The Index Option or the Portfolio?

Abstract

This article theoretically compares the index option with the corresponding-weight portfolio of individual options on the index's constituent stocks in different cases to discuss when and whether the index option is worth purchasing. Theoretically we prove that in the normal cases when the expected return rate of the index is larger than the risk-less interest rate, the index option is more worth purchasing than the portfolio for call options, but the index option is less worth purchasing than the portfolio for put options. To give the consequences, we prove the monotonic relations between correlations and an index option's expected option return. The relations show that in the normal cases, lower correlations make the index call options more worth purchasing. However, unlike the usual perception, lower correlations make the actively traded index put options less worth purchasing. By applying mathematical analysis methods, the consequences are proved uniformly for any correlations and any strike level for all in-the-money, at-the-money, out-of-the-money options. We also prove the consequences of the cases when the expected return rate of the index is lower than or equal to the risk-less interest rate. The consequences are also uniform. The monotonic relations can also be generated to the monotonic relations between the volatility and a general option's expected option return. Our results can also explain and provide theoretical evidences on empirical findings in the real option market.