260 Frequently Asked Questions In Quantitative Finance
The derivation is based on the analysis of a stop-loss
strategy in which one attempts to hedge a call by selling
one share short if the stock is above the present value
of the strike, and holding nothing if the stock is below
the present value of the strike. Although at expiration the
call payoff and the stock position will cancel each other
exactly, this is not a strategy that eliminates risk. Na ̈ıvely
you might think that this strategy would work, after all
when you sell short one of the stock as it passes through
the present value of the strike you will neither make nor
lose money (assuming there are no transaction costs). But
if that were the case then an option initially with strike
above the forward stock price should have zero value. So
clearly something is wrong here.
To see what goes wrong you have to look more closely
at what happens as the stock goes through the present
value of the strike. In particular, look at discrete moves
in the stock price.
As the forward stock price goes fromKtoK+ sell one
share and buyKbonds. And then every time the stock
falls below the present value of the strike you reverse
this. Even in the absence of transaction costs, there will
be a slippage in this process. And the total slippage
will depend on how often the stock crosses this point.
Herein lies the rub. This happens an infinite number of
times in continuous Brownian motion.
IfU( ) is the number of times the forward price moves
fromKtoK+ , which will be finite since is finite,
then the financing cost of this strategy is
U( ).
Now take the limit as →0 and this becomes the
quantity known as local time. This local-time term is
what explains the apparent paradox with the above
example of the call with zero value.
