Chapter 2: FAQs 171
money in either case (or even if you hedge using a
volatility somewhere in the 20 to 40 range) as long as
you are right about the 40% and you hedge continu-
ously. There will however be a big impact on your P&L
depending on which volatility you input.
If you use the actual volatility of 40% then you are guar-
anteed to make a profit that is the difference between
the Black–Scholes formula using 40% and the Black–
Scholes formula using 20%.
V(S,t;σ)−V(S,t; ̃σ),whereV(S,t;σ) is the Black–Scholes formula for the call
option andσdenotes actual volatility andσ ̃is implied
volatility.
That profit is realized in a stochastic manner, so that
on a marked-to-market basis your profit will be random
each day. This is not immediately obvious, neverthe-
less it is the case that each day you make a random
profit or loss, both equally likely, but by expiration your
total profit is a guaranteed number that was known at
the outset. Most traders dislike the potentially large
P&L swings that you get by hedging using the forecast
volatility that they hedge using implied volatility.
When you hedge with implied volatility, even though
it is wrong compared with your forecast, you will still
make money. But in this case the profit each day is
non negative and smooth, so much nicer than when
you hedge using forecast volatility. The downside is
that the final profit depends on the path taken by the
underlying. If the stock stays close to the strike then
you will make a lot of money. If the stock goes quickly
far into or out of the money then your profit will be
small. Hedging using implied volatility gives you a nice,