State prices

Consider a world that can be in two distinct states tomorrow. The realization of tomorrow's state affects the agent's utility, not necessarily in a financial way. We will denote for convenience the two states with $u$ -standing for 'up', and $d$ -standing for 'down'. We also make the assumption that if the agent could choose tomorrow's state, she would choose 'up'. For example, such states could be

$$\begin{tabular}{cc} \textbf{'up' state} & \textbf{'down' state} \... ... Tottenham wins & Tottenham loses \\ LSE expands & LSE crashes \end{tabular}$$

We will denote the random variable which represents tomorrow's state with $%% \omega _{1}$, the number '1' meaning that we are talking about tomorrow. Therefore, $\omega _{1}$ can take two distinct values: $\omega _{1}=u$ or $%% \omega _{1}=d$. Of course, today's state, $\omega _{0}$, is known, since it is directly observed. The above are summarized in the tree of figure 4.1

Figure 4.1: The simple tree one-period tree

It is important to observe that we have not mentioned at all the probabilities of ending up 'up' or 'down' tomorrow.

Now we turn to state dependent claims. Say that there exists some kind of security, which would pay off £1 if the state tomorrow is 'up' and nothing otherwise. We will denote the price of such a security with $q_{u}$. We can easily define $q_{d}$ in a similar way. These prices are called the state prices. The name is straightforward, for instance $q_{d}$ can be thought of as the insurance premium that the agent is prepared to pay, in order for her to enjoy £1 if $\omega _{1}=d$. It is important to investigate the factors that will affect the state price $q_{d}$.

• The likelihood of $\omega _{1}=d$ or $\omega _{1}=u$. It is easy to see that the higher the probability of moving 'down' [resp. 'up'], the higher the state price $q_{d}$ [resp. $q_u$] would be. In an efficient economy, the seller of this security would demand more, and the buyer would be ready to pay more.

• The preferences. Suppose that the agents utility, which has to be state-dependent, is of the standard form, illustrated in figure 4.2. The utility has to be increasing and concave. Make the assumption that the absolute effects of the state changes on the consumption itself^4.1 are equal. The money the agent looses if the state is 'down' is the same as the money she would make if the state was 'up'. Also assume that the probabilities are equal, in order to eliminate the aforementioned likelihood effect. Even in this [perfectly symmetric] case, the special utility shape implies that the absolute utility changes are not equal: The drop of the utility in the 'down' state is higher than the utility gains in the 'up' state. A rational agent would therefore value the 'down' state with a higher state price: more money would be happily paid out in order to ensure a higher consumption in the 'down' state.

  Figure 4.2: Effects of utility on state prices

  

Now consider that the agent buys both securities. She will enjoy £1 if $\omega _{1}=u$, and £1 if $\omega _{1}=d$. She has paid $%% q_{u}+q_{d}$, and has secured £1 in the next period. In fact, she has bought an one-period bond, implying that

$$q_{u}+q_{d}=b_{0}=\frac{1}{1+r}$$.

In such an economy, the state prices determine the bond prices.

We will now consider a security that has state dependent cash flows. It will pay $c_{k}$ if $\omega _{1}=k$, which means that it pays $c_{u}$ if the state of the world moves 'up' and $c_{d}$ if it moves 'down'. Such a security could be a stock that can go up or down, an option that pays off under circumstances, a bond that is affected by the short rate, etc. All financial assets can be thought of as such. What will be the price of such a security today? Obviously

$$c_{0}=q_{u}c_{u}+q_{d}c_{d}$$.

The state prices and the possible cash flows determine the asset prices.

If there were more than two states in the world, one can easily see that the bond and asset prices are given by

$$b_{0} = \sum_{k}q_{k}$$

$$c_{0} = \sum_{k}q_{k}c_{k}$$