Dealers` Hedging of Interest Rate Options in the U.S. Dollar Fixed

Dealers` Hedging of Interest Rate Options in the U.S. Dollar Fixed
Dealers’ Hedging of Interest
Rate Options in the U.S. Dollar
Fixed-Income Market
John E. Kambhu
A
s derivatives markets have grown, the
scope of financial intermediation has
evolved beyond credit intermediation to
cover a wide variety of risks. Financial
derivatives allow dealers to intermediate the risk management needs of their customers by unbundling customer
exposures and reallocating them through the derivatives markets. In this way, a customer’s unwanted risks
can be traded away or hedged, while other exposures
are retained. For example, borrowers and lenders can
separate a loan’s interest rate risk from its credit risk
by using an interest rate swap to pass the interest rate
risk to a third party. In another example of unbundling, an option allows an investor to acquire exposure
to a change in asset prices in one direction without
incurring exposure to a move in asset prices in the
opposite direction.
John E. Kambhu is an assistant vice president at the Federal Reserve
Bank of New York.
The derivatives markets’ rapid growth has been
driven by a number of developments. In addition to
advances in finance and computing technology, the rough
balance of customer needs on the buy and sell sides of the
market has contributed to this expansion. This balance
allows dealers to intermediate customer demands by passing
exposures from some customers to others without assuming excessive risk themselves. Without this ability to pass
exposures back into the market, the markets’ growth
would be constrained by dealers’ limited ability to absorb
customers’ unwanted risks.
The balance between customer needs on both
sides of the market is most apparent in the swaps market, the largest of the derivatives markets, where only a
small amount of residual risk remains with dealers.1 In
the over-the-counter U.S. dollar interest rate options
market, however, significant residual risks are concentrated among dealers, who have sold 50 percent more
options to customers than they have purchased (Table 1,
top panel). This imbalance has left dealers with signifi-
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
35
cant net exposure to price risk that must be hedged in
the underlying fixed-income markets.
Until now, the scale of hedging across all dealers
in the over-the-counter interest rate options market has
not been studied in the literature. The concentration of
sold options among dealers, however, makes it an ideal
place to explore how dealers’ hedging of options affects
underlying markets. Using data from a global survey of
derivatives dealers and other sources, this article estimates the volume and potential impact of such hedging
by U.S. dollar interest rate options dealers. In our analysis,
we address two questions: First, are dealers’ hedge
adjustments large enough to affect trading volume and
liquidity in the most common hedging instruments?
Second, what effects might potential hedging difficulties
have on risk premia in options prices and the structure
of the market for over-the-counter interest rate options?
In addressing these questions, we also consider whether
dealers’ dynamic hedging transactions have the potential
to amplify price shocks.
We find that, on the whole, transaction volume in
volume in the most liquid hedging instruments is more
the underlying fixed-income markets is large enough to
analysis of trading volume suggests that hedging difficulties
enable dealers to manage the risks incurred through
might occur. This pattern in the term structure of
their intermediation of price risk in the interest rate
options premia suggests that the liquidity risk in
options market. Indeed, at shorter maturities, turnover
dynamic hedging may influence options pricing.
Table 1
OVER-THE-COUNTER INTEREST
than large enough to absorb the transaction volume generated by dealers’ dynamic hedging. For medium-term
maturities, however, an unusually large interest rate shock
could cause the hedging of exposures in this segment of
the yield curve to generate trading demand that is high
relative to turnover volume in the more liquid trading
instruments. Dealers then face a risk management tradeoff between reducing price risk or incurring the liquidity
costs of immediately rebalancing their hedge positions.
However, only very large interest rate shocks, such as
those occurring during a currency crisis or a period of
high inflation, are likely to present dealers with this
hedging problem.
In addition to analyzing hedging volume, we
examine the term structure of options premia to assess
whether option prices show any sign of potential hedging
difficulties. We find an apparent risk premium in
options prices at the medium-term segment of the yield
curve that corresponds to the maturity range where our
RATE OPTIONS DATA
NOTIONAL AMOUNTS REPORTED BY DEALERS, IN BILLIONS OF U.S. DOLLARS
Bought Options
Contracts with
U.S. Dollar Interest Rates Other Interest Rates
Other dealers
529.4
726.5
Customers
431.6
340.6
Total
961.1
1,067.1
Total
1,255.9
772.2
2,028.1
Sold Options
U.S. Dollar Interest Rates Other Interest Rates
576.1
681.9
690.4
398.1
1,266.5
1,080.0
Total
1,258.1
1,088.4
2,346.5
MARKET VALUES REPORTED BY DEALERS, IN BILLIONS OF U.S. DOLLARS
Bought Options
Contracts with
U.S. Dollar Interest Rates Other Interest Rates
Other dealers
—
—
Customers
—
—
Total
20.8
16.7
Total
22.4
15.2
37.6
Sold Options
U.S. Dollar Interest Rates Other Interest Rates
—
—
—
—
19.4
16.8
Total
21.6
14.6
36.2
MATURITY DISTRIBUTION OF U.S. DOLLAR INTEREST RATE OPTIONS, IN PERCENT
Bought Options
Sold Options
Up to one year
30
29
More than one year and up to five years
58
56
More than five years
12
15
Source: Bank for International Settlements (1996).
36
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
DYNAMIC HEDGING, VOLATILITY OF
FINANCIAL ASSET PRICES, AND
MARKET LIQUIDITY
An important question in any discussion of options hedging is
whether the dynamic hedging of options in response to a price
shock can introduce transactions large enough to amplify the
initial price shock or to affect market liquidity. In asset price
dynamics, such “positive feedback” occurs when an initial
price change causes a shift in investor or trader demand that
leads to a further change in price in the same direction. For
example, a shift in investor sentiment in response to a sharp
price decline can cause the sell-off of assets or widespread
hedging of open positions—outcomes that would drive prices
down further. The hedging of options also has the potential to
cause positive feedback because dealers typically adjust their
hedge positions by selling (buying) the underlying asset after
its price falls (rises). These dynamic hedge adjustments in
response to a fall in prices could introduce further downward
pressure on prices.
Some observers cite the stock market crash of
1987—which occurred in the absence of any significant
change in economic fundamentals—as an example of positive
feedback dynamics. These observers suggest that the sharp fall
in stock prices was intensified by portfolio insurance trading
strategies that prescribe the sale (purchase) of stocks when
prices fall (rise).2 Although no empirical proof exists that
positive feedback affects market prices, a number of papers
(for example, Bank for International Settlements [1986],
Grossman [1988], Gennotte and Leland [1990], and Pritsker
[1997]) have suggested that dynamic hedging can cause positive feedback. In addition, Fernald, Keane, and Mosser (1994)
discuss a possible example of positive feedback in the behavior
of the term structure of interest rates.
If positive feedback is more than a theoretical possibility, then dynamic hedging would have the potential
to amplify the volatility of asset prices when prices fall
abruptly. Higher price volatility can in turn introduce
other problems in financial markets. Most significantly,
volatility can heighten uncertainty about credit risks
and disrupt the intermediation of credit. For example,
during the 1987 stock market crash, the increase of
credit exposures in securities and margin settlement
caused liquidity and funding problems for securities
firms (see Bernanke [1990]). The potential for such
financial market disruptions makes it worthwhile to consider the relationship between dynamic hedging and
positive feedback in asset prices.
Dynamic hedging can also have implications for
market liquidity. The financial innovations that have
broadened the scope of financial intermediation to
include the intermediation of price risks are positive
developments that might be expected to lower risk premia
in asset prices. Some of these forms of intermediation,
however, rely on the ability of dealers to manage their
risks dynamically. In the absence of market liquidity—
which makes dynamic risk management possible—dealers
would exact higher premia for their intermediation services. Some investors and fund managers may also rely on
market liquidity in their investment and risk management strategies. If significant numbers of economic
agents are relying on the liquidity of the core trading
markets, either directly or indirectly, then part of the
risk premia in financial asset prices might depend on
assumptions about the robustness of that market liquidity.
A sudden realization by investors and dealers that expec-
An important question in any discussion of
options hedging is whether the dynamic hedging of
options in response to a price shock can introduce
transactions large enough to amplify the initial
price shock or to affect market liquidity.
tations of market liquidity were overly optimistic could
lead to a sharp adjustment in asset prices. For this reason,
assessments of the potential impact of dynamic hedging
and risk management strategies on market liquidity are
particularly useful. A related question is whether such
dynamic risk management strategies by individual risk
managers would be feasible in the aggregate during
periods of extreme price volatility.3
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
37
A BRIEF DESCRIPTION OF THE ESTIMATION
The analysis in this paper is based on data from the 1995
Central Bank Survey of Foreign Exchange and Derivatives
Market Activity (Bank for International Settlements 1996),
market growth data from the surveys of the International
Swaps and Derivatives Association (ISDA), and historical
interest rate data. The central bank survey reports the global
market totals of outstanding derivatives contracts at the end of
March 1995. The over-the-counter options data in the survey
include notional amounts and market values of outstanding
contracts, broken down by bought and sold options. A key
part of our analysis is the derivation of strike prices that are
consistent with the notional amount and market value data
from the survey.4
Our estimation of dealers’ hedging transactions
has three principal steps. First, using the notional amount
and rough maturity data from the central bank survey and
market growth rates from the ISDA surveys, we estimate
the distribution of notional amounts over maturities and
origination dates. Next, we combine the estimated
notional amounts at each origination date with historical
interest rate data to estimate options strike prices that are
consistent with the market values reported in the central
bank survey and with historical interest rates. Finally, we
use these strike prices to estimate the price sensitivity of a
portfolio consisting of all dealers’ interest rate options.
