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A Closed-Form Approximation for Black–Scholes Implied Volatility

Article Quant Q&A · Author: R.G.
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Summary

The document asks how to adapt a closed-form implied volatility estimator from a call option to a put. It presents an equation attributed to Hallerbach (2004) for estimating volatility from a call price, spot price, strike, and time to expiry. This makes the post relevant to option valuation, although the requested put version is not actually derived in the exchange.

The response points readers toward other explicit implied volatility formulas and tighter bounds as alternatives, but gives no equations, comparisons, or numerical evidence for them. Consequently, the material offers a starting reference for approximation methods rather than a complete put estimator or a tested recommendation. Any implementation would need to consult the cited research and check the estimator's assumptions and accuracy across option inputs; those limits are not examined in the post.

Key ideas

  • The post presents a closed-form approximation for Black–Scholes implied volatility from a call price.
  • It asks how the call estimator should be adapted for a put option.
  • The response points to alternative explicit formulas and tighter bounds without explaining them.
  • The exchange does not provide a put equation or evidence comparing estimator accuracy.

Tags

OptionsVolatilityDerivatives pricing
Full text
# What is The Closed-Form Implied Volatility Estimator (As Defined by Hallerbach 2004) for A Put Option?


# What is The Closed-Form Implied Volatility Estimator (As Defined by Hallerbach 2004) for A Put Option?












"An Improved Estimator For Black-Scholes-Merton Implied Volatility" by Hallerbach (2004) (link to article) provides an equation (Eq. 24, Page 13, and below) for the implied volatility of a call option. What would be the equation for a put option? Thank you!

$$\sigma\sqrt{T} = \frac{\sqrt{2\pi}}{2(S+X)}\Biggl[2C+X-S+\sqrt{(2C+X-S)^2-1.85\frac{(S+X)(X-S)^2}{\pi\sqrt{XS}}}\Biggl]$$

## Answer by Tulio Carnelossi (score 2)

https://quant.stackexchange.com/a/35522

I would look to these papers below by Dan Stefanica et al. Very easy to code and yields better results.

An Explicit Implied Volatility Formula

Tighter Bounds for Implied Volatility

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