1

Dynamic Hedge Ratio for Stock Index Futures: Application of Threshold

VECM

Written by Ming-Yuan Leon Li

Applied Economics, (SSCI journal), published onlineJuly, 2007

2

3

Arbitrage Threshold? From a theoretical point of view, the stock index futures,

in the long run, will eliminate the possibility of arbitrage, equaling the spot index

However, plenty of prior studies announced that the index-futures arbitrageurs only enter into the market if the deviation from the equilibrium relationship is sufficiently large to compensate for transaction costs, as well as risk and price premiums

In other words, for speculators to profit, the difference in the futures and spot prices must be large enough to account the associated costs

4

Arbitrage Threshold?

Balke and Formby (1997) serve as one of the first papers to introduce the threshold cointegration model to capture the nonlinear adjustment behaviors of the spot-futures markets.

5

Plenty of Prior Studies Yadav et al. (1994), Martens er al. (1998) and Lin,

Cheng and Hwang(2003) for the spot-futures relationship

Anderson (1997) for the yields of T-Bills Michael et al. (1997) and O’Connell (1998) for the

exchange rates Balke and Wohar (1998) for examining interest rate

parity Obstfeld and Taylor (1997), Baum et al. (2001),

Enders and Falk (1998), Lo and Zivot (2001) as well as Taylor (2001) for examining purchasing power parity

Chung et al. (2005) and Li (2007) for ADRs.

6

Unlike the above Studies…

Adopt a new approach to questions regarding the link between the idea of arbitrage threshold and the establishment of dynamic stock index futures hedge ratio

7

Two Key Elements for Hedge Ratio

Variances Covariance or correlation

8

Nonlinear Approaches for Hedge Ratio

Bivariate GARCH by Baillie and Myers (1991), Kroner and Sultan (1993), Park and Switzer (1995), Gagnon and Lypny (1995, 1997) and Kavussanos and Nomikos (2000)

Chen et al. (2001) adopted mean-GSV (generalized semi-variance) framework

Miffre (2004) employed conditional OLS approach

Alizadeh and Nomikos (2004) using Markov-switching technique.

9

Unlike the above Studies…

Key questions include: Spot and futures prices are more or

less correlated? Volatility/stability of the spot and

futures markets? Design a more efficient hedge ratio? U.S. S&P 500 versus Hungarian BSI

10

Unlike the above Studies… The comparative analysis is meaningful. Briefly, a key feature of the non-linear threshold

model is to capture extreme, rare, large deviations from the futures-spot equilibrium relationship, which cause pronounced effects on the hedging ratio design.

In contrast, traditional linear approaches provide a poor description of extreme events.

11

Unlike the above Studies…

In general, emerging stock markets experience more extreme crisis events compared to mature stock markets

A derivative question is: would the non-linear threshold system show better/poorer performance on picturing emerging/mature stock markets?

12

Unlike the above Studies…

To our knowledge, few studies have studied the aforementioned non-trivial issues relating to research on risk hedging design.

13

The Optimal Hedge Ratio

Hedge ratio that minimizes the variance of spot positions:

FF

SSSF

t

tt

FVar

FSCovHR

)(

),(

14

Establishing Optimal Hedging Ratio via a No-Threshold System

OLS (Ordinary Least Squares)

VECM (Vector Error Correction Model)

15

OLS (Ordinary Least Squares)

OLS (Ordinary Least Squares)

;ttt uFS

HR

FF

SSSF

t

tt

FVar

FSCovHR

)(

),(

16

OLS (Ordinary Least Squares)

Weaknesses of OLS Constant variances and correlations Fail to account for the concept of

cointergration

17

VECM (Vector Error Correction Model)

VECM (Vector Error Correction Model)

tS

q

jjtjSS

p

iitiSFtSSt

tF

q

jjtjFS

p

iitiFFtFFt

uSFZS

uSFZF

,1

,1

,1

,1

,1

,1

Set up the Zt-1 to be (Ft-1-λ0-λ1 S‧ t-1) which represents the one-period-ahead disequilibrium between futures (Ft-1) and spot (St-1) prices

