A closed-form model-free implied volatility formula through delta families

Document Type

Article

Publication Date

6-1-2021

Abstract

In this article, we derive a closed-form explicit model-free formula for the (Black-Scholes) implied volatility. The method is based on the novel use of the Dirac Delta function, corresponding delta families, and the change of variable technique. The formula is expressed through either a limit or as an infinite series of elementary functions, and we establish that the proposed formula converges to the true implied volatility value. In numerical experiments, we verify the convergence of the formula, and consider several benchmark cases, for which the data-generating processes are respectively the stochastic volatility inspired model, and the stochastic alpha beta rho model. We also establish an explicit formula for the implied volatility expressed directly in terms of respective model parameters, and use the Heston model to illustrate this idea. The delta family and change of variable technique that we develop are of independent interest and can be used to solve inverse problems arising in other applications.

Identifier

85107870672 (Scopus)

Publication Title

Journal of Derivatives

External Full Text Location

https://doi.org/10.3905/JOD.2020.1.127

ISSN

10741240

First Page

111

Last Page

127

Issue

4

Volume

28

Grant

19-28231X

Fund Ref

Grantová Agentura České Republiky

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