Modeling the FTSE 100 index futures mispricing (Incomplete)

Modeling the FTSE 100 index futures mispricing

(Incomplete)

I. A. Venetis

a,∗

, D. A. Peel

b

and N. Taylor

b

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

• 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

estimate 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 nearnear-est 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 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 (421 × 77) 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.

high 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

z

_t= ln

F

_t,T − ln

F

_t,T∗ denote mispricing where

F

_t,T∗ is the cost-of-carry model theoretical (or fair) stock index futures price observed at time

t

for delivery at time

T

, and

F

t,T is the market price of the futures contract. In order to measure mispricing

z

_tin basis points, (bp, ₁₀₀1

th

of1%) we multiplied

z

_twith 10000. The resulting series is plotted in figure 1.

Figure 1. Mispricing series

z

t(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

transaction 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:

z

t =

φ

0+

φ

1

z

t−1 (1) +(

φ

2+

φ

3

z

t−1)(1 − exp
−

γ

(

z

t−1−

_σ

φ

₂0−

φ

2)2 z  ) +

u

t The transition function,

F

(

γ,φ

₀

,φ

₂) = 1 − exp

 −

γ

(zt−1−φ0−φ2)2 σ2 z 

,

is bounded from zero (when

z

_t−1− (

φ

₀ +

φ

₂) = 0) to unity (when |

z

_t−1 − (

φ

₀ +

φ

₂)| 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

z

t=

φ

0+

φ

1

z

t−1+

u

t (2)

and two outer regimes with dynamics described by

z

t= (

φ

0+

φ

2) + (

φ

1+

φ

3)

z

t−1+

u

t (3) At all other times, coefficients(

φ

₀+

φ

₂

F

) and (

φ

₁+

φ

₃

F

) reflect a mixture of arbitrage activity. Given the magnitude of the distance|

z

t−1−

φ

0−

φ

2| and the variance of the underlying series, the normalized speed of transition parameter

γ

σ2_z 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

φ

₀ = 0

, φ

₁ = 1 whereas when there is full arbitrage time-dependency is measured by

φ

₁+

φ

₃= 1 +

φ

₃ where

φ

₃

<

0 and

φ

₂ = 0

.

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

φ

₂ deserves special attention. Ideally, we would expect

φ

₂ = 0 so that the arbitrage process initiates immediately when

z

_t−1
= 0

.

However, preliminary descriptive statistics implied that the mispricing is positively skewed with arithmetic mean ¯

z

t= 2

.

84 bp,

median

(

z

t) = 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

φ

₀ parameter and setting

φ

₁ = 1 (ˆ

φ

₁ = 0

.

997

with standard error0

.

008) the estimated model is:

z

t=

z

t−1+  1

.

512 [0.426]− 0[0.102]

.

331

z

t−1  (1 − exp
−0

.

105 [0.041](

z

t−1− 1[0.426]

.

512) 2_{) +}

e

t (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

(ˆ

γ,

ˆ

φ

₀

,

ˆ

φ

₂) (vertical axis) versus the mispricing

z

t−1− ˆ

φ

2 (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 (

φ

₁) being unity while the estimated full-arbitrage coefficient

3

It is true that the exponential transition model does not reduce to a threshold model ifγ → ∞. 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.

(1+ˆ

φ

₃= 0

.

667) 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 autocorreladistribu-tions using a Monte Carlo simuladistribu-tion with 500 repetitions. We assume that mispricing follows a random walk in the absence of arbitrage, thus, we set

φ

₁ = 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 autocorrelationˆ

ρ

_j(

z

t) of

z

talong with the averaged auto-correlations¯

ρ

_{j,EST AR}(

z

_t) 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 autocorcor-relation structure of mispricing changes∆

z

t

.

The empirical autocorrelation of ∆

z

talong with the averaged autocorrelations for orders up to 36 are plotted in right panel of figure 3.