18

VECM (Vector Error Correction Model)

VECM (Vector Error Correction Model)

SS

FF

SF

SF

SS

FF

tS

tF iidu

u

0

0

1

1

0

0,0~

,

,

FF

SSSFHR

19

VECM (Vector Error Correction Model)

Weaknesses of VECM Constant variances and correlations Not consider the idea of arbitrage

threshold

20

Threshold VECM

Threshold VECM

KtS

q

jjt

KjSS

p

iit

KiSFt

KS

KSt

KtF

q

jjt

KjFS

p

iit

KiFFt

KF

KFt

uSFZS

uSFZF

,1

,1,1

,1,1

,1

,1,1

,1,1

KSS

KFF

KSF

KSF

KSS

KFF

KtS

KtF iid

u

u

0

0

1

1

0

0,0~

,

,

Observable State Variable with Discrete Values: K=1, 2, 3…

21

Threshold VECM Threshold VECM with Symmetric

Threshold Parameters

Regime 1 or Central Regime (namely k=1), if |Zt-1| θ≦

Regime 2 or Outer Regime (namely k=2), if |Zt-1|>θ

22

How to estimate the threshold parameter?

(1) Ft is regressed on St and then the observations of equilibrium error, Zt are obtained

(2) a series of arranged error term is established that orders the observations of Zt according to the value of Zt-1, rather than according to time.

(3) by assigning two small numbers to serve as the initial value of θ and –θ, for example 0.005 and -0.005, the series of arranged error terms can be split into two different regime areas: inside/outside the thresholds.

23

(4) the regressions are estimated for each regime area and the residual sum of square RSS is calculated and saves.

(5) the values of θ and –θ are increased using one grid with very small values of 0.0001 and -0.0001, and the above fourth procedure is then repeated for the new values of θ and –θ.

(6) Procedures 4 and 5 are then repeated and the RSS value is derived for each value of θ and choose the value of θ for which the RSS is minimum.

24

This paper uses the values of 2% and 20% percentiles of the error correction term, namely Zt-1 as the boundary values of the threshold parameters.

That is, the observation percentage for the outer regime is allowed to range from 2% to 20%.

For each repeated estimation work with 1,500 daily data, there are 30 (300) observations for the outer regime at least (at most)

25

Threshold VECM

Regime-varying Hedge Ratio

1

111

KFF

KSSK

SFkHR

2

222

KFF

KSSK

SFkHR

26

Threshold VECM The Superiority of Threshold System:

Consider the point of arbitrage threshold Non-constant correlation and volatility A dynamic hedging ratio approach via state-

varying framework Objectively identify the market regime at each

time point (Remember Dummy Variable?) The threshold parameter, namely the θ, could

be estimated by data itself Non-normality problem

27

Why Do We Use State-varying Models?

0.00

0.01

0.02

-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.27 1.02 1.77 2.52 3.27 4.02 4.770.00

0.01

0.02

-5 -4.2 -3.4 -2.6 -1.8 -1 -0.2 0.62 1.42 2.22 3.02 3.82 4.62

0.00

0.01

0.02

-5 -4.2 -3.5 -2.7 -2 -1.2 -0.5 0.27 1.02 1.77 2.52 3.27 4.024.77

x11,x12,x13,x14,..

x21

x22

x23

…

x21

x22

x23

x11,x12, .……………… x13,x14

-----Distribution 2: A high Volatility

Distribution

_____Distribution 1: A Low Volatility

Distribution

---- Distribution 2___ Distribution 1

28

Data

The daily stock index futures and spot U.S. S&P500 Hungary BSI

January 3. 1996 to December 30, 2005 (2610 observations)

All data is obtained from Datastream database.