Specifically, given the estimated strike prices, we calculate
the delta of the global portfolio, that is, the change in the
portfolio’s value relative to changes in forward interest
rates. The delta and its sensitivity to interest rate changes
give us an estimate of dealers’ hedge demands and dealers’
hedge adjustments to interest rate shocks. (For a detailed
description of the data and estimation, see the appendix.)
indicated changes in interest rates are the option values
calculated from the estimated strike prices. Chart 1 also
shows, as a mirror image, the value of a hedge position
that provides a delta-neutral hedge of the options at the
initial interest rates (the dashed line). The hedge position
is derived by using the estimated strike prices to calculate the price sensitivity (the delta) of the options portfolio. The estimated price sensitivity is used to
construct a hedge position in fixed-income securities
whose gain or loss in value offsets the change in value of
the options portfolio for small changes in interest rates
in either direction. The chart reveals a number of interesting facts about the dealers’ portfolio of options.
First, at prevailing interest rates, the net value of
the dealers’ portfolio is positive. Although in notional
amounts dealers sell more options than they purchase, at
prevailing interest rates the bought options have higher
market values than the sold options (Table 1, top and middle
panels). This relationship between the notional amounts and
the market values implies that the options sold to customers
have a lower degree of moneyness than options purchased
from customers (for definitions of terms, see box). The strike
prices we estimate show the same relationship: relative to
Chart 1
Options and Hedge Values as a Function
of Interest Rate Changes
Billions of U.S. dollars
2
0
Hedge value
-2
-4
ESTIMATED PRICE RISK IN THE GLOBAL
DEALER PORTFOLIO
We begin our analysis by using the estimated strike prices
to derive the value of the global dealer portfolio of options
at different interest rates (the solid line in Chart 1). The
value of the options portfolio at the prevailing interest
rates is the net market value reported in the central
bank survey (Table 1, middle panel). The values at the
38
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
Options value
-6
-8
-4
-3
-2
-1
0
1
2
3
Percentage point change in interest rates
4
Source: Author’s caculations.
Notes: The hedge value is the mirror image of the value of a hedge position
that provides the dealer with a delta-neutral position at the initial interest rate.
The hedged portfolio has a positive value when the option value (the solid line)
is above the hedge value (the dashed line).
swap interest rates at origination, sold options have estimated
strikes that are out-of-the-money, while options purchased
from customers are slightly in-the-money (see appendix).
Because dealers are net sellers of options, however, large
interest rate shocks will drive the sold options into-the-
money, causing them to gain value, and as a result, the total
value of the sold options will exceed the bought options’
value. Hence, if the portfolio is not hedged, the aggregate
dealers’ portfolio value becomes negative when interest rates
rise more than 125 basis points.
OPTIONS TERMS AND CONCEPTS
MONEYNESS
An option’s moneyness is a measure of its payoff at expiration. An option’s payoff is defined relative to a specified level
of the underlying asset’s price called the strike price. For a call
(put) option, the option is in-the-money at expiration when
the asset price is above (below) the strike price, and the
option pays the difference. When the asset price is below
(above) the strike price at expiration, the call (put) option
pays nothing, and the option is said to be out-of-the-money. An
option’s varying sensitivity to price risk is a result of the
asymmetry in an option’s payoff. An option is at-the-money
when the underlying asset’s price is equal to the strike price,
and a call (put) option’s moneyness is higher when the
underlying asset’s price is higher (lower).
INTEREST RATE CAPS
Caps and floors are options on interest rates. In an interest rate
cap (floor), if the interest rate at expiration of the contract is
above (below) the strike rate specified in the contract, the buyer
receives the difference, and nothing otherwise. Caps and floors
are variations of call and put options. In terms of fixed-income
securities, a cap is equivalent to a put option on a bond; in terms
of interest rates, a cap is equivalent to a call option on interest
rates. A cap (put option on a bond) gains value when interest
rates rise (bond prices fall).
VARYING SENSITIVITY TO PRICE RISK AND POSITIVE FEEDBACK
A call option’s value increases by an amount smaller than
the increase in the value of the underlying asset because
there is always some probability that the price of the underlying asset will fall below the strike price at expiration,
rendering the option worthless. As the underlying asset’s
price rises, this probability becomes smaller, and the value
of the option becomes more sensitive to changes in the
underlying asset’s price. To compensate for this increase in
the price sensitivity of a call option, a hedge position in the
underlying asset must be made larger after the price of the
underlying asset rises. This adjustment in the hedge position introduces the potential for positive feedback in price
dynamics because the hedge adjustment is to buy (sell) the
underlying asset after its price rises (falls).
HEDGE ADJUSTMENTS AND OPTIONS PRICES
As the value of the underlying asset rises, the writer of a call
option must make the hedge position larger to ensure that its
value is sufficient to cover the rising option exposure. As the value
of the underlying asset falls, the hedge position must be reduced
in size to ensure that the writer of the option is not left holding
the underlying asset when the option expires out-of-the-money.
Thus, the hedge adjustments in dynamic hedging involve buying
the underlying asset after the price goes up and selling it after the
price goes down. The cumulative cost of these “buy high, sell
low” hedge adjustments equals the value of the option (for further
discussion of option hedging, see Hull [1993]).
VOLATILITY AND OPTIONS RISK
The path-breaking option-pricing models developed more
than two decades ago rely on continuous hedge adjustments to
construct a dynamically hedged portfolio of underlying assets
that perfectly replicates the payoff of an option (under the
assumption that volatility remains constant). This ability to
replicate the option means that the option does not contain
unique risks, and, therefore, its value can be derived straightforwardly from the probability distribution of the underlying
assets by using a risk-neutral expected value calculation. In
practice, however, continuous hedge adjustments are not possible, and the difficulty in constructing a hedge portfolio that
would perfectly replicate an option leaves the writer of an
option with a unique and unhedgeable volatility risk.
IMPLIED VOLATILITY AND VOLATILITY SMILES
Market prices of options differ in characteristic ways from theoretical prices derived from benchmark pricing models, depending on the options’ moneyness. These differences are manifested
as differences in the implied volatility of the underlying asset
when the benchmark model is used to infer the volatility of the
underlying asset from the observed market price of the option.
Options that are either deep out-of-the-money or deep in-themoney typically are priced in the market as if they had higher
volatility in the log-normal distribution embedded in the
benchmark pricing model. (This implied volatility pattern is
called the volatility smile.) By incorporating these implied volatility differences in the benchmark pricing model, analysts can
use the model to generate observed market prices.
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
39
Second, Chart 1 shows that a static hedge can only
protect against small interest rate shocks. For a small change
in interest rates, the change in the value of the options portfolio is offset by a corresponding change in the value of the
hedge position. For a large interest rate change, however, the
change in the value of the hedge position cannot offset the
change in value of the options portfolio. Indeed, the hedged
portfolio value turns negative where the options value and
the hedge value intersect. Thus, with only a static hedge in
place, the value of the hedged portfolio will turn negative
after a large interest rate shock—specifically, an interest rate
increase of more than 175 basis points. Dynamic adjustments to the hedge position as interest rates change,
however, can prevent such an adverse outcome.
Third, to hedge against interest rate changes fully,
dealers must adjust their hedge position after an interest rate
shock. This adjustment compensates for the fact that the
option portfolio’s value falls at an increasing rate as interest
rates rise.5 As Chart 1 shows, without the hedge adjustment,
the gain in value of the initial hedge position will no
longer compensate for the decline in value of the option
portfolio if interest rates continue to rise. This need to
adjust the hedge position dynamically as interest rates
change introduces the potential for positive feedback.
Because the required hedge is a short position in fixedincome securities, the hedge adjustment to an increase
To hedge against interest rate changes fully,
dealers must adjust their hedge position after an
interest rate shock. This adjustment compensates
for the fact that the option portfolio’s value falls
at an increasing rate as interest rates rise.
in interest rates will introduce additional sales into the
fixed-income market and may contribute further upward
pressure on interest rates (by driving bond prices lower).
Finally, Chart 1 suggests that not all dealers can
hedge their options exposures with offsetting exposures
40
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
within their firms. The conventional view of financial institutions’ interest rate risk profiles holds that these firms have
a structural long position in the fixed-income market. That
is, they have a firmwide exposure to rising rates. The negative
slope of the options value curve at the prevailing forward
rates, however, shows that the aggregate dealer portfolio of
options has an exposure to rising interest rates as well. Thus,
because the options portfolio and the other portfolios are
exposed to rising rates, dealers as a group cannot hedge their
net options exposures with offsetting structural positions in
other parts of their firms. Although some dealers may rely
on offsetting exposures elsewhere in their firms to hedge
their options position, Chart 1 suggests that dealers as a
group cannot hedge internally.
