Finding American Option Implied Volatility with Binomial Pricing

Summary

The discussion explains how to infer implied volatility for an American-style option by repeatedly pricing it with a binomial model and adjusting volatility until the model value matches the observed option price. Bisection can be used in the same way as for a European option when the contract is in a region where early exercise is not optimal. The resulting implied volatility can differ from the European-style estimate. The cited reference discusses applying bisection to American options in a binomial framework.

The answers compare numerical solvers. Bisection is straightforward and robust but converges linearly. Newton’s method can converge faster, but needs vega, which may have to be estimated numerically in a binomial model, and can be sensitive to its starting value. Brent’s method is suggested when Newton’s method fails; one proposed workflow starts with a closed-form estimate, tries Newton, then falls back to Brent. The notes do not specify a complete production implementation or address commodity-futures contract details, so model and exercise assumptions still matter.

Key ideas

• Implied volatility can be found by adjusting binomial-model volatility until the model price matches the observed American option price.
• Bisection is applicable when early exercise is not optimal, though the resulting implied volatility can differ from a European option estimate.
• Newton’s method may converge faster but depends on a suitable starting estimate and access to vega.
• Brent’s method is presented as a fallback when Newton’s method fails to converge.

Tags

# Implied Volatility from American options (binomial)


# Implied Volatility from American options (binomial)












I am trying to get the implied volatility from options on commodity futures and I know it's possible to get it from the binomial american options (on an non-dividend paying stock).

I believe it is done using the bisection method. However, I can't seem to find a paper illustrating this.

Can someone point me to relevant paper/book?

## Answer by Alexey Kalmykov (score 7)

https://quant.stackexchange.com/a/3276

I guess if your American-style option is in no-exercise region, you can use exactly the same bisection method as for European option.The implied volatility will be different, but the method is still the same. See for example, here, chapter 9.3.3. The applicability of bisection method for American-style options is discussed in the book "Binomial Models in Finance" (John Van Der Hoek, Robert James), around page 274.

Also you may find useful: How should I calculate the implied volatility of an American option in a real-time production environment?

## Answer by Dmitry (score 3)

https://quant.stackexchange.com/a/3917

Bisection method is rather fast but it has only linear convergence. Newton's method offers quadratic convergence but it requires the knowledge of Vega (which AFAIK is only accessible numerically with binomial model). However, the convergence of Newton's method can suffer from poor initial approximation. In this case Brent's method tends to perform better.

Personally I found the following scheme quite useful: analytical closed-form solution result serves as an initial approximation for Newton's method and if it fails to converge (which is unlikely and pretty rare), Brent's method is used.

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