The Financial Structure

Remember that in our model, a financial asset $k$ is represented by a list of gross-return rates $\left(r_k^{s_t}\right)$, where $s_t$ is a state that occurs in period $t$ and $r_k^{s_t}$ is the gross rate of return. Also remember that by convention, the gross return of the asset is $-1$ at the state in which the asset is issued.

There are three periods and no uncertainty. The states of the economy are just the time periods.

There are three financial assets. Two short-term bonds and one long-term bond:

1. Short-term bond 1: pays $r_1$ per dollar invested from $t_0$ to $t_1$
2. Short-term bond 2: pays $r_2$ per dollar invested from $t_1$ to $t_2$
3. Long-term (zero-coupon) bond: pays $r_3$ per dollar invested from $t_0$ to $t_2$

Let's check whether the previous financial structure admits an arbitrage opportunity. The returns of the assets are:

1. $\mathbf{r}_1 = (-1,\; r_1,\; 0)$
2. $\mathbf{r}_2 = (0,\; -1,\; r_2)$
3. $\mathbf{r}_3 = (-1,\; 0,\; r_3)$

Let $\mathbf{w} = (w_1, w_2, w_3)$ be a portfolio. $\mathbf{w}$ is an arbitrage opportunity if and only if:

and at least one of these inequalities is strict.

There are two periods and uncertainty about the state in period 1. The states of the economy are $t_0$, $s_1$ and $s_2$.

There are three financial assets:

1. Bond: pays $r_0$ per dollar invested in both states
2. Asset 1: pays $r_1$ per dollar invested only in state $s_1$
3. Asset 2: pays $r_2$ per dollar invested only in state $s_2$

Let's check whether this financial structure admits an arbitrage opportunity. The returns of the assets are:

1. $\mathbf{r}_0 = (-1,\; r_0,\; r_0)$
2. $\mathbf{r}_1 = (-1,\; r_1,\; 0)$
3. $\mathbf{r}_2 = (-1,\; 0,\; r_2)$

Let $\mathbf{w} = (w_0, w_1, w_2)$ be a portfolio. $\mathbf{w}$ is an arbitrage opportunity if and only if:

and at least one of these inequalities is strict.

Consider the following portfolio $\mathbf{w}=(1, r_1, -1)$, which corresponds to the following dynamic strategy:

1. Buy $1$ dollar worth of asset 1 and sell $1$ dollar worth of asset 3 (the strategy is self-financing).
2. Re-invest the returns of asset 1 in asset 2: buy $r_1$ dollars worth of asset 2, so the strategy is also self-financing in period 1.
3. In the last period, the strategy yields $r_2 \cdot r_1$ (return to investment in asset 2) minus $r_3$ (payment to the buyers who bought the asset from the agent in period 0).

If $r_1 r_2 - r_3 \gt 0$, this strategy is an arbitrage opportunity. If $r_1 r_2 - r_3 \lt 0$, the portfolio $-\mathbf{w} = (-1, -r_1, 1)$ is an arbitrage opportunity. Finally, if $r_1 r_2 = r_3$, there are no arbitrage opportunities.

Consider the following portfolio $\mathbf{w} = (-r_1 r_2/r_0,\; r_2,\; r_1)$, which corresponds to the following strategy:

0. Buy $r_2$ dollars of asset 1 and $r_1$ dollars of asset 2. This strategy is not self-financing at $t_0$: it generates a net cash flow of $r_1 r_2/r_0 - r_1 - r_2$.
1. Sell $r_1 r_2/r_0$ dollars of the bond to offset all future payoffs: the strategy yields zero in both states $s_1$ and $s_2$.

If $r_1 r_2/r_0 - r_1 - r_2 \gt 0$, this strategy is an arbitrage opportunity. If $r_1 r_2/r_0 - r_1 - r_2 \lt 0$, the portfolio $-\mathbf{w} = (r_1 r_2/r_0,\; -r_2,\; -r_1)$ is an arbitrage opportunity. Finally, if $r_1 r_2 = r_0(r_1 + r_2)$, there are no arbitrage opportunities.

Notice that if there exists one arbitrage opportunity then there are an infinite number of arbitrage opportunities.

In cases like the above examples (economies with three states and three assets), try the following method. Fix any asset—let's call it asset $k$. Then create two portfolios:

$\mathbf{v}\rightarrow$

invests one dollar in asset $k$ and zero in the other assets.

$\mathbf{z}\rightarrow$

invests in the other assets but not in asset $k$. The payoff of $\mathbf{z}$ in the states in which asset $k$ is not issued coincides with the payoffs of $\mathbf{v}$.

Let $\mathbf{w} = \mathbf{v} - \mathbf{z}$. Portfolio $\mathbf{w}$ yields zero payoffs in every state except the state in which asset $k$ is issued.

If the payoff of $\mathbf{w}$ in that state is positive, then $\mathbf{w}$ is an arbitrage opportunity. If the payoff is negative, then $-\mathbf{w}$ is an arbitrage opportunity. Finally, if the payoff is zero—for every choice of $k$—there are no arbitrage opportunities in this economy.

Try to practice this intuition by playing with the Arbitrage app.

When markets are complete, the equilibrium allocation (equilibrium consumption baskets) in the financial economy and in the contingent goods economy are the same.

Intuition: If we can transfer income freely across states then it is as if we have only one budget constraint as in the case of contingent goods markets.

Suppose the only available asset is asset 1, the short-term bond from $t_0$ to $t_1$:

For instance, it is impossible to transfer income from $t_1$ to $t_2$.

Suppose the only available asset is asset 0, the bond:

For instance, it is impossible to transfer income from $s_1$ to $s_2$ while keeping income at $t_0$ constant.