ESTIMATED SCALE OF DEALERS’
DYNAMIC HEDGING
A comparison of the size of dealers’ hedge adjustments and
transaction volume in the most common hedging instruments enables us to assess the market impact of dealers’
hedging. As an option’s moneyness increases after a price
shock, the sensitivity of its value to further changes in prices
increases. Thus, to maintain an option portfolio’s exposure to
price risk within a given limit, a dealer must adjust the
hedge position after a price shock to allow for the change in
the options’ price sensitivity. For a given interest rate shock,
we estimate the change in the hedge position required to
restore the portfolio’s price sensitivity (the delta) to its initial
level. This hedge adjustment is the incremental demand of
dealers for hedge instruments after an interest rate shock, if
we assume that dealers maintain their exposure to price risk
at some initial comfort level.
In our analysis of dealers’ dynamic hedging, we make
a number of assumptions. First, we assume that customers do
not dynamically hedge their options positions because
doing so would negate the investment or hedging objective
that motivated the purchase of the option. Thus, we need
consider only dealers’ hedging demands. Second, we
assume that interdealer options do not result in a net
increase in dealers’ hedge demands because they create only
offsetting exposures among dealers.6 Thus, we calculate
dealers’ net hedge requirements from dealers’ contracts
with customers. Finally, to calculate our benchmarks of
dealers’ hedging demands, we assume that dealers match
the maturity of an option’s interest rate exposure with the
interest rate maturity of its hedge. For this reason, our
benchmark hedge estimates are exact hedges that do not
have yield curve or correlation risk. (For further discussion
of these assumptions, see the appendix.)
The interest rate shocks we use in our estimates
are increases in forward interest rates of 25 and 75 basis
points. These interest rate changes are consistent with historical experience. For example, consider forward interest rates in
the four-to-seven-year segment of the yield curve during the
period 1991-95. For that period, a change of 25 basis points is
slightly less than the largest daily change, and a change of
75 basis points is slightly less than the largest two-week
change. During the last two decades, the ten largest daily
changes in forward rates in the medium-term segment of
the yield curve ranged from 60 to 100 basis points. At the
short-term end of the yield curve, the ten largest daily
changes in the three-month Treasury bill rate ranged from
80 to 130 basis points. These episodes of extreme volatility
occurred between 1979 and 1981.
ESTIMATES FOR THE MOST COMMON
HEDGING INSTRUMENTS
In the U.S. dollar fixed-income market, options dealers
executing their hedges can choose from a wide range of
fixed-income instruments such as futures contracts, forward
rate agreements (FRAs), interest rate swaps, interbank
deposits, and Treasury and other bonds. These instruments,
however, are imperfect substitutes because they have
different credit risks, liquidity, and transaction costs. These
differences create a need and an opportunity for intermediation. Dealers who provide risk management services to the
markets take on and manage the risks and costs resulting
from holding portfolios of such imperfect substitutes. When
dealers have enough time to hedge a position or replace an
initial hedge with a cheaper or better instrument, they can
usually keep their exposure to price risk within manageable
limits while still earning a profit from intermediation.
When an immediate hedge adjustment in large volume is
needed, however, dealers’ hedging alternatives are more
limited. For example, although the market for interest rate
swaps is very large and becoming increasingly liquid, the
daily turnover volume of swaps is still very small relative to
outstanding contracts. The turnover of swaps is also small
compared with turnover in the Eurodollar futures markets.7
This difference in turnover volume suggests that swaps are
more likely used to hedge structural or longer term
exposures than to hedge positions that require frequent
adjustment. Consequently, for dynamic hedge adjustments,
For dynamic hedge adjustments, dealers are
likely to use the most liquid instruments as
hedging vehicles. In the U.S. dollar fixedincome market, these instruments are Eurodollar
futures, Treasury securities, and Treasury futures.
dealers are likely to use the most liquid instruments as
hedging vehicles. In the U.S. dollar fixed-income market,
these instruments are Eurodollar futures, Treasury securities, and
Treasury futures.8
Hedging with Eurodollar Futures
Our estimate of dealers’ hedging demands suggests that at
shorter maturities the Eurodollar futures market is more than
large enough to accommodate dealers’ hedging—even when
large interest rate shocks occur. For the hedging of longer
maturity exposures, however, the Eurodollar futures market
appears able to accommodate only the hedging of residual
exposures, that is, marginal hedge adjustments and exposures
that remain after the use of other hedging instruments.
As Tables 2 and 3 show, at maturities of up to one
year, daily turnover volume exceeds the estimated hedge
adjustment even when forward rates increase by as much as
75 basis points.9 At longer maturities, however, the estimated
hedge adjustments are sometimes larger than turnover volume. For a 25-basis-point change in forward rates, the
largest daily turnover volume of Eurodollar futures contracts exceeds the estimated hedge adjustments for maturities
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
41
out to five years and for maturities between eight and ten
years (Table 2). For a 75-basis-point change in forward
rates, however, the largest daily trading volume exceeds the
estimated hedge adjustment out to only two years’ maturity
(Table 3). For small interest rate changes of, say, 10 basis
points, daily turnover volume exceeds the estimated hedge
adjustment at all maturities.
A comparison of the hedge estimates to another
benchmark—the difference between the market’s largest
daily volume and its average volume—yields a similar conclusion. For maturities of up to two or three years, the
surge in the largest daily volume exceeds the estimated
hedge adjustment for a 25-basis-point change in forward
rates (Table 2). For a 75-basis-point change in forward
rates, however, the surge in volume exceeds the estimated
hedge adjustment only out to maturities of a year and a
half (Table 3). Thus, in response to a large interest rate
shock, hedging volume at maturities beyond two years
would be larger than daily turnover volume.
The stock of outstanding Eurodollar futures contracts also suggests that the market can support dealers’
hedge adjustments. Our estimated hedge adjustments are
smaller than the stock of outstanding futures contracts at all
maturities. Even in the case of hedge adjustments in
response to a 75-basis-point change in forward rates, the
estimated hedge adjustment at most maturities is much less
than half the outstanding futures contracts (Table 4). This
result, along with our analysis of turnover volume, suggests
that difficulties executing hedge adjustments are likely to be
liquidity problems. That is, at the medium-term maturities,
the Eurodollar futures market would have difficulty accommodating the entire hedging volume immediately, but the
hedge adjustments could be absorbed over time.
So far, we have considered whether turnover volume
is large enough to absorb transactions from adjustments to
hedge positions in response to a price shock. Another consideration, however, is how large the hedge position is relative to
outstanding contracts in the market. For the estimated
Table 2
CHANGE IN HEDGE POSITION FROM 25-BASIS-POINT INCREASE IN FORWARD
AND THE DAILY TURNOVER VOLUME OF EURODOLLAR FUTURES
Billions of U.S. Dollars
Largest Daily Volume
Maturity
(Years)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Change in Hedge Position
-6.3
-9.2
-7.7
-5.7
-4.6
-3.7
-3.1
-2.6
-2.1
-1.9
-1.6
-1.4
-1.2
-1.0
-0.9
-0.6
-0.4
-0.3
-0.1
—
First Contract
374.0
260.9
55.1
26.9
9.2
7.3
3.9
2.7
2.4
2.0
1.3
1.3
1.0
3.3
0.6
0.8
1.2
1.2
1.0
1.2
Second Contract
334.1
135.2
39.7
18.9
7.5
4.5
2.6
3.3
2.3
1.4
2.4
1.3
1.2
0.7
1.2
3.7
1.2
1.7
0.7
1.0
RATES
Average Daily Volume
First Contract
115.7
92.1
20.0
9.4
4.0
2.7
1.5
1.2
0.9
0.8
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
Second Contract
148.4
35.8
14.0
6.0
3.3
1.9
1.3
1.1
0.8
0.5
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.0
Difference between Largest
and Average Daily Volume
First Contract
258.2
168.8
35.1
17.5
5.2
4.6
2.4
1.5
1.5
1.2
1.1
1.0
0.9
3.1
0.5
0.7
1.1
1.1
0.9
1.1
Second Contract
185.7
99.4
25.7
13.0
4.3
2.5
1.3
2.2
1.5
1.0
2.2
1.1
1.1
0.6
1.1
3.5
1.1
1.6
0.6
1.0
Source: Author’s calculations.
Notes: Hedge estimates are based on data as of the end of March 1995. Contract volume is for the first half of 1995. Bold type indicates that the contract volume exceeds
the change in hedge position. Negative values indicate a short position. Because the futures contracts are contracts on a three-month interest rate, the hedge for each sixmonth exposure requires two back-to-back contracts (“first contract” and “second contract” in the table).
42
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
hedge position at longer maturities, the Eurodollar futures
market is not large enough to accommodate all of the hedge
demands that would be generated by a fully delta-neutral
hedging strategy, particularly for exposures beyond four or
five years (Table 4). Rather, at longer maturities, the Eurodollar futures market can accommodate only marginal hedge
adjustments and the hedging of exposures that remain after
the use of other hedging instruments.
