Modeling the FTSE 100 index futures mispricing
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Modeling the FTSE 100 index futures mispricing
(Incomplete)
I. A. Venetisa, ∗, D. A. Peelb and N. Taylorb
a Centre of Planning and Economic Research (KEPE), Hippokratous 22, Athens 106 80, Greece
b Cardiff Business School, Aberconway Building, Colum Drive, Cardiff, United Kingdom, CF103EU.
February 20, 2004
Abstract
The present paper examines empirically in a time series perspective how
well certain types of nonlinear models as well as a linear long memory model
match the observed correlation function of FTSE 100 index futures mispric-
ing.
∗ Corresponding author. E-mail: ivenetis@kepe.gr, tel:+30-210-3676423 Pre-
liminary versions of this work were presented at seminars in the Department of Accounting & Fi-
nance at the Athens University of Economics and Business and in the Department of Economics at
the University of Crete.
1• Purpose: model the correlation structure of mispricing and mispricing changes.
Garrett and Taylor (2001) successfully model the first order autocorrelation
in FTSE 100 futures index mispricing using a threshold model. They show
that microstructure effects are not responsible for the observed first order cor-
relation implying arbitrage induced persistence in mispricing. The threshold
model assumes all investors have the same trading costs and market con-
straints and that all investors exploit arbitrage opportunities simultaneously
whenever the mispricing exceeds the threshold.
• We allow heterogeneity in investors due to differing transaction costs, objec-
tives, perceived risks, capital constraints etc. as in Tse (2001). The speed
of adjustment towards “equilibrium” mispricing varies directly with the mis-
pricing itself. Heterogeneity among investors suggests that the proportion of
investors who would capitalize on the mispricing gradually increases with
the extent of mispricing. This results in a smooth transition between regimes
with strong aggregate adjustment and those with weak or no adjustment.
Tse (2001) provides a comprehensive justification for the existence of het-
erogeneity among agents participating in intraday index arbitrage of the Dow
Jones Industrial Average (DJIA).
• Given preceding considerations, we use the Granger and Terasvirta (1993)
ESTAR model to describe mispricing (van Dijk et al., 2000 provide a rig-
orous survey on recent developments in smooth transition models) and a
variation the ARB-STR model of Peel and Venetis (2004). The nonlinear
models produce supportive evidence for the arbitrage induced nonlinearity
in the mispricing dynamics, nevertheless do not adequately account for ob-
served correlation
• The next step is to assume an ARFIMA model that could account for the
empirical correlation structure. Indeed, results are satisfactory, nevertheless
the linearity of the model has some inherent weaknesses. Final step is to
estimate a nonlinear long memory model. FISTAR.
• Implications: ARFIMA model produces impulse responses to shocks with
realistic half life of 4 minutes. Of course due to linearity different speeds are
not allowed. Combination of long memory and smooth transition seems to
be the required model with half life varying from **** to ****
1 Introduction
We apply the nonlinear exponential smooth transition model of Granger and Terasvirta
(1993) and Terasvirta (1994) and a recently proposed nonlinear model by Peel and
Venetis (2003) in an attempt to model the empirical correlation structure of the
FTSE 100 futures index mispricing. Data driven considerations also forced us to
2estimate a linear parametric long memory model. We found that although non-
linear models are theoretically appealing and provide fast responses to mispricing
shocks, they are not capable to replicate the observe correlation structure.
2 Data
We analyze the relationship between spot and futures prices using intraday data.
