1. 1 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

2. 2 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

3. 3 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.

4. 4 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.

5. 5 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

6. 6 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.

7. 7 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

8. 8 The Optimal Hedge Ratio Hedge ratio that minimizes the variance of spot positions:

9. 9 Establishing Optimal Hedging Ratio via a No-Threshold System OLS (Ordinary Least Squares) VECM (Vector Error Correction Model)

10. 10 OLS (Ordinary Least Squares)

11. 11 OLS (Ordinary Least Squares) Weaknesses of OLS Constant variances and correlations Fail to account for the concept of cointergration

12. 12 VECM (Vector Error Correction Model) Set up the Z t-1 to be (F t-1 -λ 0 -λ 1 ‧ S t-1 ) which represents the one-period-ahead disequilibrium between futures (F t-1 ) and spot (S t-1 ) prices

13. 13 VECM (Vector Error Correction Model)

14. 14 VECM (Vector Error Correction Model) Weaknesses of VECM Constant variances and correlations Not consider the idea of arbitrage threshold

15. 15 Threshold VECM Observable State Variable with Discrete Values: K=1, 2, 3…

16. 16 Threshold VECM Threshold VECM with Symmetric Threshold Parameters Regime 1 or Central Regime (namely k=1), if |Z t-1 | ≦ θ Regime 2 or Outer Regime (namely k=2), if |Z t-1 |>θ

17. 17 Threshold VECM Regime-varying Hedge Ratio

18. 18 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

19. 19 Why Do We Use State- varying Models? x 11,x 12,x 13,x 14,.. x 21 x 22 x 23 … x 21 x 22 x 23 x 11,x 12,.……………… x 13,x 14 -----Distribution 2: A high Volatility Distribution _____ Distribution 1: A Low Volatility Distribution ---- Distribution 2 ___ Distribution 1

20. 20 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.

21. 21 Data

22. 22 Data

23. 23 Horse race via a rolling- estimation process Arbitrage Threshold and Three Key Parameters of Hedge Ratio Hedging Effectiveness Comparison of Various Alternatives

24. 24 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

25. 25 Three Key Parameters for Hedging Ratios Threshold VECM

26. 26 Three Key Parameters for Hedging Ratios Regime 1 or Central Regime (namely k=1), if |Z t-1 | ≦ θ Regime 2 or Outer Regime (namely k=2), if |Z t-1 |>θ

27. 27 Threshold Parameter Estimates,θ

28. 28 Observation Percentage of Outer Regime,|Z t-1 |>θ

29. 29 Correlation Coefficient, ρ K S,F

30. 30 Standard Error of Futures Position, σ K FF

31. 31 Standard Error of Spot Position, σ K SS

32. 32 Relative Standard Error of Spot to Futures, (σ K SS /σ K FF )

33. 33 Hedge Ratio Estimates, HR

34. 34 Three Key Parameters for HR

35. 35 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

36. 36 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

37. 37 Hedging Effectiveness Comparison The variance (namely, Var) of hedged spot position with index futures can be presented as:

38. 38 Hedging Effectiveness Comparison

39. 39 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

40. 40 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

41. 41 Why??? Hungarian BSI : HR k=2 is 0.4775 and HR k=1 = 0.7825 The difference %=64% ((0.7825- 0.4775)/0.4775) U.S. S&P 500 HR k=2 is 0.9158 and HR k=1 =0.9430 The difference %=2.96% ((0.9430- 0.9158)/0.9158)

42. 42 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