Hedging with Treasury Securities and Treasury Futures
The Treasury securities market and the market for futures contracts on Treasuries—which are both large and highly
liquid—are ideal complements to the Eurodollar futures
market for dealers’ hedging needs. To estimate the hedge
of an exposure to forward rates between five and ten years’
maturity, we assume that dealers’ hedges consist of a short
position (the sale of a borrowed security) in the ten-year note
and a long position in the five-year note. This hedge can be
executed with either cash market securities or futures con-
tracts. The position is a hedge of the exposure between five
and ten years’ maturity because the long and short positions
extinguish exposures to forward rates below five years, leaving
only the exposure to forward rates beyond five years.
The Treasury cash and futures markets are generally
large enough to accommodate dealers’ dynamic hedging
(Table 5). The estimated hedge adjustment is less than the
combined daily turnover volume of on-the-run securities10
and Treasury futures even for large interest rate shocks. However, dealers’ hedging demand could be significant relative to
the size of the markets. For example, the estimated hedge
adjustment to a 75-basis-point shock could be as large as
21 percent of the combined average daily turnover in the
Treasury futures and interdealer on-the-run cash markets, and
almost 10 percent of the outstanding stocks of the on-the-run
securities and futures contracts. Moreover, the estimated
hedge position could be as large as a third of total outstanding
contracts in the two markets. In sum, the Treasury cash and
futures markets significantly expand the pool of fixed-income
Table 3
CHANGE IN HEDGE POSITION FROM 75-BASIS-POINT INCREASE IN FORWARD
AND THE DAILY TURNOVER VOLUME OF EURODOLLAR FUTURES
Billions of U.S. Dollars
Largest Daily Volume
Maturity
(Years)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Change in Hedge Position
-31.9
-31.2
-23.7
-17.2
-13.6
-11.0
-9.0
-7.6
-6.2
-5.5
-4.7
-4.1
-3.5
-3.0
-2.4
-1.9
-1.3
-0.7
-0.3
—
First Contract
374.0
260.9
55.1
26.9
9.2
7.3
3.9
2.7
2.4
2.0
1.3
1.3
1.0
3.3
0.6
0.8
1.2
1.2
1.0
1.2
Second Contract
334.1
135.2
39.7
18.9
7.5
4.5
2.6
3.3
2.3
1.4
2.4
1.3
1.2
0.7
1.2
3.7
1.2
1.7
0.7
1.0
RATES
Average Daily Volume
First Contract
115.7
92.1
20.0
9.4
4.0
2.7
1.5
1.2
0.9
0.8
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
Second Contract
148.4
35.8
14.0
6.0
3.3
1.9
1.3
1.1
0.8
0.5
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.1
0.1
0.0
Difference between Largest and
Average Daily Volume
First Contract
258.2
168.8
35.1
17.5
5.2
4.6
2.4
1.5
1.5
1.2
1.1
1.0
0.9
3.1
0.5
0.7
1.1
1.1
0.9
1.1
Second Contract
185.7
99.4
25.7
13.0
4.3
2.5
1.3
2.2
1.5
1.0
2.2
1.1
1.1
0.6
1.1
3.5
1.1
1.6
0.6
1.0
Source: Author’s calculations.
Notes: Hedge estimates are based on data as of the end of March 1995. Contract volume is for the first half of 1995. Bold type indicates that the contract volume exceeds
the change in hedge position. Negative values indicate a short position. Because the futures contracts are contracts on a three-month interest rate, the hedge for each sixmonth exposure requires two back-to-back contracts (“first contract” and “second contract” in the table).
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
43
instruments available to options dealers for their hedging
needs. However, dealers’ hedge demands could amount to a
significant share of turnover volume and outstanding contracts in the Treasury on-the-run cash and futures markets.11
Alternative Hedging Estimates
The hedge estimates above do not account for changes in the
volatility of interest rates. An option’s hedge position against
Table 4
DELTA-NEUTRAL HEDGE POSITION IN EURODOLLAR FUTURES
CONTRACTS AND EURODOLLAR FUTURES CONTRACTS
OUTSTANDING
Billions of U.S. Dollars
Open Interest
Maturity
(Years)
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Hedge
Position
38.3
23.9
2.8
-4.0
-9.8
-13.4
-16.4
-17.9
-20.2
-18.9
-18.8
-18.4
-17.5
-15.1
-12.6
-9.6
-6.2
-3.4
-1.4
—
First
Contract
561.9
279.7
174.0
114.2
84.9
60.3
49.5
34.4
22.6
12.9
7.5
6.2
6.7
6.8
3.8
1.6
1.8
1.7
0.8
0.8
Second
Contract
366.4
222.0
145.4
96.3
68.6
54.8
38.8
27.2
14.5
9.5
7.7
5.9
6.8
4.5
2.5
2.2
1.8
2.0
0.9
0.0
Change in Hedge from
a 75-Basis-Point
Shock
-31.9
-31.2
-23.7
-17.2
-13.6
-11.0
-9.0
-7.6
-6.2
-5.5
-4.7
-4.1
-3.5
-3.0
-2.4
-1.9
-1.3
-0.7
-0.3
—
Source: Author’s calculations.
Notes: The table reports delta-neutral hedge estimates and open interest as of the
end of March 1995. Bold type indicates that the contract volume exceeds the
change in hedge position. Negative values indicate a short position. Because the
futures contracts are contracts on a three-month interest rate, the hedge for each
six-month exposure requires two back-to-back contracts (“first contract” and
“second contract” in the table).
Table 5
HEDGE POSITION IN
Billions of U.S. Dollars
Five-year
Ten-year
changes in the underlying asset’s price, however, depends on
the underlying asset’s price volatility as well as on the asset’s
price level. Moreover, large changes in asset prices are often
associated with higher implied volatilities in options. For this
reason, alternative hedge adjustments were estimated
assuming simultaneous volatility and interest rate level
shocks (Table 6). Although the estimated hedge adjustment
is larger, the difference does not appreciably change the conclusions because the difference from the base case is small
relative to the turnover volume in the hedge instruments.
DEALERS’ HEDGE ADJUSTMENTS
AND MARKET LIQUIDITY
Our estimate of dealers’ hedging demands suggests that
dealers might encounter hedging difficulties only for
exposures beyond three or five years’ maturity when large
interest rate shocks occur. Together, the Eurodollar futures,
on-the-run Treasury securities, and Treasury futures markets can
absorb hedge adjustments to interest rate shocks as large as
25 basis points along the entire term structure. For example,
for exposures between five and ten years’ maturity, the
estimated hedge adjustment to a 25-basis-point shock is
only 7 percent of the combined turnover in the Treasury
futures and interdealer on-the-run cash markets (Table 5).
For a large interest rate shock, however, such as a
75-basis-point shock to forward rates, dealers’ dynamic
hedge adjustments in the medium-term segment of the
yield curve would generate significant demand relative to
turnover in these hedging instruments. This demand would
amount to 21 percent of the combined turnover in the
Treasury futures and interdealer on-the-run cash markets
(Table 5). In addition, the hedge adjustment in the three-
TREASURY SECURITIES AND FUTURES
Hedge Position
13.0
-13.0
Change in Hedge Position from an Interest Rate Shock of
10 Basis Points 25 Basis Points 75 Basis Points
0.4
1.0
2.9
-0.4
-1.1
-3.3
On-the-Run Treasury Securitiesa
Outstanding
Daily Volume
13.2
9.0
13.8
6.0
Treasury Futuresb
Outstanding
Daily Volume
19.7
5.1
25.8
9.2
Source: Author’s calculations.
Notes: Hedge estimates are based on data as of the end of March 1995. Negative values indicate a short position.
a
Outstanding amount as of the end of March 1995. Daily volume is estimated from interdealer trading volume (Fleming 1997).
b
Five- and ten-year note contracts. Outstanding contracts are as of the end of March 1995. Daily volume is for the first half of 1995.
44
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
to-five-year maturity segment would be large relative to
Eurodollar futures turnover volume (Table 3). If all dealers
executed their hedge adjustments simultaneously, these
transactions could have an impact on turnover volume and
affect market liquidity. Moreover, in the presence of a large
interest rate shock, other traders and investors might under-
Turnover volume in standard hedging
instruments appears large enough to accommodate
dealers’ dynamic hedging in all but the most
extreme periods of interest rate volatility.
take transactions in the same direction as options dealers’
hedge adjustments. All these demands together suggest that
dealers wishing to adjust their hedge positions immediately
could indeed encounter market liquidity problems.
Dealers can manage the impact on market liquidity
by trading off price risk against the cost of immediacy or
liquidity. For example, only part of the exposure opened up by
a large interest rate shock might be hedged initially, with the
remainder hedged over time. Dealers can spread the hedge
adjustment over a number of days by executing a series of
transactions that are small relative to daily turnover in the
hedge instruments. This strategy reduces the market impact
of the hedge adjustment but leaves the dealer exposed to
some price risk until the hedging transactions are completed. Alternatively, by assuming some correlation risk, a
dealer could also hedge the longer maturity exposures with
the first three near-term futures contracts. The volume of these
shortest maturity contracts is large enough to accommodate
the hedging of longer maturity exposures easily, but
returns on these contracts are less than perfectly correlated
with longer maturity interest rates.