The futures price of the nearest FTSE 100 contract is obtained for every transac-
tion carried out between January 5 and April 24, 1998. These data were obtained
from LIFFE. The contract is changed when the volume of trading in the next near-
est contract is greater than the volume of trading in the nearest contract.1 To syn-
chronize the futures and spot prices, the futures price series is converted to a price
series with a frequency of one minute. The (spot) level of the FTSE 100 index
was obtained from FTSE International. The trading hours of the futures market
and the spot market are, 8.35am to 4.10pm and 8.00am to 4.30pm, respectively.2
Thus one can obtain overlapping futures and spot data covering the period, 8.35am
to 4.10pm. We only consider the period between 9.00am and 4.00pm a) to avoid
any contaminating effects from factors such as stocks going ex div overnight on
421 × 77
opening prices and b) to enable an hour-by-hour analysis of arbitrage behavior. A
total of 77 trading days are considered. This gives a total of 32,417 ( ) one
minute frequency observations.
The validity of the constructed mispricing series relies heavily on the use of
appropriate ex ante dividends and interest rates. To this end we make use of data
supplied by Goldman Sachs. These data are used by arbitragers employed by Gold-
man Sachs when making judgements about the mispricing (or otherwise) of FTSE
100 futures contracts. Goldman Sachs construct ex ante dividends by making in-
dividual forecasts for each of the dividends paid by companies in the FTSE 100
index and then weight these by market capitalization. The interest rate applica-
ble over the contract life used by Goldman Sachs is the interpolated LIBOR rate.
For instance, if a 25 day interest rate is required then Goldman Sachs interpolate
between the two week and the four week rates.
Following Garrett and Taylor (2001) we will base our analysis on the afternoon
period (12:01 P.M. to 4:00 P.M.) with a total of 18480 observations. The significant
negative autocorrelation for the first few trading hours in the mispricing series is
most likely the result of the high bid-ask spreads observed during the beginning of
trading in the spot market. For instance, Naik and Yadav (1999) found evidence of
1
The volume cross-over method of changing futures contracts results in one change. The change
involves a switch from the March 1998 contract to the June 1998 contract on March 11, 1998. On
this day the volume of trading in the March contract was 6,312 contracts and the volume of trading
in the June contract was 13,355 contracts.
2
The futures market re-opens at 4.32pm under the Automated Pit Trading (APT) system. How-
ever, this additional period of trading is not considered because of the lack of data between 4.11pm
to 4.31pm.
3high bid-ask spreads on all stocks making up the FTSE 100 index during the first
few hours of the trading day. These high spreads are most likely the result of a lack
of liquidity during this period. Shah (1999) argued that the lack of liquidity during
the early morning period may be due to the absence of an opening auction facility
for market on open orders.
3 Alternative models of arbitrage behavior
3.1 The exponential smooth transition model
Let zt = ln Ft,T − ln Ft,T
∗ denote mispricing where F ∗ is the cost-of-carry model
t,T
theoretical (or fair) stock index futures price observed at time t for delivery at
time T , and Ft,T is the market price of the futures contract. In order to measure
mispricing zt in basis points, (bp, 1001
1%)
th of we multiplied zt with 10000. The
resulting series is plotted in figure 1.
Figure 1. Mispricing series zt (12:01 - 16:00) data, 18480 observations.
Recent literature on arbitrage activity has employed smooth transition autore-
gressive (STAR) models (see Anderson, 1997, and Taylor, van Dijk, Franses and
Lucas, 2000) to model mispricing dynamics. These models allow for a continuum
of regimes through a smooth parametric function and they allow heterogeneous
4transaction costs. Depending on the form the function takes the most popular mod-
els have been the logistic STAR (LSTAR) and the exponential STAR (ESTAR).
The threshold autoregressive (TAR) framework has also been employed success-
fully by Garret and Taylor (2001) in an attempt to identify the cause of the observed
negative first order autocorrelation in mispricing. Nevertheless, the model imposes
homogeneous transaction costs and it will not be further considered unless there is
favorable statistical evidence.
