

Market participants can compare their estimate of
volatility with the implied volatility in the market and
the historical volatility. If implied volatility
significantly diverges from the historical average,
then one of two conclusions are possible. First, the
effect is transitory and market implied volatilities will
return to historical averages. Second, something
structural has changed in the marketplace so market
participants demand more or less insurance than
usual. An example of this latter case is the drive
toward EMU, Market participants quickly realised
that if a number of currencies were to be combined,
then volatility of those exchange rates will tend to
Seasonals in Currency Volatility: Exchange rate volatility is sometimes subject to seasonal volatility
patterns because of variations in corporate or other demand for foreign exchange. For example,
multinational corporations may repatriate their overseas revenues shortly before financial year-end for
accounting reasons. Unless these flows are balanced by speculative supply, market volatility can rise
during these periods. Traders should consider these patterns when estimating the future exchange rate
volatility and assessing relative volatility across markets subject to these patterns.

                                                         To better understand seasonal currency volatility
                                                         patterns, we computed the average, maximum,
                                                         and minimum monthly volatility of US$/DM and
                                                         US$/JPY for historical prices going back to 1975
                                                         and for options prices back to 1985. We found that
                                                         both US$/DM and US$/JPY implied volatility
                                                         tends to rise twice a year-around March and April
                                                         and between September and October. These
                                                         months are associated with the fiscal year-end and
                                                         half year-end of Japanese corporations. These are
                                                         also periods when there are the bi-annual and
                                                         annual IMF, and since 1985, G7 meetings.
                                                         Interestingly enough, we did notfind any monthly
                                                         seasonal pattern in US$/JPY returns over these
                                                         periods. This exercise demonstrates that exchange
                                                         rate volatility is not homogenous over the calendar.

                                                         Correlation and Volatility: Another common
                                                         technique to establish relative value in currency
                                                         options markets is to look at the relationship
                                                         between correlation and volatility. When looking at
                                                         three currency pairs such as the US$, DM, and
                                                         JPY, the following statistical identity holds:

ζJPY/DM = √ (ζ2 DM/US$ + ζ2 JPY/US$ - 2ρ JPY/DM * ζ DM/US$ * ζ JPY/US$)


ζJPY/DM = the volatility or standard deviation of JPY/DM,
ζUS$/DM = the variance of US$/DM,
ζDUS$/JPY = the variance of US$/JPY,
and ρ JPY/DM = the correlation US$/DM and US$/JPY.

Several implications about relative value can be deduced from this identity. First, if either US$/DM or
US$/JPY volatility rises, then JPY/DM volatility will also rise (holding the correlation term constant). In
addition, if the correlation between US$/DM and US$/JPY returns rises, then JPY/DM volatility will
decline. The US$ volatilities are additive in the equation while the correlation term is negative. Thus, an
investor trading the relative spread between two options markets must expect that volatility will rise or that
correlation will diminish.
                                                         The variance-correlation identity is also useful for
                                                         taking positions on inter-market correlations. One
                                                         can easily use market prices to solve for the implied
                                                         correlation and take positions on the basis that the
                                                         actual correlation will be different than that
                                                         estimate. Academic analysis of the implied
                                                         correlation estimate in currency options reveals that
                                                         it is a better estimate of future correlation than
                                                         historical correlation, a moving average of past
                                                         correlation, or a statistical time series correlation

                                                         Modelling the Term Structure of Volatility:
Options at different dates or tenors reveal a term structure of implied volatilities that is a useful tool in
relative value analysis. Long-dated options are priced relative to the value of short-dated options. The
effect is similar to that observed in fixed income yield curves. Long-dated implied volatilities do not fully
respond to transitory movements in short-dated implied volatility because of the mean-reverting nature of
the series. A shock to short-dated volatility is likely to dissipate over a longer time interval as the market
reverts to its "normal" or average level. Finally, greater volatility of short-dated volatility reduces the value
of short-dated volatility as a benchmark for valuing long-dated volatility, so it tends to lower long-dated
implied volatilities.

Using this intuition, we model the term structure of implied currency volatility as a function of two factors:
short-dated volatility and the volatility of short-dated volatility. We estimate the following statistical model:

ζt1 = α + β1ζt2 + β2(ζt2)2 + ε,


ζt1 = the volatility of a long-dated option,
ζt2 = the volatility of a short-dated option,
and α , β1, and β2 = coefficients or parameters of the model.

In words, long-dated implied volatility is a function of a constant, short-dated volatility, the volatility of
short-dated volatility and an error term. Deviations
from the expected values from this econometric
model suggest a useful approach to determining
whether implied volatilities in a specific portion of
the yield curve are expensive or cheap.

