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Unformatted text preview: Estimating Future Hedge Ratio: A Generalized Hyperbolic Distribution Approach Cheng-Few Lee Institute of Finance, National Chiao-Tung University, Hsinchu, Taiwan TEL: +886-3-5712121x57071 E-mail: email@example.com Jang-Yi Lee Department of Mathematics, Tunghai University TEL:+886-6-2695543 E-mail:firstname.lastname@example.org Kehluh Wang Institute of Finance, National Chiao-Tung University, Hsinchu, Taiwan TEL: +886-3-5731760 E-mail: email@example.com Yuan-Chung Sheu Department of Applied mathematics, National Chiao-Tung University, Hsinchu, Taiwan TEL: +886-3-5712121x56428 E-mail: firstname.lastname@example.org Abtract Under martingale and joint-normality assumptions, various optimal hedge ratios are iden- tical to the minimum variance hedge ratio. As empirical studies usually reject the joint- normality assumption, we propose the generalized hyperbolic distribution as the joint log-return distribution of the spot and futures. Using the parameters in this distribution, we estimate several most widely-used optimal hedge ratios: minimum variance, maximum Sharpe measure and minimum generalized semivariance. Under mild assumptions on the parameters, we find that these hedge ratios are identical. Empirical studies show that our proposed models fit the TAIEX futures and S & P 500 futures very well. Numerical results for different optimal hedge ratios also verify our theoretical observations. Regarding the equivalence of these three optimal hedge ratios, our analysis suggests that the martingale property plays a much important role than the joint distribution assumption. JEL Classification: G11, C130 Running title: Hyperbolic Distribution in Futures Hedge 1 Introduction Because of their low transaction cost, high liquidity, high leverage and ease of short position, stock index futures are among the most successful innovations in the financial markets. Besides the speculative trading, they are widely used to hedge against the market risk of the spot position. One of the most important issues for investors and portfolio managers is to calculate the optimal futures hedge ratio, the proportion of the position taken in futures to the size of the spot so that the risk exposure can be minimized. The optimal hedge ratios typically depend on the objective functions under con- sideration. In literature on futures hedging, there are two different types of objective functions: the risk function to be minimized, and the utility function to be maximized. Johnson (1960) obtains the minimum variance hedge ratio by minimizing the variance of the change in the value of the hedged portfolios. On the other hand, as Adams and Mon- tesi (1995) indicate, corporate managers are more concerned with the downside risk rather than the upside variation. A measure of the downside risk is the generalized semivariance (GSV) where the risk is computed from the expectation of a power function of short- falls from the target return (Bawa 1975, 1978; Fishburn 1977). De Jong, De Roon and Veld (1997) and Lien and Tse (1998, 2000, 2001) have calculated several GSV-minimizing...
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