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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?

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2 Answers 2

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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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?

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Both sources you provide either apply the bisection method on Black-Scholes or on European-style binomial trees. They do mention that it is applicable to American-style options but provide no illustration. What happens when the American-style option is in an exercise region so there is an early exercise premium? how does this affect the implied volatility? –  user2301 Apr 18 '12 at 13:33
    
You just need to replace in this method the European BS pricing by American option pricing algorithm. If we are in exercise region, American option price is its intrinsic value. So you can get your implied volatility, however it looks a bit useless because option must be exercised immediately anyway. –  Alexey Kalmykov Apr 18 '12 at 14:10

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