As an illustration, we present the results for our
model for US$/JPY options. A 100 basis point (bp)
increase in one-month at-the-money-forward
(ATMF) implied volatility US$/JPY translated on
average into a 49 bp move in comparable one-year
implied volatility. As we expected, an increase in
the volatility of short-dated volatility tends to
reduce long-dated volatility. Both of these terms are statistically significant and the model explains 53% of
the variation in long-dated volatility.

Box 1: Modeling the Term Structure of US$/JPY Implied Volatility.

As described in the text, we estimated an equation for one-year ATMF US$/JPY implied volatility as a
function of one-month implied volatility and the volatility of one-month implied volatility. Below are the
results from this exercise:

              Dependent Variable: One-Year US$/JPY Implied Volatility
 Independent Variables:    Coefficient                Standard Error
 Constant                  6.589                      0.502
 One-Month Volatility      0.494                      0.051
 (One-Month Volatility)    -0.012                     0.004

All of the coefficients are statistically significant. The standard error of estimate was 1.05 and Rbar2 was
0.53. The sample has 92 monthly observations between 1990:1 and 1997:7.

As expected, we found that long-dated volatility does not fully adjust to changes in short-dated volatility. In
addition, we observed a negative relationship between the volatility of short-dated volatility and long-dated


Central to trading options is some understanding of the fundamental forces that affect market volatility. In
this section, we highlight several of our theories concerning the nature of currency volatility. We begin by
describing the volatility process as being mean-reverting. While this would suggest that it is advantageous
to sell volatility when it reaches an extreme value, we find that this strategy is not always effective. Next,
we discuss some of the important short-term economic and political influences on currency volatility. We
conclude with a discussion of several structural factors that influence long-run trends in exchange rate

Mean Reversion: Volatility in financial markets tends to be a mean-reverting process. This implies that
the further away volatility gets from its long-term average or "normal' level, the greater the likelihood that
it will move back towards its mean. This property violates the constant volatility assumption of most major
options valuation models, including B-S. Thus, an understanding of the mean reverting nature of the
volatility process enables someone taking an options position to recover the market's implicit beliefs about
the dynamics of volatility as the option moves through time.

Currency volatility is mean-reverting for two reasons: First, the economic factors generating financial
market volatility are also mean-reverting. If policy makers mismanage their economy and the risk premium
on their nation's assets rises, then the volatility of their currency will also naturally rise.

The fact that there is a positive relationship between relative government bond yield spreads and currency
volatility supports this proposition. For example, when the US Federal Reserve began raising short-term
interest rates in 1994, investors were uncertain about what the peak in short rates would be, so the US$
traded with above-normal volatility. When this uncertainty was resolved in early 1995 and US economic
activity began to slow, US$ volatility reverted back to its normal level.

Second, volatility may be mean-reverting because it cannot be explosive. Unless an economy is
experiencing hyperinflation, the spot value of a currency is bounded in terms of how far it can deviate
from its equilibrium value. If the price of the underlying asset is bounded, then a derivative of the asset is
also bounded and implied volatilities cannot deviate too far from their central tendency.

Case Study: Buying Volatility at Extremes If currency volatility is a mean-reverting process, then
perhaps buying or selling it at extreme levels should be a profitable trading strategy. We examine this
proposition using historical data on US$/JPY In every case in which six-month US$/JPY
at-the-money-forward (ATMF) implied volatility has been above 12.67% in the past five years (one
standard deviation above the long-run mean), it has returned to cross the long-run mean implied volatility at
least once over the course of the next three months. In this subsection, we investigate whether selling
volatility when it reaches an extreme peak is a profitable trading strategy.

One experiment we conducted was to examine the profits generated from selling an ATMF six-month
straddle and buying it back three months later. Effectively an investor using this strategy would be short
volatility, hoping to purchases it back for less in three months time. In the 12 instances in which this rule
was triggered since 1991, four cases were profitable and eight were unprofitable. This strategy's average
return was -0.47%; its high positive return was 2.95% and the largest drawdown was -4.53%. Hence, the
strategy is not successful. There is also insufficient return in the reverse strategy, being long volatility
when volatility is high.

One reason that this strategy failed to generate a positive expected return is that an options position has
both a spot directional and a volatility component. While volatility dropped, as it should in a mean-reverting
process, the move in the spot price in the intervening period rendered the position unprofitable. In addition,
implied volatilities have to drop faster than the forward volatility curve anticipates for the position to make
money. Of course, this trial used a stylised currency pair, exit period, and strike price, etc., so our ability to
generalise the results are limited.

Short-Term Influences on Currency Volatility: A number of economic and political factors can affect
the short-term dynamics of currency volatility.

