A Closed-form Model-free Implied Volatility Formula through Delta Families

Journal of Derivatives, forthcoming

24 Pages Posted: 6 May 2020 Last revised: 24 Jul 2020

See all articles by Zhenyu Cui

Zhenyu Cui

Stevens Institute of Technology - School of Business

Justin Kirkby

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE)

Duy Nguyen

Marist College - Department of Mathematics

Stephen Michael Taylor

New Jersey Institute of Technology

Date Written: July 23, 2020

Abstract

In this paper, 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 (SVI) model, and the stochastic alpha beta rho (SABR) 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.

Keywords: Dirac Delta function, delta sequence, implied volatility, model-free, SVI, SABR, Heston

JEL Classification: G12, G13, G14, C58

Suggested Citation

Cui, Zhenyu and Kirkby, Justin and Nguyen, Duy and Taylor, Stephen Michael, A Closed-form Model-free Implied Volatility Formula through Delta Families (July 23, 2020). Journal of Derivatives, forthcoming, Available at SSRN: https://ssrn.com/abstract=3573239 or http://dx.doi.org/10.2139/ssrn.3573239

Zhenyu Cui

Stevens Institute of Technology - School of Business ( email )

Hoboken, NJ 07030
United States

HOME PAGE: http://sites.google.com/site/zhenyucui86/publications

Justin Kirkby

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE) ( email )

765 Ferst Drive
Atlanta, GA 30332-0205
United States

Duy Nguyen

Marist College - Department of Mathematics ( email )

NY
United States

HOME PAGE: http://sites.google.com/site/nducduy/

Stephen Michael Taylor (Contact Author)

New Jersey Institute of Technology ( email )

University Heights
Newark, NJ 07102
United States

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
538
Abstract Views
1,596
rank
64,663
PlumX Metrics