TY - JOUR

T1 - A closed-form model-free implied volatility formula through delta families

AU - Nguyen, Duy

AU - Taylor, Stephen

AU - Cui, Zhenyu

AU - Kirkby, Justin

N1 - Funding Information:
We thank the editor and the anonymous referee for detailed comments that greatly improve the article. The usual disclaimer applies. S. Taylor was partially sponsored by the Grant Agency of the Czech Republic, grant 19-28231X.
Publisher Copyright:
© 2021 Portfolio Management Research. All rights reserved.

PY - 2021/6

Y1 - 2021/6

N2 - 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.

AB - 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.

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U2 - 10.3905/JOD.2020.1.127

DO - 10.3905/JOD.2020.1.127

M3 - Article

AN - SCOPUS:85107870672

SN - 1074-1240

VL - 28

SP - 111

EP - 127

JO - Journal of Derivatives

JF - Journal of Derivatives

IS - 4

ER -