In another alternative, dealers could use an interest
rate term structure model to design a hedge that avoids concentrated transactions at yield curve sectors with liquidity
problems. For example, using a two-factor interest rate term
structure model, a dealer could construct a hedge of exposures
between five and ten years using a position in one-year bills
and thirty-year bonds that replicates the exposure to the term
structure factors that drive forward rates between five and
ten years. Such hedges, however, are vulnerable to atypical
price shocks not accounted for by the correlations in the term
structure model.
Regardless of how the trade-off between price risk
and the cost of immediacy or liquidity is executed, the
terms of the trade-off depend on the volatility of interest
rates. If volatility rises at the same time that liquidity is
most impaired, then these hedging strategies could leave
the firm exposed to higher than usual price risk.
These results suggest that transaction volume in the
underlying fixed-income markets is large enough to enable
dealers to manage the risks acquired from the intermediation
of price risk in the interest rate options market. Turnover
volume in standard hedging instruments appears large
enough to accommodate dealers’ dynamic hedging in all but
the most extreme periods of interest rate volatility. For very
large interest rate shocks, however, the hedging of exposures
in the medium-term segment of the yield curve could lead to
trading demand that is large relative to turnover volume in
the more liquid trading instruments. Thus, for large interest
Table 6
CHANGE IN REQUIRED HEDGE POSITION FROM SIMULTANEOUS
FORWARD AND VOLATILITY RATE SHOCKS
Billions of U.S. Dollars
Maturity
(Years)
Interest Rate
Shock
EURODOLLAR FUTURES
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-31.9
-31.2
-23.7
-17.2
-13.6
-11.0
-9.0
-7.6
-6.2
TREASURY SECURITIES AND FUTURES
5
2.9
10
-3.3
Volatility
Shock
Interest Rate and
Volatility Shocks
-6.0
-9.7
-8.7
-7.7
-6.2
-4.9
-3.9
-3.0
-2.2
-40.7
-38.7
-29.4
-22.6
-17.9
-14.4
-11.6
-9.5
-7.5
0.8
-0.8
3.4
-3.8
Source: Author’s calculations.
Notes: Hedge estimates are based on data as of the end of March 1995. The table
assumes a 75-basis-point increase in forward rates. Volatility is assumed to
increase by 25 percent relative to initial volatility levels at six months’ maturity
and by 8 percent at ten years’ maturity. Negative values indicate a short position.
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
45
rate shocks, dealers’ risk management decisions appear to be
driven by a trade-off between price risk and the liquidity costs
of immediate hedge rebalancing. Even so, for interest rate
shocks typical of a low-inflation environment, the trade-off
would need to be managed only for a short period of time.
THE EFFECT OF DYNAMIC HEDGING ON
THE PRICES OF UNDERLYING ASSETS
Our results on the impact of dealers’ dynamic hedging on
prices in the fixed-income market are less clear. Any comprehensive assessment would need to account for the demands of
other market participants as well. For example, investors
whose demands are driven by macroeconomic fundamentals
might undertake transactions in the opposite direction of
dealers’ dynamic hedging flows if interest rates were driven to
unreasonable levels. If these investors constitute a sufficiently
large part of the market, then their transactions could stabilize
prices and reduce or even eliminate positive feedback dynamics
(Pritsker 1997). These stabilizing investors, however, are not
the only players. Traders who follow short-term market trends
in “technical trading” strategies and speculators who
anticipate the impact of positive feedback trading also
participate in the market. These short-term traders could
amplify the price impact of dealers’ dynamic hedging
because they would trade in the same direction as dealers’
hedging transactions (see DeLong et al. [1990]). The ultimate impact of dealers’ dynamic hedging would depend on
the relative size of different types of market participants.
For this reason, our analysis of the volume of dealers’ hedging
demands provides only a preliminary assessment of the
potential for positive feedback because we have data on the
hedging demands of dealers exclusively.
At shorter maturities, both the transaction volume
and the outstanding stock of the most liquid trading instruments are much larger than dealers’ dynamic hedging flows,
so that the occurrence of positive feedback from dealers’
dynamic hedging seems unlikely, even for very large interest
rate shocks. At maturities beyond three years, however, if
dealers fully rebalance their hedge positions, dynamic hedging
in response to a large interest rate shock could be of significant
volume relative to transaction volume and outstanding
46
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
contracts in the most liquid trading instruments. At
this segment of the yield curve, the potential for positive
feedback when a very large interest rate shock occurs cannot
be dismissed. The volume of dynamic hedging in response to
an unusually large interest rate shock could be large enough
to affect order flows and might temporarily affect the
medium-term segment of the yield curve.
THE EFFECTS OF HEDGING DIFFICULTIES
ON OPTIONS PRICING AND
MARKET STRUCTURE
Our results suggest that the hedge adjustments of dealers in
aggregate could be large relative to the size of the market for
hedge instruments at the medium-term segment of the yield
curve. Given this potential market impact, we consider
whether market prices of interest rate options with mediumterm maturities contain a premium to cover potential hedging
difficulties. To look for evidence of a premium, we compare
the implied volatility of interest rates derived from the market
prices of options to historical interest rate volatility. An
option’s implied volatility is a volatility parameter in an
options-pricing model that causes the model’s option price to
equal the option’s actual price. If the market pricing of options
contains a premium, we would expect the implied volatility to
be large relative to other measures of the underlying asset’s
price volatility. Although by no means a comprehensive test,
the simple comparison of the term structures of implied
volatilities and historical volatilities provides a quick assessment of the possible existence of such a premium.
The difference between the term structure of
implied volatility and the term structure of the historical
volatility of forward Eurodollar interest rates is shown in
Chart 2. Notably, the difference is greatest at the mediumterm maturities (three to seven years), where the estimated
hedge adjustments are large relative to the transaction volume of the hedge instruments. By contrast, the difference
between implied volatility and historical volatility is
smallest at short-term maturities (under two years). The
estimated hedge adjustment relative to transaction volume
in hedge instruments is also smallest at this maturity
range. Although the difference between the historical and
implied volatility term structures could reflect uncertainty
about interest rate volatility in the medium-term segment
of the yield curve, its shape is consistent with the existence
of a premium for hedging difficulties. This apparent consistency between the term structure of options premia and
our analysis of hedging volumes suggests a need for further
research on how potential hedging difficulties may affect
the term structure of interest rate options prices.
VOLATILITY RISK
AND
HEDGING C OSTS
The change in the cost of hedging when interest rate volatility changes also affects the value of an option. Although the
Chart 2
Term Structures of Forward Interest Rate Volatility
February 1, 1994–January 31,1995
Percent
15
Difference between Implied and Historical Volatility
Percent
30
Volatility Term Structures
25
Implied volatility on July 5, 1994
20
10
15
Historical volatility
10
5
5
0
0
February 1, 1995–January 31,1996
30
15
Volatility Term Structures
Difference between Implied and Historical Volatility
25
20
Implied volatility on July 24, 1995
10
15
Historical volatility
10
5
5
0
0
February 1, 1996–January 31,1997
30
15
Volatility Term Structures
Difference between Implied and Historical Volatility
25
10
Implied volatility on July 29, 1996
20
15
5
Historical volatility
10
0
5
0
-5
1
2
3
4
5
6
7
Maturity (years)
8
9
10
1
2
3
4
5
6
7
Maturity (years)
8
9
10
Sources: Historical volatilities were derived using the yields on Eurodollar futures contracts as reported by DRI/McGraw-Hill. Implied volatilities are from
Derivatives Week. In each panel, the historical volatility is for the period indicated.
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
47
exposure of an option’s value to changes in the level of the
underlying asset’s price can be hedged, exposure to changes
in the volatility of the underlying asset is not hedgeable with
a linear fixed-income instrument such as a bond or a futures
contract. Rather, an option’s volatility risk can be hedged
fully only with another option. Given that dealers as a group
are net writers of options, their exposure to volatility risk is
significant. If most customer options in the over-the-counter
market are held to maturity, changes in volatility would
To look for evidence of a premium, we compare
the implied volatility of interest rates derived
from the market prices of options to historical
interest rate volatility.
affect dealers through changes in the cost of hedging over
the life of an option. Higher volatility would raise these
hedging costs because it amplifies the costs of adjusting
hedge ratios.12 When volatility changes, the change in an
option’s value is equal to the expected change in hedging
costs over the life of the option.
An estimate of the sensitivity to volatility shocks
of the global dealer portfolio of interest rate options is
shown in Chart 3. In the chart, the estimated strike prices
are used to revalue the dealers’ options portfolio for the
indicated changes in volatility. An increase in volatility of
approximately 35 to 40 percent causes the portfolio value
to turn negative.13 This change in value of the dealers’
options portfolio is a measure of the volatility risk incurred
by options dealers.
earlier, dealers of U.S. dollar interest rate options have sold
about 50 percent more options to customers than they have
bought from customers. Thus, dealers are more willing than
other investors to take on the volatility risk in selling an
option. Given the wide range of financial assets and risks that
investors are willing to acquire, why do they leave interest rate
option exposure to dealers?