Figure 3. (Left diagram) Empirical mispricing autocorrelations and ESTAR produced averaged autocorrelations for ordersj = 1, ..., 36. (Right diagram) Empirical mispricing changes autocorrelations and ESTAR produced averaged

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 variablez_t−1with the conditional expectationE_t−1z_t. Thus, the speed of mean reversion does not depend on the level of mispricing but on the agents expectations at timet −

1

regarding mispricing at timet. Peel and Venetis (2003) showed that in the special case where φ3

=

−φ1we can solve with respect toEt−1ztand estimate4the following model:

zt

=

φ0

+

φ2

+ (

φ1zt−1− φ2

)

(5) ×

exp

−

1

_{2 LambertW}

2

γ

(

φ2− φ1zt−1

)

2 σ2_z 

+

ut where

LambertW(

.

)

is the Lambert’s-W function. The transition function

F

(

γ, φ1, φ2

) = exp

−

1

_{2 LambertW}

2

γ

(

φ2− φ_σ1₂zt−1

)

2 z 

= 1

whenzt−1−

(

φ2/φ1

) = 0

andF

(

.

)

→

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,φ₀was insignificant):

zt

= 2

.

785

[0.442]

+ (

zt−1−

2

[0.442].

785

)

(6) ×

exp

    −

1

2 LambertW

    

2

×

0

[0.003].

036

× 

2

.

785

[0.442]− zt−1 2 σ2 z          

+

et with regression standard errors.e

= 4

.

698

bp. The model estimate ofγ is statis-tically significant. The constantφ₂ was again statistically different from zero and positiveφ

ˆ

₂

= 2

.

785

. Notice that model (5) is theoretically more appealing as it imposes the restrictionφ₃

=

−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-tionF

(ˆ

γ,φ

ˆ

₁,φ

ˆ

₂

)

(vertical axis) versus the mispricingz_t−1−φ

ˆ

₂ (horizontal axis).

4

Details on how to approximate the Lambert’s-W function for estimation purposes can be found in Peel and Venetis (2003).

Figure 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, functionF

(ˆ

γ,φ

ˆ

₁,φ

ˆ

₂

)

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.

Figure 5. (Left diagram) Empirical mispricing autocorrelations and ARB-STR produced averaged autocorrelations for ordersj = 1, ..., 36. (Right diagram) Empirical mispricing changes autocorrelations and ARB-STR produced averaged

As with the ESTAR model, figure 5 (left panel) illustrates the empirical auto-correlationˆρ_j(z_t) of z_talong with the averaged autocorrelations¯ρ_{j,ARB−ST R}(z_t) 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 autocorcor-relation 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. Notwithstandmispric-ing 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, the theoretical price at timet (F_t,T∗ ) of an index futures contract maturing at timeT equals the opportunity cost of keeping a basket replicating the spot index between t and T :

F∗

t.T = Ste(r−d)(T −t) (7)

whereStis 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

model. 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−1exp{vtet} (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

F∗

t = Ptc (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∗htexp{ut} (10) The additional noise termexp{u_t} could arise as futures attract additional traders to the market. The extra volatility termhtreflects a common finding in the litera-ture that fulitera-tures 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 serieszt = ln(Ft/Ft∗) = log(ht) + utcan be seen as a log volatility proxy. If spot index returns follow a typical GARCH model then the conditional upon timet − 1 variance of ln(F_t/F_t−1) is given by v_t2+ E_t−1(∆ log(h_t))2. *** more here ***

Under the preceding assumptions the log theoretical mispricing could exhibit long memory iflog(h_t) 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 forzt which should provide a close approximation to the observed correlation structure.

The ARFIMA(p,d,q) model is written as: (1 − L)d_Φ(L)(z

whereE(u_t) = 0, E(u2_t) = σ2_uandΦ(L) = 1+ p
i=1ϕiL i_{, Θ(L) = 1+}
p i=1θiL i_are lag polynomials of order p,q respectively with roots outside the unit circle. Using the Akaike model selection criterion forp, q = 0, 1, 2, 3 we chose p = 2, q = 0 as the most suitable lag order for the ARMA structure of (11). Estimated parameters (using exact maximum likelihood6) are presented below

(1 − L)[0.016]0.447 1 − 0.415 [0.018]L −[0.007]0.059L2   zt− _[9.663]3.030  = ˆut (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-tocorrelationˆρ_j(zt) of ztalong with the averaged autocorrelations¯ρ_{j,ARF IMA}(zt) of orderj = 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∆z_t(right panel)

6

ARFIMA estimation was performed using the ARFIMA 1.01 package for Ox. See also Ooms and Doornik (1998).

