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THE JUMP COMPONENT OF THE VOLATILITY STRUCTURE OF INTEREST RATE FUTURES MARKETS: AN INTERNATIONAL COMPARISON
THE JUMP COMPONENT OF THE VOLATILITY STRUCTURE
OF INTEREST RATE FUTURES MARKETS:
AN INTERNATIONAL COMPARISON
CARL CHIARELLA* AND THUY-DUONG T
O**
School of Finance and Economics
University of Technology, Sydney
PO Box 123
Broadway NSW 2007
Australia
*Carl.Chiarella@uts.edu.au
**Thuy.D.To@uts.edu.au
A
BSTRACT
. We propose a generalization of the Shirakawa (1991) model to capture
the jump component in xed income markets. The model is formulated under the
Heath, Jarrow and Morton (1992) framework, and allows the presence of a Wiener
noise and a nite number of Poisson noises, each associated with a time deterministic
volatility function. We derive the evolution of the futures price and use this evolution
to estimate the model parameters via the likelihood transformation technique of Duan
(1994). We apply the method to the short term futures contracts traded on CME,
SFE, LIFFE and TIFFE, and nd that each market is characterized by very different
behaviour.
Key words: Term structure; Heath-Jarrow-Morton; Jump-diffusion; FIML; Likeli-
hood transformation; Interest rate futures;
JEL classications: C51, E43, G12, G13
Date: Current revision: Jan 6, 2003.
** Corresponding author
The authors would like to thank helpful comments from participants at the 2002 Quantitative Method in
Finance Conference and the 15th Australasian Finance and Banking Conference. All remaining errors
are our own responsibility.
1 JUMP COMPONENT - INTEREST RATE FUTURES MARKETS
2
1. I
NTRODUCTION
The nancial market is inevitably affected by information arrivals. As argued in
Merton (1976) in relation to the jump-diffusion stock market model, the release of
routine trading information will result in smooth changes in the underlying stock re-
turn process whereas bursts of information are often reected in price behaviour as
discontinuous jumps. The existence of jump components has been documented in the
literature for the stock market and the exchange rate market, for example in Ball and
Torous (1983, 1985), Jorion (1988), and recently in Bates (1996, 2000), Jiang (1999)
and Pan (2002).
The xed-income market is equally substantially impacted by information surprises.
As cited in Das (2002), a number of researchers have found that economic news an-
nouncements and other releases of information have an impact on the Treasury bond
market. Such ndings include those of Hardouvelis (1988), Dwyer and Hafer (1989)
and more recently, Balduzzi, Green and Elton (1998) and Green (1998). Therefore,
jump-diffusion models for interest rates are proposed and analyzed in Babbs and Web-
ber (1995), Naik and Lee (1995), El-Jahel, Lindberg and Perraudin (1997), Piazzesi
(1999) and Das (2002), albeit the specication of the jump models are different. The
general models of Chacko (1998), Chacko and Das (2002) and Singleton (2001) also
contain jumps.
The above-mentioned works have extensively focused on the spot rate of interest. In
this paper, we study jump-diffusion interest rate models under the general framework
of Heath, Jarrow and Morton (1992) (hereafter HJM), which starts with specication
of the dynamics of the instantaneous forward rate, under which the spot rate is one par-
ticular rate, ie. the instantaneous forward rate with zero maturity. The HJM model is
an arbitrage-free model which matches the current term structure by construction, and
relies on no assumption about investor preferences. The model offers a parsimonious
representation of the market dynamics and requires only specication of the form of
the forward rate volatility function.
The HJM model in the form of a diffusion process is difcult to estimate due to its
(in general) non-Markovian nature. When jumps are introduced into the model, the
degree of complexity is magnied. In this paper, we only consider time-deterministic
diffusion volatility processes. Unlike previous studies on jump diffusion of the spot JUMP COMPONENT - INTEREST RATE FUTURES MARKETS
3
rate of interest, where the jump size is assumed to be drawn from a continuous distribu-
tion, we use multiple jump processes, each of which is associated with a constant jump
value scaled by a time-deterministic jump volatility. The advantage of this approach is
that if the jump size is distributed with an innite number of possible realizations (as in
the case of a continuous distribution such as the normal distribution), under the HJM
framework, the market will not be complete (see Bj �ork, Kabanov and Runggaldier
(1997) for a detailed discussion of this issue), and therefore contingent claim prices
are not uniquely dened. Our approach is based on Shirakawas (1991) extension of
the HJM model, in which a Wiener noise is generalized into a Wiener-Poisson noise,
with a nite number of possible jumps. Shirakawa (1991) assumes the existence of a
sufcient number of bonds to hedge away all of the jump risks, and so obtains a unique
arbitrage-free pricing measure. Our specication here is slightly more general in that
it allows the generalized noise to be maturity-dependent.
