What's Going On?

In this lesson, we return to the question posed earlier about call options with the same price despite the fact that there are different probabilities of the underlying stock moving up in price or down in price.

We had two securities that were identical but they had different probabilities of going up and down:

The option prices for both of these stocks is the same = $4.80.

We saw earlier that the calculation to get the option price are as follows:

The true probabilities p and (1-p) don't appear in this formula, so it appears that p does not matter.  However, this only happens because we are asking the wrong question.  The real question to ask is why would we find two stocks in the same economy with such different results.  So the problem isn't with the option pricing theory or the option pricing formula above, but is actually why there would ever be two securities like this in the same economy.  This is the source of the problem, not the option pricing theory.


Another result that may seem incorrect or surprising to people is that the price for options will change when the risk-free rate R changes.  When the risk-free rate R increases, so do the option prices.  This is opposite of what you would expect in the real world.  We would expect that as R increases, the present value of the cash flow would decrease because the discount rate would decrease as well.



We then discussed risk-neutral probabilities and their implications for no-arbitrage.
Remember that for no-arbitrage, d < R < u.  If there exists a risk-neutral distribution Q such that the following formula holds true:

then arbitrage cannot exist.

For example:
(i). Suppose there is a model with m states, w(1...m).  The payoff for each state is non-negative, but the payoff is strictly positive in 1 state.  There are also risk neutral probabilities q(1...m) that correspond to each state.  The expected value of the cost of this model is q(1)*w(1) + ... + q(m)*w(m) which must be strictly greater than 0, so there cannot be Type B arbitrage.

(ii). Similarly, it is impossible to get a Type A arbitrage by stating that all the states, w(1...m) are non-negative.  The expected cost of this model is non-negative as well.  This means that it cannot have a negative cost so Type A arbitrage cannot exist.

The reverse statement also holds true: if there is no arbitrage, then a risk-neutral distribution of q values also exists.

Together, these two statements are known as the first fundamental theorem of asset pricing.

The first fundamental theorem of asset pricing: the existence of risk-neutral probabilities and no-arbitrage are equivalent with one another.