## Implied Biasedness

Usually, forecast unbiasedness is not an overriding issue in any forecasting exercise. Forecast bias can be estimated and corrected if the degree of bias remains stable through time. Testing for biasedness ^usually carried out using the regression equation (2.3), where Xi = Xt is the implied forecast of period t volatility. For a forecast to be unbiased, one would require a = 0 and P = 1. Implied forecast is upwardly biased if a > 0 and P = 1, or a = 0 and P > 1. In the case where a > 0 and P < 1, which is the most common scenario, implied underforecasts low volatility and overforecasts high volatility.

It has been argued that implied bias will persist only if it is difficult to perform arbitrage trades that are needed to remove the mispricing. This is more likely in the case of stock index options and less likely for futures options. Stocks and stock options are traded in different markets. Since trading of a basket of stocks is cumbersome, arbitrage trades in relation to a mispriced stock index option may have to be done indirectly via index futures. On the other hand, futures and futures options are traded alongside each other. Trading in these two contracts are highly liquid. Despite these differences in trading friction, implied biasedness is reported in both the S&P100 OEX market (Canina and Figlewski, 1993; Christensen and Prabhala, 1998; Fleming, Ostdiek and Whaley, 1995; Fleming, 1998) and the S&P500 futures options market (Feinstein, 1989b; Ederington and Guan, 1999, 2002).

Biasedness is equally widespread among implied volatilities of currency options (see Guo, 1996b; Jorion, 1995; Li, 2002; Scott and Tucker, 1989; Wei and Frankel, 1991). The only exception is Jorion (1996) who cannot reject the null hypothesis that the one-day-ahead forecasts from implied are unbiased. The five studies listed earlier use implied to forecast exchange rate volatility over a much longer horizon ranging from one to nine months.

Unbiasedness of implied forecast was not rejected in the Swedish market (Frennberg and Hansson, 1996). Unbiasedness of implied forecast was rejected for UK stock options (Gemmill, 1986), US stock options (Lamoureux and Lastrapes, 1993), options and futures options across a range of assets in Australia (Edey and Elliot, 1992) and for 35 futures options contracts traded over nine markets ranging from interest rate to livestock futures (Szakmary, Ors, Kim and Davidson, 2002). On the other hand, Amin and Ng (1997) find the hypothesis that a = 0 and P = 1 cannot be rejected for the Eurodollar futures options market.

Where unbiasedness was rejected, the bias in all but two cases was due to a > 0 and p < 1. These two exceptions are Fleming (1998) who reports a = 0 and P < 1 for S&P100 OEX options, and Day and Lewis (1993) who find a > 0 and P = 1 for distant-term oil futures options contracts.

Christensen and Prabhala (1998) argue that implied is biased because of error-in-variable caused by measurement errors. Using last period implied and last period historical volatility as instrumental variables to correct for these measurement errors, Christensen and Prabhala (1998) find unbiasedness cannot be rejected for implied volatility of the S&P100 OEX option. Ederington and Guan (1999, 2002) find bias in S&P500 futures options implied also disappeared when similar instrument variables were used.

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