The following version of the exponential STAR model (see Terasvirta, 1994,
for further details) will be considered:
zt = φ0 + φ1zt−1 (z − φ − φ )2 (1)
+(φ2 + φ3zt−1)(1 − exp −γ t−1 σ20 2 ) + ut
z
−
The transition function, F (γ, φ0 , φ2 ) = 1 exp γ φ φ
σ2z − z
( t−1 − 0 − 2 )2
, is bounded
−
from zero (when zt−1 (φ0 + φ2) = 0) to unity (when zt−1 | − (φ0 + φ2)| is
large). The higher the level of mispricing in the previous period, the higher the
value of the transition function. In the limits, model (1) describes three regimes.
The mid-regime where F . ()=0 corresponds to
zt = φ + φ zt− + ut
0 1 1 (2)
and two outer regimes with dynamics described by
zt = (φ0 + φ2) + (φ1 + φ3)zt−1 + ut (3)
( + ) ( +
At all other times, coefficients φ0 φ2 F and φ1 φ3 F refl ect a mixture of)
| − − |
arbitrage activity. Given the magnitude of the distance zt−1 φ0 φ2 and the
variance of the underlying series, the normalized speed of transition parameter
γ
σ2z is unit free and expresses how fast the mispricing is “covered” by the market.
Ideally, when there is no arbitrage, mispricing would follow a driftless random
walk with φ0 =0 =1
, φ1 whereas when there is full arbitrage time-dependency
+ = 1+
is measured by φ1 φ3 0 =0
φ3 where φ3 < and φ2 . That is, the arbitrage
process forces mispricing towards equilibrium thus in the outer regimes the process
is not allowed to wonder following a stationary autoregressive model. Parameter φ2
deserves special attention. Ideally, we would expect φ2 =0
so that the arbitrage
process initiates immediately when zt−1 = 0
. However, preliminary descriptive
statistics implied that the mispricing is positively skewed with arithmetic mean
= 2 84
z̄t . bp, median zt ( ) = 2 67
. bp and skewness equal to 0.019.
The specification given in (1) is estimated using the level of mispricing ob-
served between 12.01pm and 4.00pm on each trading day producing 18480 obser-
vations. Ignoring the insignificant φ0 parameter and setting φ1 (φ̂1 =1 . = 0 997
5with standard error 0.008) the estimated model is:
zt = zt−1 + 1[0..512
426]
− 0[0..331
102]
zt−1 (1 − exp −0..105(zt − 1..512) ) + et
[0 041]
−1
[0 426]
2
(4)
with regression standard error s.e = 4 698
. bp. Heteroscedastic consistent standard
errors are reported in squared brackets. Figure 2 illustrates a scatter diagram of
the transition function F (γ̂, φ̂0 , φ̂2 (vertical axis) versus the mispricing zt−1 φ̂2
) −
(horizontal axis).
Figure 2. Estimated transition function vs mispricing for the ESTAR model.
The first point to note is that γ is significantly different from zero, suggesting
that the smooth transition model is an appropriate way of modeling mispricing.
The linear autoregressive model (γ =0) or the threshold model3 (γ →∞
) are not
supported. The autoregressive parameters take on reasonable values with the no-
arbitrage coefficient (φ1) being unity while the estimated full-arbitrage coefficient
3
It is true that the exponential transition model does not reduce to a threshold model if γ→∞.
However, estimation of a logistic STAR of order 2 that closely approximates ESTAR and encom-
passes TAR as a limit case was also performed. The results, that are available upon request, were
qualitatively similar to those reported from the ESTAR specification.
61+ = 0 667
( φ̂3 . ) is positive and relatively small suggesting rather weak persistence.
The half life of shocks given that the level of mispricing is such as to justify action
by all potential arbitragers is implied to be only 2 minutes.