First, exchange rate volatility tends to rise when two nation's economic policies diverge. This phenomenon
is usually associated with two countries being at different points in the business cycle. Economic policy
may diverge because one nation's central bank may be expected to pursue a very restrictive monetary
policy while another is anticipated to maintain an accommodative monetary policy. The impact of these
divergent economic policies is generally felt in currency markets as exchange rates adjustment to these
expectations and corporate and/or investor uncertainty raises hedging demand.

The divergence between British and German economic policies in 1996 provides an excellent example
how such a move can affect volatility. The UK economy began to accelerate in the middle of 1996 while
the German economy was mired in a period of sluggishness. Between July 1996 and July 1997, the GBP
appreciated by over 30% against the DM. At the same time, one-month ATMF DM/GBP implied volatility
more than doubled from 5% to 11%.
Exchange rate volatility is also sensitive to turning points in monetary policy. When investors believe that a
shift in economic policy is near, they seek to hedge the currency exposure of their foreign holdings with
options. This demand tends to drive up options implied volatilities.

When the US Federal Reserve began raising short-term interest rates in February 1994, US$/JPY implied
volatilities moved sharply higher. One-month ATMF US$/JPY implied volatilities rose from 9% in January
1994 to 16% by mid-February. Of course at the time, US and Japanese political officials were also battling
over trade issues-a factor that contributed to a sharp down-move in the US$/JPY exchange rate.

A third factor that induces short-term exchange rate volatility is a change in either credit or inflation risk
premia. Options are an investment vehicle that enables investors to transfer risk. When investors demand
a higher risk premium on a nation's assets, this development tends to be reflected in a greater price for
currency risk as well.

In the case of Italy, this mechanism has recently worked in reverse. Since 1996, the Italian government
has committed itself to meeting the rigid Maastricht inflation and budget targets. Success at bringing down
inflation and reducing its budget deficit has resulted in a virtuous circle of lower interest rates, lower risk
premia on Italian assets, an appreciating currency, and falling long-dated currency implied volatility. Bond
spreads best reflect these risk premia. Italian BTP versus German bund spreads have fallen over 350 bps
in the past 18 months. One-year ATMF ITL/DM implied volatility has fallen from over 12% at the
beginning of 1996 to under 5 % today.

Political uncertainty is a fourth short-term factor influencing currency volatility. The prospect of a
                                                     governmental regime change generally boosts the
                                                     demand for hedging instruments and raises currency
                                                     implied volatilities. However, when there is little
                                                     uncertainty about an election's outcome, then there
                                                     is usually little response from currency markets as

                                                        Canadian financial markets are periodically buffeted
                                                        by prospects that a major province, Quebec, may
                                                        separate from Canada. The referendum in October
                                                        1995 was closely contested and financial markets
                                                        were uncertain about the outcome throughout.
One-month ATMF C$/US$ implied volatilities began rising in August 1995 as uncertainty mounted.
Options implied volatilities jumped to over 14% at the time of the election in October, up from under 6%,
its long-run mean, in July.

Finally, central banks often attempt to dampen
"excessive" swings in the value of their currency.
Movements in a nation's exchange rate can have'
important ramifications for monetary policy.
Therefore, if market participants push a currency's
value to levels that the central bank feels are too
restrictive or accommodative given their growth and
inflation objectives, then they may try to counteract
the move. This effort usually means that the central
bank intervenes in the currency market, or, if that
fails, they adjust interest rates to offset the
undesired change in monetary conditions. These
policies can sometimes effectively minimize
currency volatility.

Modelling Short-Term Currency Implied Volatility: Many of the factors described above are difficult to
place in a simple linear econometric model. A mean-reverting econometric model of currency volatility
simply relates changes in volatility in a given period with the difference between spot volatility and its
long-run mean. This model forms the baseline specification for our short-term currency volatility

Vt – Vt-1 = α * (c – V t-1) + ε t1

where V is volatility and c is the series mean. This equation can be transformed to:

ΔVt = γ + δV t-1 + ε t2

Where α = -δ is the speed of adjustment back to the mean and c = -γ/δ.

Following the discussion above, divergent or changing economic policies can best be observed by their
impact on spot exchange rates. Rising investor expectations of a tightening in monetary policy, for
example, can push an exchange rate well above its long-run average value. Hence, we include the
absolute value of the percentage deviation between the spot exchange rate and its 200-day moving
average in our specification. As the spot exchange rate wanders further from its long-term value, implied
volatility should rise.

Finally, we use long-term government bond yields to capture investors' perceptions concerning both
economic and political risk premia. Rather than impose a specific functional form on our equation, we
include both country's bond yields in the specification. Higher economic or political risk premia should be
associated with greater currency volatility.