The concentration of interest rate option exposure
among dealers implies that sellers of these options bear
unique risks that are not present in the returns of underlying
assets and that dealers are more willing to bear those risks.
Volatility risk is one risk that is unique to options. Another
is the difficulty in adjusting hedge positions as rapidly as
required for the accurate hedging of an option’s price
risk.15 The fact that dealers are more willing than other
investors to sell interest rate options suggests that dealers
are in a better position to bear the options’ volatility and
hedging risks. Dealers have two possible advantages in this
area. First, they may be able to execute hedging transactions
faster and at lower costs than other investors. Second, they
may have other sources of income that offset the volatility
risk in an option position. If dealers’ income from market
making in products other than options rises during periods
of higher volatility, then that income will offset the increase
in volatility risk from selling options. While some of that
higher income would be compensation for the higher risk
that dealers incur in making prices in volatile markets, any
Chart 3
Options Values as a Function of Volatility Changes
Billions of U.S. dollars
4
2
HEDGING DIFFICULTIES, VOLATILITY RISK,
AND MARKET STRUCTURE
Volatility risk and potential hedging difficulties may also
affect the structure of the interest rate options market. In other
derivatives markets, end-user needs are roughly balanced
across buyers and sellers,14 but in the over-the-counter interest
rate options market, end-users are mostly buyers. As we noted
48
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
0
-2
-60
-45
-30
-15
0
15
30
Percentage change in volatility
Source: Author’s calculations.
45
60
remaining excess returns would offset the higher volatility
risk from the options book. Such offsetting risks may
explain why large interest rate options dealers also are
market makers in a broad array of fixed-income products,
and, for that reason, are willing to bear volatility risk at a
smaller premium than other investors.
The evidence that market making in options and
other products provides dealers with offsetting exposures
to changes in volatility is not strong, however. For
instance, even though the turnover volume of derivatives
grew rapidly during 1994, dealers’ trading income suffered
from the bond market turbulence that occurred in that
year. It has been reported that a significant part of the
1994 earnings decline occurred in dealers’ bond and proprietary trading positions and not in their market-making
activity.16 Nevertheless, we lack detailed data on marketmaking income that would enable us to resolve with any certainty the question of offsetting exposures to volatility.
CONCLUSION
Our analysis suggests that transaction volume in underlying
markets is large enough for dealers to manage the price and
liquidity risks they incur through the intermediation of price
risk in selling interest rate options. With the possible
exception of the medium-term segment of the term structure, turnover volume in the most liquid hedging instruments
is large enough to absorb dealers’ dynamic hedging.
In the case of an unusually large interest rate shock
at the medium-term segment of the term structure, the full
rebalancing of hedge positions would generate hedging
transactions that would be large relative to daily transaction
volume in the most liquid medium-term instruments. In
this case, dealers’ risk management decisions would appear
to be driven by a trade-off between price risk and the
liquidity costs of immediate hedge rebalancing. For interest
rate shocks of the size experienced in the last five years,
dealers’ hedge adjustments would be a small proportion of
only a few days’ worth of turnover volume, and dealers
would need to manage the trade-off between liquidity and
price risks only for a short period of time. For large interest
rate shocks, however, such as those experienced by a country
in the midst of a currency crisis or a period of high inflation, the hedging of exposures in the medium-term
segment of the yield curve could lead to trading demand
that is large relative to turnover volume in the more liquid
trading instruments.
The ratios of estimated hedge adjustments to
transaction volume in trading instruments at different
maturities are consistent with the pattern we find in the
term structure of option premia. The term structure of
implied volatility shows an apparent risk premium for
options at the medium-term segment of the yield curve, a
segment that corresponds to the maturity range where
hedging difficulties might occur. The structure of the overthe-counter interest rate options market is also consistent
with the hypothesis that such hedging problems may exist.
Despite investors’ willingness to hold a wide variety of
financial assets and risks, they choose to leave interest rate
options exposures in the hands of dealers. This preference
suggests that interest rate options sellers are exposed to
risks that are not present in the returns of underlying
assets. These risks are likely volatility and hedging-related
risks, which may be managed more effectively by dealers
than by other market participants.
The results presented in this article provide a preliminary assessment of the impact of dynamic hedging on
market liquidity and price dynamics in the fixed-income
market. As the appendix makes clear, limitations of the
data make further investigation worthwhile. In addition,
an estimate of the market excess demand function and the
relationship between prices and quantities would be useful.
Such an analysis, however, would require data that do not
currently exist on investors’ demands in addition to dealers’ hedging demands. Nonetheless, comparing potential
hedging demand with transaction volume in typical
hedging instruments is useful in assessing the likelihood
of positive feedback.
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
49
APPENDIX: THE ESTIMATION
THE DATA
Our primary source of data is the 1995 Central Bank Survey of
Foreign Exchange and Derivatives Market Activity (Bank for
International Settlements 1996). This survey of derivatives
dealers worldwide reports global market totals of outstanding
contracts as of the end of March 1995. The over-the-counter
options market is a dealer market where all options contracts
involve a dealer on at least one side of the contract. An option
contract can either be a transaction between two dealers or a
contract between a dealer and a customer. The central bank survey captured the entire over-the-counter options market by collecting data from the dealers that executed all contracts.17 The
options data in the survey include notional amounts, market
values, and maturity data, broken down by bought and sold
options, as shown in Table 1.18 The options were also broken
down by the survey’s three counterparty categories: interdealer
options, options bought from customers by dealers, and options
sold to customers by dealers. Because reporters in the survey
were derivatives dealers, interdealer transactions appear as both
bought and sold options. In other words, an option bought by
one dealer from another was reported as a bought option by one
dealer and as a sold option by the other.19
OPTION VALUATION
All options in the estimation are caps and floors on sixmonth interest rates. In accordance with the data from
the International Swaps and Derivatives Association (ISDA),
73 percent of the options in the estimation are assumed to be
caps, and the remainder are assumed to be floors.
Although a small proportion of interest rate options are
swaptions (19 percent at year-end 1994 in the ISDA data), for
simplicity, we treat all options as either caps or floors.20 Option
values are calculated using Black’s forward contract option
model, the benchmark model used for implied volatility quotes
for interest rate options (see Hull [1993]). The valuation uses
50
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
the term structure of forward rates and the term structure of
implied volatilities coinciding with the central bank survey
data (end of March 1995).21 To test the valuations, we also calculate the option values using different volatility structures. The
baseline case assumes that caps and floors have identical implied
volatilities that do not vary with moneyness. Alternative valuations using a volatility smile for options with different degrees
of moneyness and higher implied volatilities for floors relative
to caps did not affect our conclusions (see table below).
MATURITY DISTRIBUTION
To estimate the distribution of notional amounts over the
remaining maturity and origination dates, we fit a quadratic
function defined over original maturities to the remaining
maturity data from the central bank survey and the ISDA
VOLATILITY ASSUMPTIONS: CHANGE IN HEDGE POSITION
FROM 75-BASIS-POINT INCREASE IN FORWARD RATES
Billions of U.S. Dollars
Assumption
Maturity
(Years)
Base
Cap/Floor Volatility Volatility Smile
EURODOLLAR FUTURES
0.5
-31.9
1
-31.2
1.5
-23.7
2
-17.2
2.5
-13.6
3
-11.0
3.5
-9.0
4
-7.6
4.5
-6.2
-31.5
-31.2
-23.7
-17.2
-13.6
-10.9
-9.0
-7.5
-6.2
TREASURY SECURITIES AND FUTURES
5
2.9
2.9
10
-3.3
-3.3
Cap/Floor Volatility
and Smile
-27.2
-27.8
-21.1
-14.6
-11.4
-9.1
-7.4
-6.2
-5.3
-26.8
-27.7
-21.0
-14.5
-11.3
-9.0
-7.4
-6.2
-5.3
2.6
-2.9
2.6
-2.9
Source: Author’s calculations.
Notes: Hedge estimates are based on data at the end of March 1995. Negative
values indicate an increase in a short position. The base is the estimate using the
assumptions in the text. For the cap/floor volatility assumption, option values
were calculated with higher volatility for floors using cap and floor implied volatility
differences reported by DRI/McGraw-Hill. For the volatility smile assumption,
option values were calculated using a volatility smile derived from Eurodollar
futures options prices.
APPENDIX
APPENDIX: THE ESTIMATION (Continued)
market growth data. The three maturity categories in the
central bank survey provide the three equations required to
estimate the three parameters of the quadratic function. In
the estimation, the market growth rates between origination
dates are applied as a scaling factor to the quadratic function.
We estimate separate maturity distributions for options purchased from customers and options sold to customers.22
In the estimated distribution, most outstanding
contracts have less than four years’ remaining maturity and
have origination dates that fall within three years of the central bank survey date. The estimated distribution has a
trough along the diagonal for caps with maturities at origination of between five and ten years. This feature of the distribution suggests that long maturity caps are originated at
discrete maturities, specifically at the ten-year maturity.23
STRIKE PRICES
Strike prices are derived from historical yield curves and
assigned to the options using the estimated distribution of
notional amounts over origination dates. Because separate market values are not available for caps and floors, the estimation
requires that a relationship between the strikes of caps and
floors be imposed. The relationship assumed is that buyers (or
sellers) of caps and floors have similar preferences regarding
their options’ moneyness. Under this assumption, if buyers of
caps desire out-of-the money options because of their cheaper
premia, then buyers of floors will also.
