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?
Brenner and Subrahmanyam (1988) provided a closed form estimate of IV, you can use it as the initial estimate:
$$ \sigma \approx \sqrt{\cfrac{2\pi}{T}} . \cfrac{C}{S} $$
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.
It is a very simple procedure and yes, Newton-Raphson is used because it converges sufficiently quickly:
the following contains a very simple example of how you derive the implied vol from an option price: http://risklearn.com/estimating-implied-volatility-with-the-newton-raphson-method/
You can also derive implied volatility through a "rational approximation" approach (closed form approach -> faster), which can be used exclusively if you are fine with the approximation error or as a hybrid in combination with a few iterations of NR (better initial guess -> less iterations). Here a reference: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=952727
There are some references on this topic. You may find them helpful.
Peter Jaeckel has articles named "By Implication (2006)" and "Let's be rational (2013)"
Li and Lee (2009) [download] An adaptive successive over-relaxation method for computing the Black–Scholes implied volatility
Stefanica and Radoicic (2017) An Explicit Implied Volatility Formula
The bisection method, Brent's method, and other algorithms should work well. But here is a very recent paper that gives an explicit representation of IV in terms of call prices through (Dirac) delta sequences:
Cui et al. (2020) - A closed-form model-free implied volatility formula through delta sequences
To get IV I do the following: 1) change sig many times and calculate C in BS formula every time. That can be done with OIC calculator All other parameters are kept constant in BS call price calculations. The sig that corresponds to C value closest to the call market value is probably right. 2) without OIC calculator for every chosen sig I am using old approach: calculate d1, d2, Nd1, Nd2 and BS option value. Again calculated BS value closest to the market value probably correspond to correct IV.
