A Closed-Form Model-Free Implied Volatility Formula through Delta Families

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 (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.

TOPIC: Derivatives

Key Findings

• ▪ A novel closed-form representation of the Black-Scholes implied volatility is developed by utilizing a delta-family technique.

• ▪ Convergence and error analyses of approximate forms of this representations are presented.

• ▪ This technique is applied to the parametric SVI and SABR models as well as the stochastic volatility Heston model.