We use the estimated parameters from equation (4) to generate the distribu-
tion of the first 36-order autocorrelations using a Monte Carlo simulation with 500
repetitions. We assume that mispricing follows a random walk in the absence of
arbitrage, thus, we set φ1 =1 . The number of observations used in each repetition
is 18,480 and the errors are drawn from a NID , . (0 4 698)distribution. Figure 3
( )
illustrates the empirical autocorrelation ρ̂j zt of zt along with the averaged auto-
( )
correlations ρ̄j,EST AR zt of order j =1 36
, ..., produced from the simulation of
model (4). It is apparent that the ESTAR model cannot replicate the empirical cor-
relation structure of mispricing. The same is true for the autocorrelation structure
∆ ∆
of mispricing changes zt . The empirical autocorrelation of zt along with the
averaged autocorrelations for orders up to 36 are plotted in right panel of figure 3.
7Figure 3. (Left diagram) Empirical mispricing autocorrelations and ESTAR
produced averaged autocorrelations for orders j = 1, ..., 36. (Right diagram)
Empirical mispricing changes autocorrelations and ESTAR produced averaged
mispricing changes autocorrelations for orders j = 1, ..., 36.3.2 The arbitrage consistent smooth transition model
In this section we model mispricing using the arbitrage consistent smooth transition
model (ARB-STR) of Peel and Venetis (2003). The ARB-STR model arises as
a special case of model (1) when we replace the transition variable zt−1 with the
conditional expectation Et−1 zt . Thus, the speed of mean reversion does not depend
on the level of mispricing but on the agents expectations at time t 1 regarding −
mispricing at time t. Peel and Venetis (2003) showed that in the special case where
−
φ3 = φ1 we can solve with respect to Et−1 zt and estimate4 the following model:
z
t = φ0 + φ2 + (φ1 z −1 − φ2 )
t
(5)
1 (φ2 − φ1 z −1 )2
× exp − 2 LambertW 2γ σ2 z
t
+u t
where LambertW(.) is the Lambert’s-W function. The transition function
1 (φ − φ1 z −1 )2
F (γ, φ1, φ2) = exp − LambertW 2γ 2 t
=1
2 σ2 z
−
when zt−1 (φ2/φ1 ) = 0 and F (.) →
0 as mispricing increases. Using the level
of mispricing observed between 12.01pm and 4.00pm on each trading day with
zt measured in basis points, we obtained the following results (heteroscedastic
consistent standard errors are reported in squared brackets, φ0 was insignificant):
z t = 2.785 + (z −1 − 2.785)
t (6)
[0.442]
[0.442]
2
2.785 − z −1
1
× exp − 2 LambertW 2 × 0[0.036 ×
+e
t
[0 442]
.
003] . σ2 z
t
with regression standard error s.e = 4.698 bp. The model estimate of γ is statis-
tically significant. The constant φ2 was again statistically different from zero and
positive φ̂2 = 2.785. Notice that model (5) is theoretically more appealing as it
−
imposes the restriction φ3 = φ1 before estimating the transition speed. Ideally,
if the expectations about the one period ahead mispricing are large enough then
arbitragers will completely eliminate the opportunity within one time period and
mispricing will not follow a stationary autoregressive model. At the limit it reduces
to a pure noise process. Figure 4 illustrates a scatter diagram of the transition func-
−
tion F (γ̂, φ̂1, φ̂2) (vertical axis) versus the mispricing zt−1 φ̂2 (horizontal axis).
4
Details on how to approximate the Lambert’s-W function for estimation purposes can be found
in Peel and Venetis (2003).
9Figure 4. Estimated transition function vs mispricing for the ARB-STR model.
Given that the estimated function represents the autoregressive coefficient mag-
nitude, figure 4 suggests that the ARB-STR model would not exhibit significant dif-
ferences from the ESTAR model. Based on the sample data, function F (γ̂, φ̂1 , φ̂2 )
is restricted in the range 1 to 0.62 which is quite close to the range of the autore-
gressive coefficient implied by the ESTAR model.
