2.7
RiskNeutral Valuation Principle
I 
n both the riskless hedge and the synthetic option approaches, you could
calculate the value of an option without referring to any of the following:
1) 
the expected return from the stock 

2) 
the expected return from the option 

3) 
the risk preferences of investors 

4) 
the probability that the stock moves up or down 

It
is sufficient to know the current stock price, and the riskfree
interest rate, and the fact that there are only two possible future
stock values.
This
observation leads to an elegant valuation argument.
Since the option value is independent of these factors, the
option price is the same whether you are in a riskaverse world or in a
riskneutral world. But in
a riskneutral world, we know how to value an asset: You simply discount
the expected future value by the riskfree interest rate.
The
question that arises, then, is whether we can transform our world into a
riskneutral world. The key
thing to keep in mind is that in such a world, the rate of return on
every asset must equal the riskfree interest rate.
In particular, the return on the stock (and on any option) must
equal the riskfree return.
Therefore,
we first need to discover whether the binomial stock model is consistent
with some riskneutral world. In
particular, can we choose the probabilities so that the expected return
on the stock equals the riskfree interest rate?
If
we can find such probabilities, called "riskneutral
probabilities," then we have a parallel world in which all the
prices are the same as in the original world.
It turns out that this is a very general principle, and that in
fact, such probabilities always exist as long as there is no arbitrage.
Deriving the RiskNeutral Probabilities
The simplest method for deriving the risk neutral probabilities is to solve for the probability of an uptick that equates the expected present value of the stocks terminal values to the current stocks price:
By rearranging and canceling the stock price S reveals that:
Alternatively you can see why such a probability is implied from a closer inspection of the synthetic option approach. In this approach, we used a stock and the riskfree asset to replicate the payoffs from a call. We then solved for the endofperiod value as:
and
to
get
and
By
substituting these values, we can rewrite this as follows:
where
Since
u > r > d, observe that both p and (1p) are
between 0 and 1. Therefore
they can be interpreted as probabilities.
We
can now provide a riskneutral interpretation for the value of the call
option:
The
numerator of the right hand side is the expected endofperiod value
where the probability used to evaluate the expectation is
p. The
denominator is the riskfree interest rate, and therefore C equals the
expected future value discounted by the riskfree rate.
Thus
in this world the expected return on the option when evaluated relative
to p
is the riskfree rate of return. To
complete the derivation, we must demonstrate that this probability has
the property that the expected return for the stock is also equal to the
riskfree rate of return. This
is because in the riskneutral world, every asset must earn the
riskfree return.
Consider:
By
canceling out common terms and dividing both sides by r,
we find that the stock also has an expected return equal to the
riskfree rate:
For
this reason, p is
called the "riskneutral probability."
You
should remember that the riskneutral valuation principle does not imply
that the true expected return on the option is equal to the riskfree
rate of interest. Indeed
this will not be the case for a capital market with riskaverse
investors. This is because
the actual expected return on the option is evaluated with respect to
the true probabilities, not the riskneutral probabilities.
Probabilities and Returns
Recall
that with respect to the riskneutral probabilities the option value is
and
the stock value is
In
a riskaverse market, the "true" probability will be some p
> p,
and thus the "true" expected returns for the stock and option
(respectively rs
and rc)
are greater than r.
It
is not always the case that we can determine the option price
independently of risk preferences; we discuss this in the next topic, Reconciliation
of RiskNeutral Valuation with CAPM.