158 Frequently Asked Questions In Quantitative Finance
Long Answer
Let us begin with how to calculate the implied volatil-
ities. Start with the prices of traded vanilla options,
usually the mid price between bid and offer, and all
other parameters needed in the Black–Scholes formulæ,
such as strikes, expirations, interest rates, dividends,
exceptfor volatilities. Now ask the question, what volatil-
ity must be used for each option series so that the
theoretical Black–Scholes price and the market price
are the same?
Although we have the Black–Scholes formula for option
values as a function of volatility, there is no formula
for the implied volatility as a function of option value,
it must be calculated using some bisection, Newton–
Raphson, or other numerical technique for finding zeros
of a function. Now plot these implied volatilities against
strike, one curve per expiration. That is the implied
volatility smile. If you plot implied volatility against
both strike and expiration, as a three-dimensional plot,
that is the implied volatility surface. Often you will find
that the smile is quite flat for long-dated options, but
getting steeper for short-dated options.
Since the Black–Scholes formulæ assume constant
volatility (or with a minor change, time-dependent
volatility) you might expect a flat implied volatility plot.
This appears not to be the case from real option-price
data. How can we explain this? Here are some questions
to ask.
- Is volatility constant?
- Are the Black–Scholes formulæ correct?
- Do option traders use the Black–Scholes formulæ?
Volatility does not appear to be constant. By this we
mean that actual volatility is not constant, actual volatil-
ity being the amount of randomness in a stock’s return.
