270 Frequently Asked Questions In Quantitative Finance
that we must change from its usual^12 σ^2 S^2 to accommo-
date new models.
Second, if we want to fudge our option prices, to mas-
sage them into line with traded prices for example, we
can only do so by fiddling with this diffusion coeffi-
cient, i.e. what we now know to be the volatility. This
derivation tells us that our only valid fudge factor is the
volatility.
Black–Scholes for Accountants
The final derivation of the Black–Scholes equation
requires very little complicated mathematics, and
doesn’t even need assumptions about Gaussian returns,
all we need is for the variance of returns to be finite.
The Black–Scholes analysis requirescontinuoushedging,
which is possible in theory but impossible, and even
undesirable, in practice. Hence one hedges in some
discrete way. Let’s assume that we hedge at equal time
periods,δt. And consider the value changes associated
with a delta-hedged option.
- We start with zero cash
- We buy an option
- We sell some stock short
- Any cash left (positive or negative) is put into a
risk-free account.
We start by borrowing some money to buy the option.
This option has a delta, and so we sell delta of the
underlying stock in order to hedge. This brings in some
money. The cash from these transactions is put in the
bank. At this point in time our net worth is zero.
