Dynamic Hedge Ratio for Stock Index Futures: Application of Threshold VECM Written by Ming-Yuan Leon Li Department of Accountancy Graduate Institute of Finance and Banking National Cheng Kung University, Taiwan July, 2007
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
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.
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) for ADRs.
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
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.
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
The Optimal Hedge Ratio • Hedge ratio that minimizes the variance of spot positions:
Establishing Optimal Hedging Ratio via a No-Threshold System • OLS (Ordinary Least Squares) • VECM (Vector Error Correction Model)
OLS (Ordinary Least Squares) • OLS (Ordinary Least Squares)
OLS (Ordinary Least Squares) • Weaknesses of OLS • Constant variances and correlations • Fail to account for the concept of cointergration
VECM (Vector Error Correction Model) • VECM (Vector Error Correction Model) Set up the Zt-1 to be (Ft-1-λ0-λ1‧St-1) which represents the one-period-ahead disequilibrium between futures (Ft-1) and spot (St-1) prices
VECM (Vector Error Correction Model) • VECM (Vector Error Correction Model)
VECM (Vector Error Correction Model) • Weaknesses of VECM • Constant variances and correlations • Not consider the idea of arbitrage threshold
Threshold VECM Observable State Variable with Discrete Values: K=1, 2, 3… • Threshold VECM
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
Threshold VECM • Regime-varying Hedge Ratio
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
Why Do We Use State-varying Models? _____Distribution 1: A Low Volatility Distribution -----Distribution 2: A high Volatility Distribution x21 x22 x23 … x11,x12,x13,x14,.. ---- Distribution 2 ___ Distribution 1 x21 x22 x23 x11,x12, .……………… x13,x14
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.
Horse race via a rolling-estimation process • Arbitrage Threshold and Three Key Parameters of Hedge Ratio • Hedging Effectiveness Comparison of Various Alternatives
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
Three Key Parameters for Hedging Ratios • Threshold VECM
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
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
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
Hedging Effectiveness Comparison • The variance (namely, Var) of hedged spot position with index futures can be presented as:
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
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 crisiscondition versus an unusualcondition
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)
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