Down-side Risk and the Puzzle of Implied Volatility Premium

Volatilities implied in options significantly stray upward from realized volatilities. Trading prices in markets are higher than theoretical prices calculated by the BS model. This paper aims to explain the implied volatility premium from the perspective of investors' loss-aversion. In practice investors pay more attention to potential losses rather than volatility of payoffs. Semi-variance is therefore used as a more plausible measure to reflect down-side risk. Due to the asymmetric payoffs of options, down side risk exposure varies between a buyer and a seller, which can not be captured in those pricing models using variance as risk measure. This paper shows that semi-variance of a short option position is higher than that of a long option position when option prices are determined by the BS model. Therefore the seller of an option asks for a higher trading price for enduring more down-side risk. That is why we observe upward deviation of market option prices away from theoretical prices, and the higher implied volatilities than realized volatilities on average. The empirical studies show the strong statistically positive correlation between deviation of market prices and the excess down-side risk exposure taken by a seller.