174 Frequently Asked Questions In Quantitative Finance
trading the underlying will increase. Can we quantify
transaction costs? Of course we can.
If we hold a short position in delta of the underlying and
then rebalance to the new delta at a timeδtlater then
we will have had to have bought or sold whatever the
change in delta was. As the stock price changes byδS
then the delta changes byδS. If we assume that costs
are proportional to the absolute value of the amount of
the underlying bought or sold, such that we pay in costs
an amountκtimes the value traded then the expected
cost eachδtwill be
κσS^2
√
δt
√
2
π
||,
where the
√
2
πappears because we have to take the
expected value of the absolute value of a normal vari-
able. Since this happens every time step, we can adjust
the Black–Scholes equation by subtracting from it the
above divided byδtto arrive at
∂V
∂t
+^12 σ^2 S^2
∂V
∂S^2
+rS
∂V
∂S
−rV−κσS^2
√
2
πδt
||= 0.
This equation is interesting for being non linear, so that
the value of a long call and a short call will be different.
The long call will be less than the Black–Scholes value
and a short call higher. The long position is worth less
because we have to allow for the cost of hedging. The
short position is even more of a liability because of costs.
Crucially we also see that the effect of costs grows
like the inverse of the square root of the time between
rehedges. As explained above if we want to halve hedg-
ing error we must hedge four times as often. But this
would double the effects of transaction costs.
In practice, people do not rehedge at fixed intervals,
except perhaps just before market close. There are
