options - A simple formula for calculating implied volatility?

We all know if you back out of the Black Scholes option pricing model you can derive what the option is "implying" about the underlyings future expected volatility.

Is there a simple, closed form, formula deriving Implied Volatility (IV)? If so can you could you direct me to the equation?

Or is IV only numerically solved?

Answer

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.