Chapter 2: FAQs 105
Delta is constantly changing so you must always be
buying or selling stock to maintain a risk-free position.
Obviously, this is not possible in practice. Second, it
hinges on the accuracy of the model. The underlying
has to be consistent with certain assumptions, such as
being Brownian motion without any jumps, and with
known volatility.
One of the most important side effects of risk-neutral
pricing is that we can value derivatives by doing simula-
tions of the risk-neutral path of underlyings, to calculate
payoffs for the derivatives. These payoffs are then
discounted to the present, and finally averaged. This
average that we find is the contract’s fair value.
Here are some further explanations of risk-neutral pricing.
Explanation 1: If you hedge correctly in a Black–Scholes
world then all risk is eliminated. If there is no risk then
we should not expect any compensation for risk. We can
therefore work under a measure in which everything
grows at the risk-free interest rate.
Explanation 2: If the model for the asset isdS=μSdt+
σSdXthen theμs cancel in the derivation of the Black–
Scholes equation.
Explanation 3: Two measures are equivalent if they have
the same sets of zero probability. Because zero proba-
bility sets don’t change, a portfolio is an arbitrage under
one measure if and only if it is one under all equivalent
measures. Therefore a price is non-arbitrageable in the
real world if and only if it is non-arbitrageable in the
risk-neutral world. The risk-neutral price is always non-
arbitrageable. If everything has a discounted asset price
process which is a martingale then there can be no
arbitrage. So if we change to a measure in which all the
fundamental assets, for example the stock and bond,