Lognormal vs Normal Volatilities and Sensitivities in Practice

20 Pages Posted: 8 Nov 2015 Last revised: 20 Mar 2016

See all articles by Georgi Dimitroff

Georgi Dimitroff

Independent

Christian P. Fries

Ludwig Maximilian University of Munich (LMU) - Faculty of Mathematics; DZ Bank AG

Mark Lichtner

Independent

Niklas Rodi

Independent

Date Written: March 19, 2016

Abstract

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.

Keywords: negative rates, low rates, lognormal volatility, normal volatility, displaced, displacement, normal delta, lognormal delta

JEL Classification: C1, C5

Suggested Citation

Dimitroff, Georgi and Fries, Christian P. and Lichtner, Mark and Rodi, Niklas, Lognormal vs Normal Volatilities and Sensitivities in Practice (March 19, 2016). Available at SSRN: https://ssrn.com/abstract=2687742 or http://dx.doi.org/10.2139/ssrn.2687742

Georgi Dimitroff

Independent ( email )

No Address Available
United States

Christian P. Fries

Ludwig Maximilian University of Munich (LMU) - Faculty of Mathematics ( email )

Theresienstrasse 39
Munich
Germany

DZ Bank AG ( email )

60265 Frankfurt am Main
Germany

Mark Lichtner (Contact Author)

Independent ( email )

No Address Available
United States

Niklas Rodi

Independent ( email )

No Address Available
United States

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