The risk-neutral probability measure

It is clear from the previous section that asset prices are not equal to the --discounted-- expectation of their cash flows. The cash flows have to be translated through the pricing kernel, before expectations are taken. It can be shown that there is a way of representing asset prices as a discounted expectation of cash flows, but expectations have to be taken under a different probability measure. This probability measure is called the risk neutral probability measure, for exactly this reason.

Keep in mind that the only things that one can observe are the bond and asset price. Perhaps one can use these in order to retrieve the state prices, if the number of states is fairly small, smaller than the number of securities. But how can the state prices be used in practice? After all, they are just prices of imaginary securities. It could be useful, if one could use the [forward looking] asset prices in order to retrieve the probabilities of the states $\pi _{k}$. Is that possible? It turns out to be an impossible task, since these probabilities are well hidden into the state prices. Since the pricing kernel is not observable neither, one cannot split the state prices into their components. This is where the risk neutral probabilities come into the game.

Consider the asset pricing equation, and divide both sides with the bond. This will give

$$c_{0}=b_{0}\sum_{k}\frac{q_{k}}{b_{0}}c_{k}$$.

Now observe the quantities $\frac{q_{k}}{b_{0}}=Q_{k}$ and recall that $%% b_{0}=\sum_{k}q_{k}$. The state prices are positive, which means that all $%% \pi _{k}^{\star }$ will be positive too. In addition, one can check that $%% \sum_{k}Q_{k}=1$ by construction. These two features make the collection $%% \left\{ Q_{k}\right\}$ look like state probabilities, although they are not the true ones. In fact, they are the risk neutral ones, and we can rewrite the asset pricing equation as

$$c_{0}=b_{0}\sum_{k}Q_{k}c_{k}=\mathbf{E}_{Q}\left[ c\right]$$   .

These risk neutral probabilities are the ones that we would retrieve, if we were just observing asset prices. The agents seem to price assets as a discounted sum of expected future cash flows by using these probabilities, not the true ones. Why? Because they have different attitude towards different states. On the other hand, the risk neutral probabilities can be extremely useful: If the agent wants to price other assets, she can use these probabilities to average the future cash flows. She shouldn't care about the true state probabilities.

Example 12 (Risk neutral measure)   Suppose that the interest rate is $r=\frac{1}{4}=25\%$. Therefore the bond will have a price equal to $b_{0}=\frac{1}{1+r}=\frac{4}{5}=0.8$. In addition say that the two states have equal probabilities $\pi _{u}=\pi _{d}=%% \frac{1}{2}$, and that the agent is indifferent between them, which implies that the state prices will be the same. Since $b_{0}=q_{u}+q_{d}$, each one will be equal to $q_{u}=q_{d}=\frac{2}{5}=0.4$ --half of the bond price. Now consider a security that has the payoff tree of figure4.3

Figure 4.3: A one-period tree problem

What will be the price of such security? Of course $%% c_{0}=q_{u}c_{u}+q_{d}c_{d}=12$. What would happen if the agent bought the bond with these $\pounds 12$? With this interest rate she would enjoy $%% \pounds 15$ in the next period, which is just the average of the two cash flows. Say that we just observed that $c_{0}=\pounds 12$, and did not know anything about state prices. What would be the risk neutral probabilities? They have to satisfy $c_{0}=Q_{u}c_{u}+\left( 1-Q_{u}\right) c_{d}$, and $%% Q_{d}=1-Q_{u}$, giving $Q_{u}=Q_{d}=\frac{1}{2}$.

Example 13 (cont. Risk neutral measure)   Now continue the same story, but assume that the agent is more afraid of losing money, than making them. Suppose that the true probabilities are the same, $\pi _{u}=\pi _{d}=\frac{1}{2}$, but the state prices are different: $%% q_{u}=0.2$ and $q_{d}=0.6$ --they still sum up to the bond price of course. The price of the security in this case is $c_{0}=q_{u}c_{u}+q_{d}c_{d}=10$, giving the tree of figure 4.4

Figure 4.4: The solution of the one-period tree

In this case the bond would have allowed the agent to enjoy $\pounds 12.5$ in the next period, far below the average cash flow. What are the risk neutral probabilities? They turn out to be $Q_{u}=\frac{1}{4}$ and $Q_{d}=%% \frac{3}{4}$. Because the agent dislikes loosing, she prices the securities as if the bad outcome was more probable. In fact the average discounted cash flow using the true probabilities is still equal to $\pounds 12$, but the agent requires a risk premium to purchase the security. This risk premium is equal to $\pounds 2$.