We implement the assumed relationship regarding
the moneyness of caps and floors in three different ways. In all
three approaches, the historical swap term structure at an
option’s origination date is our starting point. The first
method is a proportional displacement of the strike rates from
the historical swap term structure (in the same proportion,
but opposite directions, for caps and floors). The other two
methods are displacements of the strikes from the historical
APPENDIX
swap term structure with a constraint that caps and floors (of
the same maturity and origination date) have equal premia
(the second method) or equal deltas (the third method).
Under the last two methods, the strikes for caps and floors can
have different displacements from the swap term structure.
In these specifications, a cap will be out-of-themoney at origination when a floor is out-of-the-money. In
each specification, the restrictions are applied to bought and
sold options separately. The figures in the text are derived
using the first approach, but similar results followed from
the other specifications.24
ESTIMATED STRIKE PRICES AND OPTION VALUES
Given the strike price specification relative to historical yield
curves, option values are calculated as functions of the displacement of the strike prices from the historical yield curves. The
estimation then involves finding the displacement that produces option values equal to the market values observed in the
central bank survey.
The objective of the estimation is to find values of
the strike price displacement variables ( A b , As ) such that
Vb ( Ab ) + vD = vb
V s ( As ) + vD = vs ,
where v b and v s are the observed market values of U.S. dollar
options bought and sold by dealers (including interdealer
options), and v D is the market value of interdealer options. The
functions V ( A ) are the option values as functions of the displacement ( A ) of the strike prices from the historical term
structures, and the subscripts indicate options bought ( b ) and
sold ( s ) from customers.25 In the proportional displacement
specification of the strike price, the term A is a single variable.
In the other two cases, the term A is a vector with two elements—the displacement for caps and the displacement for
floors. In both cases, the additional equation required to solve
for the two displacement variables is the equal premia or equal
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
51
APPENDIX: THE ESTIMATION (Continued)
delta restriction in the specification of the strike prices.
In the option value equations above, the market
value of U.S. dollar interdealer options is not available
directly from the central bank survey data because market
values are reported in aggregate—across counterparty types
and across currencies (Table 1, middle panel). To find the
displacement variables for bought and sold options ( A b and
A s ) in the option value equations above, we first estimate
the value of U.S. dollar interdealer options.
INTERDEALER OPTIONS
Estimates of the market value of U.S. dollar interdealer options
are calculated in four different ways. In the first three methods
we make assumptions about the strike rates of interdealer
options: (1) interdealer options have strikes equal to swap
rates (at-the-money strikes relative to the swap term structure);
(2) interdealer options have the same strikes as options bought
from customers; and (3) interdealer options have the same
strikes as options sold to customers.26 In the fourth method, we
estimate the value of U.S. dollar interdealer options using the
data reported in Table 1. In this method, the market value of
interdealer options is distributed between U.S. dollar options
and options on other currencies so as to minimize the discrepancy between the ratio of market value to notional amount for
each currency and counterparty type and the ratio of the market
value to the notional amount of the margin totals in the top
and middle panels of Table 1.
The first and last alternatives produce comparable
values for U.S. dollar interdealer options, while the other
two do not. The estimation using the at-the-money
assumption produces a value of interdealer options of
$11.3 billion, while the last method results in a value of
interdealer options of $10.9 billion. Strike prices that produce
a value of $10.9 billion would be very slightly out-of-themoney at origination. The comparability of the estimates
52
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
in methods one and four implies that interdealer options
have strikes closer to at-the-money than do customer
options. This result is plausible because dealers using the
interdealer market to hedge their short volatility and
negative gamma position would obtain more hedging
benefit from at-the-money options. Such options have
larger gamma and provide the most hedging benefit relative to their premia. The results reported in the text are
derived using the fourth method. Despite the different
estimates of interdealer option values, similar hedge estimates follow from all four alternatives.
CUSTOMER OPTIONS
For options sold to customers, estimated strikes consistent with
observed market values are deep out-of-the money (relative to
swap rates of comparable maturity) at origination. This result is
plausible—customers buying options to hedge can acquire
inexpensive protection against large interest rate shocks with
deep out-of-the money options. For caps sold to customers, the
estimated strike rates are 18 percent higher than swap rates of
the same maturity at origination. The figure of 18 percent is
comparable to the standard deviation of annual changes in
interest rates, or two standard deviations of quarterly interest
rate changes (six-month LIBOR rates from January 1991 to
December 1995).
For options bought from customers, strike prices consistent with the observed market values are slightly in-themoney (relative to swap rates of comparable maturity) at origination. This relationship is the opposite of the relationship
found for options sold to customers. Although this result
might appear counterintuitive and could point to a problem in
the estimation, it is consistent with market commentary in the
early 1990s. Customers looking for “yield enhancement”
during the low-interest-rate regime of the early 1990s acquired
higher premia by selling interest rate caps with a higher degree
APPENDIX
APPENDIX: THE ESTIMATION (Continued)
of moneyness. While this higher yield is the market price or
compensation for the expected payout of the option, investors
speculating on the path of interest rates by selling options
would obtain higher investment returns (or losses) per option
by selling in-the-money options. In addition, investors who
believed that the forward curve was an overestimate of the
future path of spot rates would have sold options that were
in-the-money relative to swap rates.
HEDGING ASSUMPTIONS
The final step in the estimation of dealers’ hedge adjustments is
the calculation of the delta and the change in delta of the global
dealers’ portfolio using the estimated distribution of notional
amounts and the estimated strike prices. The analysis of dealers’
hedging behavior relies on the following assumptions:
1. After an interest rate shock, dealers restore the net
delta of their position to its initial level. Dealers
may or may not fully hedge the initial delta of the
options book, and whatever hedging is done initially may be accomplished either internally, with
offsetting positions in the firm, or externally, with
hedging transactions. These initial offsetting positions, either internal or external, are assumed to
have a small gamma, so that changes in interest
rates—and thus the options’ delta—make additional hedging transactions necessary to return the
portfolio’s net delta to its original level.
2. An option exposure to a period t interest rate is
hedged with an instrument that also has exposure to the period t interest rate—there is no
basis risk in hedged positions. Using this
assumption, we calculate a separate hedge ratio
for each maturity’s exposure.
3. Customers do not hedge their options positions.
Customers who have bought or sold options are
assumed not to hedge, because doing so would
negate whatever investment or hedging objective
the options were used for. Customers who have
sold options to dealers presumably did so for spec-
APPENDIX
ulative “yield enhancement” or intertemporal
income shifting. The costs of delta-hedging the
options would negate that investment objective.
Customers who have bought options from dealers
for hedging purposes would not hedge the option
because doing so would expose the underlying
position that the option was hedging. Thus, the
impact of dynamic hedging is assessed using the
aggregate dealers’ position.
If customers were to hedge their options, perhaps
as a result of a reassessment of risks, then the market
impact of dealers’ hedge adjustments would be
smaller because these adjustments would be offset by
customer hedges. Because most of our results support
the claim that the market impact of dealers’ hedging
is small relative to the size of the market, dropping the
assumption would strengthen our conclusion that the
markets for typical hedging instruments are sufficiently large for dealers to manage the price risk
acquired from market making in options.
4. Interdealer options have no effect on dealers’ net
demand for hedge instruments. Using this assumption, interdealer options can be ignored, and the
net hedge position and hedge adjustment of dealers
in the aggregate can be calculated from dealers’
contracts with customers. This assumption is reasonable when interdealer options are executed to
reallocate customer exposures among dealers or to
take a position in volatility risk but not directional
interest rate risk. In the first case, the interdealer
option that passes a customer exposure from one
dealer to another does not create additional net
option exposure for dealers in the aggregate. Thus,
dealers’ net hedge demands would be unaffected by
such interdealer options.
The second type of interdealer trading that is
consistent with this assumption is position taking
on changes in interest rate volatility. This trading
strategy entails the hedging of directional interest
rate risk. If executed by dealers on the two sides of
an interdealer trade, such hedges would offset
each other in the market, with no impact on the
net dealer hedge amount.
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
53
APPENDIX: THE ESTIMATION (Continued)
An important justification for the presumption that dealers’ position taking in options is a
position on volatility changes is the fact that dealers wishing to take directional exposures to interest rate risk could do so less expensively with
instruments other than options.
ROBUSTNESS CHECKS
To test our results, we derive estimates of dealers’ hedging
using alternative assumptions about implied volatility, the
structure of strike prices, and other restrictions. The variation
in hedge demands across these assumptions is small relative to
turnover volume in the hedge instruments and does not
change our conclusions. The results under these alternative
assumptions are available in Kambhu (1997, Tables 5-8).
