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:

It is sufficient to know the current stock price, and the risk-free 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 risk-averse world or in a risk-neutral world.  But in a risk-neutral world, we know how to value an asset: You simply discount the expected future value by the risk-free interest rate. 

The question that arises, then, is whether we can transform our world into a risk-neutral world.  The key thing to keep in mind is that in such a world, the rate of return on every asset must equal the risk-free interest rate.  In particular, the return on the stock (and on any option) must equal the risk-free return.

Therefore, we first need to discover whether the binomial stock model is consistent with some risk-neutral world.  In particular, can we choose the probabilities so that the expected return on the stock equals the risk-free interest rate?

If we can find such probabilities, called "risk-neutral 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 Risk-Neutral Probabilities

The simplest method for deriving the risk neutral probabilities is to solve for the probability of an up-tick 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 risk-free asset to replicate the payoffs from a call. We then solved for the end-of-period 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 (1-p) are between 0 and 1.  Therefore they can be interpreted as probabilities.

We can now provide a risk-neutral interpretation for the value of the call option:

The numerator of the right hand side is the expected end-of-period value where the probability used to evaluate the expectation is  p.  The denominator is the risk-free interest rate, and therefore C equals the expected future value discounted by the risk-free rate.

Thus in this world the expected return on the option when evaluated relative to  p is the risk-free 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 risk-free rate of return.  This is because in the risk-neutral world, every asset must earn the risk-free 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 risk-free rate:

For this reason,  p is called the "risk-neutral probability."

You should remember that the risk-neutral valuation principle does not imply that the true expected return on the option is equal to the risk-free rate of interest.  Indeed this will not be the case for a capital market with risk-averse investors.  This is because the actual expected return on the option is evaluated with respect to the true probabilities, not the risk-neutral probabilities.

Probabilities and Returns

Recall that with respect to the risk-neutral probabilities the option value is

and the stock value is

In a risk-averse 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 Risk-Neutral Valuation with CAPM.