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A hedge is a trade designed to reduce risk. The hedge ratio is another concept related to this definition. A hedge ratio is the ratio of the size of a position in a hedging instrument to the size of a position being hedged. There are many different ways to compute the hedge ratio.
In the simple case of an European call, the hedge ratio is the inverse of the delta of the call defined by: delta = e–q(T–t)N(d1), with q being the continuous dividend yield and N(.) the cumulative function of the normal distribution. There are many ways to hedge. One popular method to compute the hedge ratio has been developed by Witt et al. (1987).
This method consists in the regression of the spot price of a security over its future price: St = β0 + β1Ft + εt, where St is the spot price, Ft the future price, and εt the innovation. In this equation, β1 is the hedge ratio. In practice, the method to estimate this hedge ratio is the ordinary least squares (OLS).
Brown (1985) proposes to compute the hedge ratio using the percentage change of St and Ft in the regressions. The idea behind this procedure is that these prices are not stationary. Therefore, it seems preferable to relate these two prices by an error correction model because they seem to be cointegrated.
Wilson (1983) estimates the hedge ratio using the change in the spot and future prices. In this case, the computation of the hedge ratio corresponds to the minimization of the variance of ∆S – β1∆F, β1 being the hedge ratio.
A hedged portfolio is not necessarily a portfolio whose beta is equal to zero and there may be a nonlinear relation between the return of a portfolio and the return of the market. However, we cannot resort to the correlation coefficient to judge the relationship between two variables when there is a nonlinear association between them.