10Figure 5. (Left diagram) Empirical mispricing autocorrelations and ARB-STR
produced averaged autocorrelations for orders j = 1, ..., 36. (Right diagram)
Empirical mispricing changes autocorrelations and ARB-STR produced averaged
mispricing changes autocorrelations for orders j = 1, ..., 36.As with the ESTAR model, figure 5 (left panel) illustrates the empirical auto-
( )
correlation ρ̂j zt of zt along with the averaged autocorrelations ρ̄j,ARB −STR zt( )
of order j = 1 36, ..., produced from the simulation of model (6). Again, the
nonlinear formulation of the ARB-STR model cannot replicate the empirical cor-
relation structure of mispricing. The same is true for the autocorrelation structure
∆
of mispricing changes zt (right panel).
4 A long memory model
Let us abstract (but not abandon) for the moment from the heterogeneous costs
assumption and assume a long memory linear model for mispricing and mispric-
ing changes. Notwithstanding the fact that the assumption is data driven since, for
example a long memory ARFIMA model should be capable of replicating the ob-
served (slowly decaying) autocorrelation function, there is also a theoretical base
for such an assumption. Liu (2000) inspired by the idea that regime switching
may give rise to persistence that is observationally equivalent to a unit root and
derived a regime switching process exhibiting long memory. Davidson and Sib-
bertsen (2003) further elaborate on the model by adding a stochastic component.
They show that a process with regime switching in the conditional mean such that
the regimes’ duration is distributed in a heavy tailed fashion admits an autocorre-
lation function resembling that of a long memory process (under appropriate as-
sumptions). The feature that generates long memory is the heavy-tailed duration
distribution. Liu (2000) put this in a financial markets frame by arguing that the ar-
rival of major news triggers volatility jumps or switches in stock market volatility.
In particular, when different news arrive at the market in a heavy-tail fashion, we
observe long memory in stock market volatility.
According to the cost-of-carry valuation (the standard forward pricing model),
which assumes perfect markets and non-stochastic interest rates and dividend yields,
∗ ) of an index futures contract maturing at time T
the theoretical price at time t (Ft,T
equals the opportunity cost of keeping a basket replicating the spot index between
t and T :
∗
Ft.T = Ste r−d T −t
( )( )
(7)
where St is the index value and (r − d) is the net cost of carry associated to the un-
derlying stocks in the index, i.e., the riskless rate of return minus the dividend
yield of the stocks in the index. Under the previous assumptions, the cost-of-
carry model implies (among others) that the variance of returns in the spot market
equals the variance of returns in the futures market. However, in the presence of
market imperfections such as transactions costs, asymmetric information, capital
requirements and short-selling restrictions, there could be discrepancies between
the traded futures price and its theoretical valuation according to the cost-of-carry
12model. Indeed, there is a wealth of studies showing systematic discrepancies be-
tween the traded futures price and its theoretical price according to the cost-of-
carry valuation. (see Mackinlay and Ramaswamy, 1988; Lim, 1992; Miller et al.,
1994; Yadav and Pope, 1990, 1994; Buhler and Kempf, 1995; among others).
Following Lafuente and Novales (2003) we can model such discrepancy by
introducing a noise component specific to the derivative market. Let stock prices
evolve according to
Pt = Pt−1 exp{vt et } (8)
where the disturbance term et is i.i.d with zero mean and unit variance and vt
expresses the positive volatility process. Following the cost-of-carry model the fair
futures price is given by
Ft∗ = Pt c (9)
( )
where for simplicity we suppress the t, T notation and c denotes a positive con-
stant. Our crucial assumption is related to the empirical modelling of the observed
futures price. We assume that
Ft = Ft∗ht exp{ut} (10)
The additional noise term exp
{ut } could arise as futures attract additional traders
to the market. The extra volatility term ht reflects a common finding in the litera-
ture that futures return volatility exceeds that of the cash at all times5 . A number of
explanations have been offered for this phenomenon. For example, the difference
to market microstructure and infrequent or non-trading effects or the lower transac-
tion costs in the futures markets which makes them simply more sensitive to news
(new information is incorporated in futures prices first). Thus, the log theoretical
mispricing series zt = ln(
Ft /Ft∗ ) = log( ) +
ht ut can be seen as a log volatility
proxy. If spot index returns follow a typical GARCH model then the conditional
1 ln( )
upon time t − variance of Ft /Ft−1 is given by vt2 Et−1 + (∆ log( ))
ht 2. ***
more here ***
Under the preceding assumptions the log theoretical mispricing could exhibit
long memory if log( )
ht follows an ARFIMA model (for example see Bollerslev
and Mikkelsen, 1996) or we could adopt the Liu (2000) approach with stochastic
volatility and regime switching. Under both modelling alternatives, an elaborate
specification and estimation procedure needs to be followed something that ex-
ceeds the scope of the present paper but surely remains open for future research.
