Chapter 4: Ten Different Ways to Derive Black–Scholes 273
get the profit made from the stock move as
1
2 σ(^2) S (^2) δt.
Put these three value changes together (ignoring theδt
term which multiplies all of them) and set the resulting
expression equal to zero, to represent no arbitrage, and
you get
+^12 σ^2 S^2 +r(S−V)=0,
the Black–Scholes equation.
Now there was a bit of cheating here, since the stock
price move is really random. What we should have said
is that
1
2 σ
(^2) S (^2) δt
is the profit made from the stock moveon average.
Crucially all we need to know is that the variance of
returns is
σ^2 S^2 δt,
we don’t even need the stock returns to be normally
distributed. There is a difference between the square
of the stock price moves and its average value and this
gives rise to hedging error, something that is always
seen in practice. If you hedge discretely, as you must,
then Black–Scholes only works on average. But as you
hedge more and more frequently, going to the limit
δt=0, then the total hedging error tends to zero, so
justifying the Black–Scholes model.
References and Further Reading
Andreason, J, Jensen, B & Poulsen, R 1998 Eight Valuation
Methods in Financial Mathematics: The Black–Scholes For-
mula as an Example.Math. Scientist 23 18–40