29

Data

Table 1 Unit Root Tests Cointergration Tests of Stock Index Futures and Spot

U.S. S&P500 Hungarian BSI Futures Spot Future Spot Log levels -2.071 -2.075 -2.975 -3.051 % Returns -13.762* -13.594* -12.080* -11.629* Error Correction Term

-7.4845* -11.1488*

30

Data

Table 2 Summary Statistics of Return Rates of Stock Index Futures and Spot

U.S. S&P 500 Hungarian BSI Futures Spots Futures Spots Mean 0.0349 0.0349 0.0972 0.1001 Skewness coefficient

-0.1307 -0.1098 -0.6321 -0.9031

Minimum value -7.7621 -7.1127 -19.678 -18.034 Maximum value 5.7549 5.5732 18.773 13.616 Variance 1.3064 1.1922 4.0349 3.2608 Kurtosis coefficient

6.9938 6.5531 20.263 16.116

31

Data

The value of kurtosis coefficient is one measure of the fatness of the tails of distribution.

These present results are consistent with the notion that most markets, particularly less developed markets, display more extreme movements than would be predicted by a normal distribution.

32

Horse race via a rolling-estimation process

Arbitrage Threshold and Three Key Parameters of Hedge Ratio

Hedging Effectiveness Comparison of Various Alternatives

33

Horse race via a rolling-estimation process

Horse races with 1,500-day windows in the rolling estimation process

For each date t, we collect 1,500 pre-daily (t-1 to t-1,500) returns of stock index futures and spot, namely to estimate the parameters of various alternatives

Then we use the parameter estimates of each model to establish the out-sample hedge ratio for date t

500,1

1, iitit FS

34

Three Key Parameters for Hedging Ratios

Threshold VECM

KtS

q

jjt

KjSS

p

iit

KiSFt

KS

KSt

KtF

q

jjt

KjFS

p

iit

KiFFt

KF

KFt

uSFZS

uSFZF

,1

,1,1

,1,1

,1

,1,1

,1,1

KSS

KFF

KSF

KSF

KSS

KFF

KtS

KtF iid

u

u

0

0

1

1

0

0,0~

,

,

35

Three Key Parameters for Hedging Ratios

Regime 1 or Central Regime (namely k=1), if |Zt-1| θ≦

Regime 2 or Outer Regime (namely k=2), if |Zt-1|>θ

1

111

KFF

KSSK

SFkHR

2

222

KFF

KSSK

SFkHR

36

Threshold Parameter Estimates,θ

0

0.002

0.004

0.006

0.008

0.01

0.012

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

37

Observation Percentage of Outer

Regime,|Zt-1|>θ

0%

5%

10%

15%

20%

25%

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

38

Correlation Coefficient, ρK

S,F

0.9

0.92

0.94

0.96

0.98

1

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

39

Standard Error of Futures Position, σK

FF

0.008

0.01

0.012

0.014

0.016

0.018

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

40

Standard Error of Spot Position, σK

SS

0.008

0.01

0.012

0.014

0.016

0.018

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

41

Relative Standard Error of Spot to Futures, (σK

SS /σK

FF)

0.8

0.85

0.9

0.95

1

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

FF

SSSFHR

42

Hedge Ratio Estimates, HR

0.8

0.85

0.9

0.95

1

2001/10 2002/2 2002/6 2002/10 2003/2 2003/6 2003/10 2004/2 2004/6 2004/10 2005/2 2005/6 2005/10

FF

SSSFHR

43

Three Key Parameters for HR

U.S. S&P 500 Hungarian BSI

Outer

Regime, k=2

Central

Regime, k=1

Outer

Regime, k=2

Central

Regime, k=1

Correlation Coefficient,

ρkS,F

0.9678 0.9784* 0.5327 0.7238*

Standard Error of Futures

Position, σkFF

0.0140* 0.0129 0.0254* 0.0162

Standard Error of Spot

Position, σkSS

0.0133* 0.0124 0.0229* 0.0171

Relative Standard Error of

Spot to Futures, (σkSS /σ

kFF)

0.9461 0.9637* 0.9539 1.0786*

44

This phenomenon is explained below. According to the present empirical findings, the

outer market regime is associated with a notable arbitrage behavior, namely simultaneous short selling of the spot (future) index and purchase of the future (spot) index when the mispricing term, namely Zt-1, is negative (positive).