Bhar, Chiarella and T o (2002) demonstrate that attempts to estimate HJM models
directly from the instantaneous forward rate specication using the short term futures
yield as a proxy would result in non-negligible estimation bias. Therefore, they rst
derive the evolution of the futures prices by treating the futures contract as a derivative
instrument written on the instantaneous forward rate. In this paper we extend the
approach of Bhar, Chiarella and T o (2002) to the HJM model under jump-diffusion
and show that if the underlying market variable follows a jump-diffusion process, then
the futures prices should also experience jump components. We then use this futures
price evolution to estimate our models parameters using Duans (1994) likelihood
transformation method. Under our specication, the likelihood is well dened, details
will be discussed in later sections of the paper.
The theoretical contribution of the paper is, therefore, two-fold. The rst is to de-
rive the evolution of futures prices
1
. This evolution can then be used as an input for the
Bj�ork and Land�en (2002) framework to determine the evolution of options on futures.
2
The second contribution is to utilize this evolution to obtain a method to estimate the
parameters for all HJM models with time-deterministic diffusion and jump volatili-
ties, under the nite jump assumption. This estimation method does not rely on the
1
As far as we are aware, research so far has exclusively focused on bond and option pricing rather than
futures contracts.
2
Bj�ork and Land�en (2002) start with a marked point process for the instantaneous forward rate, and
assert a marked point process for the futures prices without establishing a link between the two sets of
parameters. JUMP COMPONENT - INTEREST RATE FUTURES MARKETS
4
Markovianization of the system comprising the spot rate of interest, and equivalently
the bond price, which may not always be possible.
We then estimate the jump-diffusion HJM model for four major futures markets
around the world, namely the Chicago Mercantile Exchange (CME), the London Inter-
national Financial and Futures Exchange (LIFFE), the Tokyo International Financial
Futures Exchange (TIFFE) and the Sydney Futures Exchange (SFE). The empirical
analysis supports the existence of jump components in the CME futures market, as
predicted by previous studies of jumps utilizing U.S. data. The SFE futures market
also experiences jumps, though the mechanism by which a surprise in information im-
pacts on the Australian market is different from that of the U.S market. On the other
hand, we do not nd empirical support for the existence of jump components in the
U.K market. In the TIFFE market, it seems that the perception of the jump risk is
negligible, since the market price of jump risk is found not to be signicantly different
from zero.
The rest of the paper is organized as follows. Section 2 discusses the class of HJM-
jump models with which we are working. The evolution of futures prices are derived
in Section 3. This evolution is subsequently used to estimate the models parameters
via the likelihood transformation method proposed by Duan (1994), as presented in
Section 4. Data and models considered are described in Section 5. We discuss the
empirical nding in Section 6. Finally, Section 7 concludes the paper. All technical
details are relegated to the Appendices.
2. M
ODEL FRAMEWORK
Consider a nancial market characterized by a probability space
(, F, P). Assume
that the probability space carries a standard Wiener process
W , and M Poisson pro-
cesses
N
1
, N
2
, . . . , N
M
, respectively associated with intensities 1
(t),
2
(t), . . . ,
M
(t).
We further assume that the Poisson processes are independent of each other and of the
Wiener process.
The dynamic evolution of the instantaneous
T -maturity forward rate f (t, T ) (for
t T R
+
) is assumed to be governed by the stochastic differential equation
df (t, T ) = (t, T )dt + (t, T )dW (t) +
M
m=1 m
(t, T )
m
dN
m
(t),
(2.1) JUMP COMPONENT - INTEREST RATE FUTURES MARKETS
5
where
(t, T ) and the (t, T ) are respectively the drift and the Wiener diffusion coef-
cient for the instantan