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

Duy Nguyen, Stephen Taylor, Zhenyu Cui, Justin Kirkby

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)111-127
Number of pages17
JournalJournal of Derivatives
Issue number4
StatePublished - Jun 2021

All Science Journal Classification (ASJC) codes

  • Finance
  • Economics and Econometrics


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