Instead we will proceed and estimate a long memory model for zt which should
provide a close approximation to the observed correlation structure.
The ARFIMA(p,d,q) model is written as:
(1 − L)dΦ(L)(zt − µ) = Θ(L)ut (11)
5
See Abhyankar et al. 1999 for recent findings and references.
13( ) = 0, E(ut ) = σu and Φ(L) = 1+ ϕiLi, Θ(L) = 1+ θiLi are
p p
where E ut 2 2
i=1 i=1
lag polynomials of order p,q respectively with roots outside the unit circle. Using
the Akaike model selection criterion for p, q =0 1 2 3
, , , we chose p ,q as =2 =0
the most suitable lag order for the ARMA structure of (11). Estimated parameters
(using exact maximum likelihood6 ) are presented below
(1 − L) 1− . L − [00..059 zt − [93..030 = ût
0. 447
0 415
[0. 016]
.
[0 018] 007]
L2 663]
(12)
with regression standard error s.e = 4 687
. . Standard errors appear in squared
brackets. The constant term is statistically insignificant.
As with the nonlinear models, figure 6 (left panel) illustrates the empirical au-
( )
tocorrelation ρ̂j zt of zt along with the averaged autocorrelations ρ̄j,ARFIMA zt ( )
of order j = 1 36
, ..., produced from the simulation of model (11). Obviously,
the estimated model can emulate successfully the empirical correlation structure
of mispricing in the FTSE 100 stock index futures. The same is also true for the
∆
autocorrelation structure of mispricing changes zt (right panel)
6
ARFIMA estimation was performed using the ARFIMA 1.01 package for Ox. See also Ooms
and Doornik (1998).
14Figure 6. (Left diagram) Empirical mispricing autocorrelations and ARFIMA produced averaged autocorrelations for orders j = 1, ..., 36. (Right diagram) Empirical mispricing changes autocorrelations and ARFIMA produced mispricing changes averaged autocorrelations for orders j = 1, ..., 36.
Empirically, of course, a number of processes can emulate long memory be-
havior. A modelling view could be that futures price follows a random walk plus
noise model. Notice that preliminary simulation results showed that if the noise
standard deviation is large enough (around twenty times the deviation of the un-
derlying noise in the random walk component) then the random walk plus noise
model produces fractional d estimates near the region of d = 0.4. Of course the
magnitude of d is inversely related to the noise variance magnitude. In such cases,
we found that a simple state space representation and estimation of the random
walk plus noise model can successfully identify the generating model. Thus, as an
empirical exercise we estimated a random walk plus noise model for the mispricing
series. The results were not supportive for the assumed model. The large variance
noise component was simply not present7 .