The arbitrage behavior clearly causes spot and futures prices to tend to move in opposite directions and thus reduces the scale of co-movement between them

45

This finding is consistent with the notion that arbitrage behavior between the futures and spot markets increases volatility in both markets.

46

Arbitrage trading increases volatility in both futures and spot markets; however, the effects are greater in the futures markets

One explanation for this phenomenon is that the futures market can be considered a superior vehicle compared to the spot markets because of lower trading costs, fewer limitations on short sales and higher leverage effect because of margin trading mechanisms and so on.

47

Three Key Parameters for HR

The setting without arbitrage threshold will…at the “outer” regime Overestimate the correlation Underestimate the volatility Overestimate the Optimal Hedge Ratio

48

Hedging Effectiveness Comparison

For each date t, we use the pre-1,500 daily data to estimate the model parameters and three key parameters of minimum-variance hedge ratio

Next, we establish the minimum-variance hedge ratio for the one-day-after observation

49

Hedging Effectiveness Comparison

The variance (namely, Var) of hedged spot position with index futures can be presented as:

)( tt FHRSVar

50

Hedging Effectiveness Comparison

Table 4 Hedging Effectiveness of Regime-switching Hedge Ratio via Threshold VECM against Alternative No-threshold Models

U.S. S&P 500 Hungarian BSI

Variance Variance

Reduction

Improvement %

Variance Variance

Reduction

Improvement %

Unhedged 1.141207 - 1.592982 -

OLS 0.043041 96.22848% 0.428891 73.0762%

VECM 0.042324# 96.29133%* 0.34013 78.6482%

Threshold

VECM 0.042629 96.2646% 0.306169# 80.78017%*

51

Hedging Effectiveness Comparison

For the case of Hungarian BSI, the threshold systems outperform other alternatives

However, for the case of U.S. S&P 500, the performances of the threshold systems are trivial

52

Why???

The θ estimates 0.0066 for U.S. S&P 500 and 0.0322 for

Hungarian BSI 4.8 (=0.0322/0.0066) times A crisis condition versus an unusual

condition

53

The more/less significant shift in the hedge ratio between two various market regimes is consistent with the finding of crisis/unusual condition in the outer regime for the case of Hungarian BSI/U.S. S&P 500.

Moreover, the above characteristics also provide explanations for remarkable/unremarkable risk hedging effectiveness performance for the threshold VECM in the case of Hungarian BSI/U.S. S&P 500.

54

Why???

Hungarian BSI : HRk=2 is 0.4775 and HRk=1= 0.7825 The difference %=64%

((0.7825-0.4775)/0.4775) U.S. S&P 500

HRk=2 is 0.9158 and HRk=1=0.9430 The difference %=2.96%

((0.9430-0.9158)/0.9158)

55

Conclusions The outer regime will be associated with a

smaller correlations, greater volatilities and a smaller value of the optimal hedge ratio

The outer regime as a crisis (unusual) state for the case of Hungarian BSI (U.S. S&P 500)

The superiority of the threshold VECM in enhancing hedging effectiveness especially for the Hungarian BSI market, but not for U.S. S&P 500 market

56

Other Applications of Threshold Models

Li, Ming-Yuan Leon* (2008) Clarifying interrelation dynamics between option and stock markets using threshold vector error correction model, Mathematics and Computers in Simulation, accepted and forthcoming 【 SCI 】

Li, Ming-Yuan Leon* (2008) Multiple asymmetries in index stock returns from boom/bust and stable/volatile markets states- An empirical study of U.S. and U.K. stock markets, Applied Economics Letters, published online 【 SSCI 】

Li, Ming-Yuan Leon* (2008) Nonlinear interrelations between ADRs and their underlying stocks revisited: Application of threshold VECM, Applied Economics Letters, accepted and forthcoming 【 SSCI 】