104 Frequently Asked Questions In Quantitative Finance
Example
A stock whose value is currently $44.75 is growing on
average by 15% per annum. Its volatility is 22%. The
interest rate is 4%. You want to value a call option with
a strike of $45, expiring in two months’ time. What can
you do?
First of all, the 15% average growth is totally irrelevant.
The stock’s growth and therefore its real direction does
not affect the value of derivatives. What you can do is
simulate many, many future paths of a stock with an
average growth of 4% per annum, since that is the risk-
free interest rate, and a 22% volatility, to find out where
it may be in two months’ time. Then calculate the call
payoff for each of these paths. Present value each of
these back to today, and calculate the average over all
paths. That’s your option value.
Long Answer
Risk-neutral valuation of derivatives exploits the per-
fect correlation between the changes in the value of an
option and its underlying asset. As long as the under-
lying is the only random factor then this correlation
should be perfect. So if an option goes up in value with
a rise in the stock then a long option and sufficiently
short stock position shouldn’t have any random fluc-
tuations, therefore the stock hedges the option. The
resulting portfolio is risk free.
Of course, you need to know the correct number of the
stock to sell short. That’s called the ‘delta’ and usu-
ally comes from a model. Because we usually need a
mathematical model to calculate the delta, and because
quantitative finance models are necessarily less than
perfect, the theoretical elimination of risk by delta
hedging is also less than perfect in practice. There are
several such imperfections with risk-neutral valuation.
First, it requires continuous rebalancing of the hedge.