Model (11) as it stands has an inherent weakness namely it does not take into
account the possibility of nonlinearity with respect to past mispricing levels. Al-
though it can originate through a stochastic volatility model, arbitrage still could
take place if mispricing wonders sufficiently far from zero. That process alone
should be able to “press” mispricing towards its theoretical fair value of zero. For
this reason, we proceed to a combination of models (1) and (11) and we estimate a
model similar to the fractionally integrated smooth transition autoregressive model
of van Dijk, Franses and Paap (2000). The new FISTAR model is written as
(1 − L) u = e
d
t t (13)
u = z − ϕ1z −1 − ϕ2z −2 − (ϕ∗0 + ϕ∗1z −1 + ϕ∗2z −2)
t t t t t t
×[1 − exp(−γ(z −1 − ϕ∗0)2/σ2 )] t Z
where the second order short run autoregression structure is reserved. Model pa-
rameters d, ϕ = (ϕ1 , ϕ2 ) , ϕ∗ = (ϕ∗0 , ϕ∗1 ϕ∗2 ) , γ were estimated using nonlinear
least squares (NLS). Defining et as the residuals from applying the FISTAR(p,d,q)
filter to zt , NLS simply maximizes:
f (d, ϕ, ϕ∗, γ) = − 21 log( T1
e2)
T
t
(14)
=1
t
The variance - covariance matrix estimate is the inverse of minus the numerical
second derivative of (14). Our results are presented below:
FISTAR. Model (13) NLS estimates and standard errors in paren-
theses
dˆ ϕ̂1 ϕ̂2 ϕ̂∗0 ϕ̂∗1 ϕ̂∗2 γ̂
0.392 0.643 0.012 1.706 -0.433 0.185 0.209
(0.017) (0.023) (0.009) (0.241) (0.027) (0.026) (0.0239)
Regression standard error: 4.644
*************
7
These results are available upon request.
165 Model implications: Impulse response functions
Impulse response functions are meant to provide a measure of the response of
zt+h to a shock or impulse ut at time t. The impulse response measure which is
commonly used in the analysis of linear models is defined as the difference between
two realizations of zt+h which start from identical histories ωt−1 In one realization,
the process is hit by a shock of size δ at time t, while in the other realization (the
benchmark profile) no shock occurs at time t. All shocks in intermediate periods
are set equal to zero in both realizations. Hence, the traditional impulse response
function IRFh is given by
IRF = E [z + | u = δ, u +1 = ... = u + = 0, ω −1]
h t h t t t h t (15)
−E[z + | u = 0, u +1 = ... = u + = 0, ω −1]
t h t t t h t
This traditional impulse response function has some characteristic properties in
the case where the underlying model is linear. For the ARFIMA model the impulse
response weights are defined by first differencing zt in (11) to obtain
(1 − L)(z − µ) = A(L)u t t
where A(L) = (1 − L)1−d Φ−1 (L)Θ(L) = 1 +
a L , a0 = 1. The impact of a
k
j
j
unit innovation at time t on the process at zt+k
=1i
is given by 1 +
a where k
j
=1
i
a =
Γ(i + d − 1) ψ
k
=0 d − 1)Γ(i + 1)
Γ(
k k −i
i
and ψj are the parameters of Ψ(L) = Φ−1 (L)Θ(L) given by
ψ0 = 1
ψ = θ +
j j
min{j,p}
ϕ ψ −1 for j = 1, 2, ..., q
i j
=1
i
min{j,p}
ψ = j
ϕ ψ −1 for j ≥ q
i j
=1
i
As a result, we obtain IRFh = 1 + a δ and we observe the following
h
j
j =1
properties: First, the impulse response is symmetric in the sense that a shock of
−δ has exactly the opposite effect as a shock of size +δ. Second, it is linear in the
sense that response is proportional to the size of the shock δ . Third, the response
is history independent. These properties do not carry over to nonlinear models.
The Generalized Impulse Response Function, GIRFh , introduced by Koop et al.