Figure 6. (Left diagram) Empirical mispricing autocorrelations and ARFIMA produced averaged autocorrelations for ordersj = 1, ..., 36. (Right diagram)

Empirical mispricing changes autocorrelations and ARFIMA produced mispricing changes averaged autocorrelations for ordersj = 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

)d

u

t =

e

t (13)

u

t =

z

t−

ϕ

1

z

t−1−

ϕ

2

z

t−2− (

ϕ

∗0+

ϕ

∗1

z

t−1+

ϕ

∗2

z

t−2) ×[1 − exp(−

γ

(

z

t−1−

ϕ

∗0)2

/σ

2Z)]

where the second order short run autoregression structure is reserved. Model pa-rameters

d,ϕ

= (

ϕ

₁

,ϕ

₂)

,ϕ

∗ = (

ϕ

∗₀

,ϕ

∗₁

ϕ

∗₂)

,γ

were estimated using nonlinear least squares (NLS). Defining

e

tas the residuals from applying the FISTAR(p,d,q) filter to

z

_t

,

NLS simply maximizes:

f

(

d,ϕ,ϕ

∗

,γ

_{) = −}1 2log(

T

1 T
t=1

e

2 t) (14)

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 ˆ

d

ϕ

ˆ₁

ϕ

ˆ₂

ϕ

ˆ∗ 0

ϕ

ˆ∗1

ϕ

ˆ∗2

γ

ˆ 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.}

5

Model implications: Impulse response functions

Impulse response functions are meant to provide a measure of the response of

z

t+h to a shock or impulse

u

tat 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

z

_t+hwhich start from identical histories

ω

_t−1In 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

IRF

his given by

IRF

h =

E

[

z

t+h |

u

t=

δ,u

t+1=

...

=

u

t+h= 0

,ω

t−1] (15) −

E

[

z

t+h |

u

t= 0

,u

t+1=

...

=

u

t+h = 0

,ω

t−1]

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

z

_tin (11) to obtain

(1 −

L

)(

z

t−

µ

) =

A

(

L

)

u

t where

A

(

L

) = (1 −

L

)1−dΦ−1(

L

)Θ(

L

) = 1 +
k

i=1

a

j

L

j

, a

_{0 = 1}

.

_{The impact of a}

unit innovation at time

t

on the process at

z

t+k is given by1 + k
i=1

a

jwhere

a

k = k
i=0 Γ(

i

+

d

− 1) Γ(

d

− 1)Γ(

i

+ 1)

ψ

k−i and

ψ

_j are the parameters ofΨ(

L

) = Φ−1(

L

)Θ(

L

) given by

ψ

0 = 1

ψ

j =

θ

j+ min
{j,p} i=1

ϕ

i

ψ

j−1for

j

= 1

,

2

,...,q

ψ

j = min
{j,p} i=1

ϕ

i

ψ

j−1for

j

≥

q

As a result, we obtain

IRF

h = 

1 +
h j=1

a

j



δ

and we observe the following 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.

(1996), provides a natural solution to the problems involved in defining impulse responses in nonlinear models. We will calculate the following

GIRF

_hfunction

GIRF

h =

E

[

z

t+h |

u

t=

δ,U,

Ωt−1] (16)

−

E

[

z

t+h |

u

t= 0

,U,

Ωt−1]

where

U

denotes that we average out the effect of shocks occurring between

t

+ 1 and

t

+

h

andΩ_t−1denotes averaging out the effect of history. That isΩ_t−1 con-sists of all

z

_t−1deviations for

t

≥ 2. Accordingly, we can define

GIRF

_h+

,

(

GIRF

_h−) with Ω+_t−1

,

(Ω−_t−1) the set of histories such that

z

t−1

>

0, (

z

t−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 histories9drawn 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

z

tis 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.

GIRF

+

h

Half lives (in minutes) ESTAR ARB-STR Shock

δ

: +1 8 8 +5 7 7 +15 5 4 +25 3 2 Table 2

GIRF

− h

Half lives (in minutes) ESTAR ARB-STR Shock

δ

: −1 10 11 −5 9 9 −15 7 6 −25 4 4 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

GIRF

_hfunction 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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