Chapter 2: FAQs 207
Volatility is constant: If volatility is time dependent then
the Black–Scholes formulæ are still valid as long as you
plug in the ‘average’ volatility over the remaining life of
the option. Here average means the root-mean-square
average since volatilities can’t be added but variances
can.
Even if volatility is stochastic we can still use basic
Black–Scholes formulæ provided the volatility process is
independent of, and uncorrelated with, the stock price.
Just plug the average variance over the option’s lifetime,
conditional upon its current value, into the formulæ.
There are no arbitrage opportunities: Even if there are arbi-
trage opportunities because implied volatility is different
from actual volatility you can still use the Black–Scholes
formulæ to tell you how much profit you can expect
to make, and use the delta formulæ to tell you how to
hedge. Moreover, if there is an arbitrage opportunity
and you don’t hedge properly, it probably won’t have
that much impact on the profit you expect to make.
The underlying is lognormally distributed: The Black–Scholes
model is often used for interest-rate products which are
clearly not lognormal. But this approximation is often
quite good, and has the advantage of being easy to
understand. This is the model commonly referred to as
Black ’76.
There are no costs associated with borrowing stock for going short:
Easily accommodated within a Black–Scholes model, all
you need to do is make an adjustment to the risk-neutral
drift rate, rather like when you have a dividend.
Returns are normally distributed: Thanks to near-continuous
hedging and theCentral Limit Theoremall you really