Chapter 2: FAQs 103
Why Does Risk-Neutral Valuation
Work?
Short Answer
Risk-neutral valuation means that you can value options
in terms of their expected payoffs, discounted from
expiration to the present, assuming that they grow on
average at the risk-free rate.Option value=Expected present value of payoff
(under a risk-neutral random walk).Therefore the real rate at which the underlying grows on
average doesn’t affect the value. Of course, the volatil-
ity, related to the standard deviation of the underlying’s
return, does matter. In practice, it’s usually much, much
harder to estimate this average growth than the volatil-
ity, so we are rather spoiled in derivatives, that we only
need to estimate the relatively stable parameter, volatil-
ity.^2 The reason that this is true is that by hedging an
option with the underlying we remove any exposure to
the direction of the stock, whether it goes up or down
ceases to matter. By eliminating risk in this way we also
remove any dependence on the value of risk. End result
is that we may as well imagine we are in a world in
which no one values risk at all, and all tradeable assets
grow at the risk-free rate on average.For any derivative product, as long as we can hedge it
dynamically and perfectly (supposing we can as in the
case of known, deterministic volatility and no defaults)
the hedged portfolio loses its randomness and behaves
like a bond.(^2) I should emphasize the word ‘relatively.’ Volatility does vary
in reality, but probably not as much as the growth rate.