It Takes Three to Smile

Abstract

An equation relating the Black-Merton-Scholes Greeks vega, vanna, and volga is derived by making use of the generalised Hull-White formula. Given only three options the equation automatically gives a robust and accurate approximation for the volatility swap strike, the variance swap strike, and a quantity which we name the adjusted correlation. Once the aforementioned quantities have been implied from the three pillar options, an approximation for the entire volatility skew can be constructed. A potentially more interesting implication of the vega-vanna-volga relationship is that the volatility skew can be deformed into a smile in a logically consistent manner. In other words we give a prescription for converting vanilla options priced in a world with non-zero correlation between the index and volatility, to prices in a world in which the correlation would be approximately zero. As valuation of volatility derivatives is significantly easier in a zero correlation world, the translation offers a simplification of the problem of valuation of volatility derivatives in addition to the fact that all the aforementioned results can be achieved starting with only three quoted options on the index.