17(1996), provides a natural solution to the problems involved in defining impulse
responses in nonlinear models. We will calculate the following GIRFh function
GIRF = E[z + | u = δ, U, Ω −1]
h t h t t (16)
−E[z + | u = 0, U, Ω −1]
t h t t
where U denotes that we average out the effect of shocks occurring between t +
1 and t + h and Ωt−1 denotes averaging out the effect of history. That is Ωt−1 con-
sists of all zt−1 deviations for t ≥ 2. Accordingly, we can define GIRFh+ , (GIRFh− )
with Ω+ t− 1
, (Ω−t−1) the set of histories such that zt−1 > 0, (zt−1 > 0) and the
initial shock set to δ, −δ respectively. In that case we are interested in possible
response speed differences when positive (negative) mispricing levels are hit by
positive (negative) shocks. Notice that for each available history we use 500 rep-
etitions8 (draws) to average out future shocks where future shocks are drawn with
replacement from the model’s residuals. Then we average the result across 1000
histories9 drawn randomly from all available histories using a random vector index
uniformly distributed. Without loss of generality, the impulse response horizon is
set to max{h} = 20 minutes in the future. We set δ = {±1, ±5, ±15, ±25, }. The
preceding choice about δ would allow us to compare and contrast the persistence
of large and small shocks. Given that zt is measured in basis points the level of
shocks corresponds to δ basis points. As in Taylor et al., (2001) and in Peel and
Venetis (2002a,b), we will report the half-lives of shocks, that is the time needed
for the impulse response function at horizon h to be less than 12 δ . Our empirical
assessment on the propagation of shocks is presented in tables 1, 2 and 3 below:
GIRF h
Half lives (in minutes)
ARFIMA ESTAR ARB-STR
Shock δ :1 4 9 9
5 4 8 8
15 4 6 6
25 4 4 4
Table 1
8
The repetition number 500 was arbitrary chosen as high enoung for the Law of Large Numbers
to produce results virtually identical to that which would result from calculating the exact response
functions analytically by multiple integration. We found out that the difference of using 5000 repeti-
tions was qualitatively unimportant and time consuming.
9
We found out that the difference between using 1000 and all histories (18479 in the case of
GIRFh ) produced qualitatively similar results whilst it was extremely time consuming.
18GIRF + h
GIRF − h
Half lives (in minutes) Half lives (in minutes)
ESTAR ARB-STR ESTAR ARB-STR
Shock δ : +1 8 8 Shock δ : −1 10 11
+5 7 7 −5 9 9
+15 5 4 −15 7 6
+25 3 2 −25 4 4
Table 2 Table 3
Table 1 shows that the half life of shocks in the ARFIMA model is 4 minutes
and is apparently faster than the half lives implied by the nonlinear models when
mispricing is close to zero. Nevertheless, the linear long memory model does not
allow for heterogeneous arbitrage costs and, based on our estimates, shocks vanish
(have a half life of) in four minutes irrespective to their magnitude, for example
the same speed is imposed for 1 bp and 25 bp shocks. Of course, nonlinearity
allows for varying speeds of mean reversion, thus, a 25 basis points shock has
a half life of 4 minutes in both the ESTAR and ARB-STR models. We see that
the assumption of (linear) long memory is not far fetched in the sense that half
lives are realistic and similar to the ones produced by nonlinearity that is arbitrage
induced. Tables 2 and 3 summarize the results for positive and negative shocks.
Clearly the half lives of positive shocks are smaller than the corresponding ones
for negative mispricing shocks. This is consistent with the existence of restrictions
in short selling the index. In the presence of short-selling restrictions, per unit
negative mispricing (which involves short selling the index) is less profitable than
per unit positive mispricing. This results in asymmetric mean reversion speeds
with negative shocks fading more slowly than positive shocks.
Finally table 4 presents the GIRFh function for the estimated FISTAR model
(13).
TABLE 4 HERE
6 Conclusion
By no means we claim that our analysis is either exhaustive or conclusive on the
empirical modelling of mispricing or that it theoretically pinpoints the behavior
of mispricing. Nevertheless our specification assumptions although based on con-
jectures seem to be supported by the empirical modelling. The no-infinite profit
opportunities are not forbidden under a long memory model of mispricing that
treats the differences of fair and spot index futures prices as a volatility proxy.
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