# Implied Volatility Calculation

We all know if you back out of the BS option pricing model you can derive and solve what the options is "implying" as its volatility. However, what is the formula used to derive IV (can anyone direct me to the equation)? Is there a closed form equation? Or is it solved numerically? I appreciate your help guys.

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I found this one via Google: Implied Volatility Formula –  chrisaycock Apr 17 '13 at 2:28
yea, saw that one too. Newton method was used here. am I right? But how is IV calculated? Anyone here use a standard procedure? –  jessica Apr 17 '13 at 2:30

The Black-Scholes option pricing model provides a closed-form pricing formula $BS(\sigma)$ for a European-exercise option with price $P$. There is no closed-form inverse for it, but because it has a closed-form vega (volatility derivative) $\nu(\sigma)$, and the derivative is nonnegative, we can use the Newton-Raphson formula with confidence.

Essentially, we choose a starting value $\sigma_0$ say from yoonkwon's post. Then, we iterate

$$\sigma_{n+1} = \sigma_n - \frac{BS(\sigma_n)-P}{\nu(\sigma_n)}$$

until we have reached a solution of sufficient accuracy.

This only works for options where the Black-Scholes model has a closed-form solution and a nice vega. When it does not, as for exotic payoffs, American-exercise options and so on, we need a more stable technique that does not depend on vega.

In these harder cases, it is typical to apply a secant method with bisective bounds checking. A favored algorithm is Brent's method since it is commonly available and quite fast.

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Brenner and Subrahmanyam (1988) provided a closed form estimate of IV, you can use it as the initial estimate:

$$\sigma \approx \sqrt{\cfrac{2\pi}{T}} . \cfrac{C}{S}$$

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If you could embed the link to the article in your answer, it would be great. –  SRKX Apr 17 '13 at 9:24
What are the definitions of T,C and S ? I'm guessing T is the Duration of the option-contract, C is the theoretical Call-value and S is the Strike-price, correct ? –  Nick Oct 9 '13 at 12:49
No, S is the current price of the underlying. However the approximation by Brenner and Subrahmanyam works best for at the money options, hence the difference should be small in that case. –  jcfrei May 9 at 14:29