Although the results are robust to alternative
assumptions, they might be influenced by certain features of
the central bank survey data. First, dealers might have had
options positions that were not reported in the central bank
survey. Index amortizing interest rate (IAR) swaps, for
example, might have been reported as swaps instead of
options. These instruments were popular in the early 1990s,
54
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
when investors were searching for yield enhancement. The
extra yield in this instrument is the premium for a written
option embedded in the instrument’s payoff structure. Most
of the volume of IAR swaps, however, was in contracts of
three years’ or shorter maturity. By the time of the survey,
outstanding volume was likely to have been too small to
affect the results.27
In addition, the timing of the central bank survey
may have caused the survey data to capture patterns in
option strike rates, the mix of bought and sold options, or
maturity that were unique to 1995. The survey in 1995 followed a period of low interest rates in the early 1990s and a
shift to tighter monetary policy in 1994. Data from the
ISDA surveys show that the over-the-counter interest rate
options market grew rapidly in 1993 and 1995. Growth,
however, was lower than usual in 1994. The interest rate
swaps market, by contrast, grew rapidly in 1994, especially
during the first half of the year. Whether these patterns
affected the survey results can best be determined by replicating the study at some future date.28
APPENDIX
ENDNOTES
The author is grateful for the helpful comments and suggestions of Young Ho Eom,
Frank Keane, James Mahoney, and participants in workshops at the Bank for
International Settlements and the Federal Reserve Bank of New York.
1. For additional discussion of the intermediation of price risk, see
Kambhu, Keane, and Benadon (1996).
2. Gennotte and Leland (1990) summarize the debate surrounding the
role of portfolio insurance in the 1987 stock market crash.
3. For further discussion of market liquidity and risk management, see
Bank for International Settlements (1995, Chap. 2).
4. The ISDA data consist of notional amounts but not market values.
As a result, the analysis in this article was not possible until the 1995
central bank survey supplied market value data as well as a breakdown of
gross options positions by bought and sold transactions.
5. The value of an interest rate cap becomes more sensitive to changes
in interest rates as rates rise (see box). In Chart 1, the dealers’ portfolio
value falls at an increasing rate because dealers are net sellers of options
and thus incur increasing option liability as rates rise.
6. This assumption is reasonable when interdealer options are executed
to reallocate customer exposures among dealers, or to take a position in
volatility risk but not directional interest rate risk. For further
explanation, see the hedging assumptions section in the appendix.
7. Turnover volume for U.S. dollar interest rate swaps at the time of the
survey was $17 billion per day (Bank for International Settlements
1996). In contrast, Eurodollar futures turnover volume was $463 billion
per day, and turnover of the five- and ten-year Treasury on-the-run
securities and futures was $29 billion per day.
8. Exchange-traded options on futures contracts are also a potential
hedging instrument. The survey data, however, show that dealers as a
group have bought and sold roughly equal amounts of exchange-traded
options. Thus, these instruments cannot be providing a net hedge to the
aggregate dealer position, and dealers as a group must be relying on other
hedging instruments.
9. In Tables 2, 3, and 4, the hedge for each six-month exposure
requires two back-to-back futures contracts because the contracts are on
a three-month interest rate. For example, in Table 2, in response to a
25-basis-point rise in interest rates, at the two-and-a-half-year
maturity the hedge adjustment comprises a sale of $4.6 billion in each
of the two back-to-back contracts that span the interval between twoand-a-half and three years. This amount is less than the turnover
NOTES
volume of $9.2 billion and $7.5 billion in the two contracts that match
the maturity of the hedge position.
10. On-the-run securities, or the most recently issued securities, are the
most liquid Treasury issues. As a security ages, a larger proportion of the
issue tends to be held in long-term investment portfolios and thus is
traded less frequently.
11. The cash market for the on-the-run Treasury security by itself
appears too small to accommodate dealers’ hedge demands. If dealers
fully hedged their exposures beyond five years using five- and ten-year
on-the-run issues, the required hedge position would be approximately
equal to the outstanding amount of the on-the-run five- and ten-year
notes (Table 5). The Treasury securities market, however, can still
accommodate a significant share of this hedging demand in two ways.
First, the lending of Treasury securities in the repo market allows a fixed
stock of on-the-run Treasury securities to meet trading demands that
exceed the size of the on-the-run issue. Through the repo market, a trader
who sells a borrowed security to establish a short position enables another
trader to establish a long position in the security. As a result, market
participants’ long positions in the security can be significantly larger
than the outstanding stock of the security. Second, off-the-run issues can
be used as long as they are available. Fleming (1997) reports that off-therun securities account for approximately 24 percent of daily turnover in
the interdealer market.
12. Dynamic hedging requires a dealer to buy the underlying asset after
its price rises and to sell it after the price falls. The cost of implementing
this “buy high, sell low” trading strategy is higher when price changes
are more volatile.
13. In the five-year period beginning in 1991, the three largest changes in
implied volatility for one-year options on Eurodollar futures were between
33 percent and 38 percent for two-week changes in implied volatility.
14. See Kambhu, Keane, and Benadon (1996).
15. For a study of how market prices of options are influenced by
volatility and hedging risks, see, for example, Jameson and Wilhelm
(1992) for the pricing of exchange-traded stock options.
16. See Risk (1994) and Swaps Monitor (1996).
17. The interest-rate-related options were predominantly caps and
floors. The central bank survey also included data for over-the-counter
options on traded interest rate securities (bond options). These options were
not included in our analysis, because they amounted to less than 8 percent of
options related to interest rates. Moreover, the bought and sold amounts
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
55
ENDNOTES (Continued)
Note 17 continued
of these options were in rough balance, leaving dealers with very little
residual hedging needs from positions in these options.
18. The notional amount of a derivative contract is a reference amount
used to calculate the size of the cash flows between the counterparties to
the contract. (These cash flows are determined by the price of an
underlying asset.) The market value of a derivative contract is the net
value of the cash flows to be exchanged between the counterparties over
the remaining life of the contract. Notional amounts are a measure of
contract size that is independent of the price of the underlying asset,
while market values are a measure of contract size that is based on the
market value of the transaction. Market values of derivative contracts are
almost always a small proportion of the notional amount.
19. Because of reporting error, the bought and sold amounts of
interdealer options reported in the survey are slightly different. To
account for this reporting error, we average the bought and sold figures
to arrive at the interdealer volume used in the estimation. This averaging
should reduce the effects of the error.
20. The exclusion of swaptions is not likely to alter the article’s
conclusions for the following reasons. If a one-year option on a five-year
swap was reported as a one-year option, then the swaptions would appear
as shorter maturity options in the data. Hence, the true exposures of
shorter maturity would be less than estimated, with the result that
hedging demand for shorter maturity instruments would be smaller than
estimated. This effect would only strengthen the conclusion that shorter
maturity hedging volumes are small relative to transaction volume in
Eurodollar futures. The swaptions, however, would add to the estimated
hedging demand at longer maturities. Nevertheless, because swaptions
make up only 19 percent of the market, the net increment to estimated
hedging demand would not significantly change our conclusions. Rather,
the effect would be to strengthen the conclusion that longer maturity
hedging demand could be significant relative to transaction flows in
longer maturity hedge instruments, but not so much larger as to
overwhelm the market.
22. For further details, see Kambhu (1997).
23. To test whether the clustering at the ten-year maturity was a
product of the quadratic function in the estimation, we derived an
alternative estimate using a linear maturity distribution out to seven
years and a separately estimated ten-year share. This alternative produced
similar results for both the maturity distribution and the hedging
volumes. In a further test, a linear distribution out to nine years with a
separate ten-year share produced nonsensical results with negative values
at the longer maturities in the linear segment of the distribution.
We also derived results with alternative maturity estimates. These
alternatives did not change our conclusions (see Kambhu [1997, Table 7]
for further details). The heavy distribution of notional amounts in the
one-year remaining maturity range, which constrains the effects of the
alternative estimation methods, may explain the robustness of the results.
24. Alternative strike price structures did not produce much variation
in the hedge estimates relative to turnover volume in the hedge
instruments and thus did not affect the conclusions. See Kambhu (1997,
Table 5) for further details.
25. These functions are defined by the strike price specification and the
estimated distribution of notional amounts. See Kambhu (1997) for
further details.
26. In the first three methods, the estimation of interdealer market
values relies on the assumption that the maturity structure of interdealer
options is equal to the average of the bought and sold options’ maturity
distributions.
27. Cumulative volume of IAR swaps originating between 1990 and
1994 was about $100 billion to $150 billion in notional principal
(Galaif 1993-94).
28. Beginning in June 1998, global derivatives market data similar to
the 1995 survey will be collected on a semiannual basis by the Group of
Ten central banks and published by the Bank for International
Settlements.
21. The implied volatility and forward interest rate data are from
Derivatives Week (1995). The Derivatives Week data on forward rates and
implied volatility are consistent with those implied by Eurodollar futures
prices and Eurodollar futures options prices.
56
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
NOTES
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The views expressed in this article are those of the author and do not necessarily reflect the position of the Federal Reserve
Bank of New York or the Federal Reserve System. The Federal Reserve Bank of New York provides no warranty, express or
implied, as to the accuracy, timeliness, completeness, merchantability, or fitness for any particular purpose of any information
contained in documents produced and provided by the Federal Reserve Bank of New York in any form or manner whatsoever.
NOTES
FRBNY ECONOMIC POLICY REVIEW / JUNE 1998
57
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