Lognormal vs Normal Volatilities and Sensitivities in Practice

The advent of close to zero or even negative rates in major currencies has made the traditional lognormal Black-Scholes-Merton volatility as a representation of option prices in the interest rate market obsolete. Recently more and more cap/floor and even swaption prices in major currencies are violating the upper no-arbitrage bound implied by the Black-Scholes-Merton model. The corresponding lognormal volatilities fail to exist and, thus, cannot be used as inputs for trading and risk systems. Consequently many market participants have resorted to either a normal or displaced lognormal volatility market data representation.Altering the quoting convention and representing option prices in displaced lognormal or normal rather than traditional lognormal volatilities may look like a technical detail. However, depending on the set-up this may have a direct impact on the definition of the risk factor coordinate system. In this case traders and risk managers do not only need to understand the levels of the new types of volatilities but also need to rebuild their intuition for the associated risk sensitivities.In this paper we present approximate and closed-form formulas to transform lognormal volatilities and sensitivities into their normal or displaced lognormal counterparts and vice versa. These results are crucial for understanding and validating the potentially severe impact on sensitivities and (delta) hedge ratios that may appear when changing the volatility market data representation