Estimating implied volatility with computer software is so commonplace that no trader even bothers to think about it even for a moment. You want the implied vol of a 3 month ATM Nikkei Call? Just check out your Bloomberg or Reuters screen and there you go. The implied vols of all traded assets and currencies are all there on your Bloomberg screen. Most traders, academicians, structurers all have simple VBA or C++ programs that calculate implied vol in less than a second.

But if we were to estimate implied volatility using a pencil and paper how would be do it quickly?

Let's say a 3 month ATM European style call option with spot of $100 is being quoted at price of $3.74. What's the implied vol of this call? Can you simply use an ordinary calculator to make a rough estimate of the implied vol? The answer is yes!

Implied vol is estimated using the quoted option value (the traded option value) and a series of iterations using some starting value. But what is the starting value? A back of the envelope starting value can estimated very easily. The spot is trading at $100, the maturity is 3 months and since it is an ATM call which is being quoted at $3.74, so using Brenner and Subrahmanyam's approximation you will get:

Of course by construction this formula exaggerates the implied vol value, but it is a great starting point and all it takes is an ordinary calculator and a piece of paper. From this starting point we need make a better estimation of the implied vol. We know that since it is an ATM call:

If interest rates are at 3% then inputting the value of volatility (as estimated above) of 18.82% in the Black-Scholes formula for Call option we get the theoretical value of the $100 ATM call option as $4.12. If we shift the volatility by another percentage point (to 19.82%) then we can calculate the vega of the call option as:

and finally the true estimate for the implied volatility of the quoted option will be

Therefore, 16.92% is more or less the correct implied volatility of the 3 month $100 ATM call option that is being quoted in the market at $3.74. Of course computer algorithms will use more number of iterations but based on the same methodology as show here. This is the basis of Newton-Raphson type algorithm.

**Two important questions:**

- Will the above algorithm work for American style options on dividend paying assets? Does it need to be modified?

- Will the above algorithm work for path dependent options, such as barrier options? If not, then why not?

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