To illustrate this model, we estimated it using quarterly data on three-month ATMF US$/DM implied
volatility between 1985Q2 and 1997Q2. As expected, we observe a strong mean-reverting tendency in
US$/DM implied volatility. For a given deviation from its long-term mean, 50 bps of volatility deviation are
retraced each quarter. In addition, for each I% that the spot exchange rate is away from its long-run
mean, implied volatility rises by 15 bps.

We do not find a significant relationship between US government bond yields and US$/DM implied
volatility; however, we do observe one for German government bond yields. Every 100 bp increase in
German government yields results in a 23 bp increase in US$/l)M implied volatility. Perhaps the liquidity
premium in US government bonds undercuts their usefulness for identifying periods in which the risk
premium on US assets changes.

Structural Factors That Influence Currency Volatility: Over long horizons, currency volatility, and
indeed all financial market volatility, is a function of the underlying variability in the economy. A nation's
long-run economic performance is subject to a number of influences including fiscal and monetary policy,
the propensity to save and invest, and external shocks that may be beyond the control of citizens and their
leaders. Naturally, these issues matter in a relative sense (one currency against another) when we analyse
long-run exchange rate volatility. In this section, we examine two structural factors that influence currency
volatility: economic variability and relative inflation histories.

Box 2: A Mean-Reverting Econometric Model of
Short-Run US$/DM Implied Volatility Fundamentals.

We model changes in quarterly three-month ATMF US$/DM implied has a function of the level of
three-month volatility in the previous period, the percentage distance between spot US$/DM and its
200-day moving average, the 10-year US government bond yield, and the 10-year German government
bond yield. Our results are presented below:

                 Dependent Variable: Change in 3-Mth US$/DM Implied Volatility
 Independent Variables:        Coefficient                   Standard Error
 Constant                      4.911                         2.329
 Level Implied Volatility (-1) -0.497                        0.121
 Distance 200-Day MA (-1)      0.146                         0.052
 US Gov't Bond Yield           -0.213                        0.151
 German Gov't Bond Yield       0.227                         0.135
All of the coefficients are statistically significant except for the US Government Bond Yield term. The
standard error of estimate was 1.09 and Rbar2 was 0.29. The sample has 42 quart erly observations between
1986:4 and 1997:1.

These results largely confirm our theoretical predictions. The negative coefficient on the lagged Level of
Implied Volatility is strongly consistent with a mean-reverting volatility process. In addition, we observe
that the distance between spot and its long-run average, which presumably reflects the degree to which
cyclical factors are driving the exchange rate away from equilibrium, is positive and significant. This
finding means that volatility tends to rise as an exchange rate moves further away from its fundamental
value. Results on the two bond yield terms were mixed. We do find a significant relationship between US
bond yields and changes in US$/DM implied volatility. However we do detect that increases in German
bond yields am associated with positive changes in US$/DM implied volatility.

To illustrate the impact of economic variability on currency volatility, we examine the US and Germany
Economic variability is measured as the three-year rolling standard deviation of quarterly real GDP
changes. We take the ratio of German to US variability because it is relative economic performance that
matters. Long-run currency volatility is the three-year rolling US$/DM volatility.

We observe a close relationship between relative long-term economic variability in Germany and the US
and US$/DM exchange rate volatility. Since 1984, the correlation between these series has been 75%.
Exchange rate volatility tends to rise when the economic variability on one country rises relative to the
other. For example in the wake of the German unification shock in mid 1990, the variability of Germany's
real GDP rose relative to US real GDP. This structural change was partially responsible for raising the
long-run volatility of US$/DM by approximately 35%, from 9.5% to 12.8%.

Relative inflation histories are a second major structural factor that influences currency volatility. Since
relative inflation rates are an important long-run factor governing exchange rate movements, exchange
rate volatility should diminish as the inflation outcomes in two countries converge. Using Germany and the
US again as an example, we observe that as the sum of the rolling three-year standard deviation of
German and US inflation has fallen in the past five years, US$/DM volatility has declined as well.

                                                      Looking forward, many major nations have adopted
                                                      economic policies seeking to limit government fiscal
                                                      deficits and maintain a steady monetary policy aimed
                                                      at keeping inflation under control. These
                                                      developments should minimise the dispersion in
                                                      economic outcomes across nations and, in turn,
                                                      dampen long-run exchange rate volatility. An
                                                      extreme example of this programme is the European
                                                      effort to merge their domestic currencies into a
                                                      single European currency. In the face of
                                                      unpredictable economic or political events, the
                                                      volatility that would have occurred in the foreign
                                                      exchange markets will necessarily be absorbed in
                                                      the entire European real economy because national
                                                      currency values cannot adjust to smooth these

                                                                                            John Simpson

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