For what fixed interest rates is a certain single-period, finite-state market arbitrage free?

Question

A single period market with three states of nature $\omega_1$, $\omega_2$ and $\omega_3$ is given, in which a single asset is available, namely a stock that is worth $8$ units today, and whose payoff tomorrow is $9$ with probability $1/4$, $11$ with probability $1/4$ or $12$ with probability $1/2$, depending on whether the state of nature is $\omega_1$, $\omega_2$ or $\omega_3$, respectively. For what fixed interest rates $r \geq 0$ is this market arbitrage free?

---

The solution

According to the official solution that is available to me, the answer is $\left(\frac{1}{8}, \frac{1}{2}\right)$. The solution emphasizes the point that when $r \in \{1/8, 1/2\}$, arbitrage opportunities exist, however no concrete examples of arbitrage-yielding portfolios are given.

---

My attempt at a solution

According to the first fundamental theorem of asset pricing, the market is arbitrage free iff there is an EMM distribution, which makes today's stock price equal to the present value of tomorrow's stock price, i.e. iff there exist non-negative numbers $p_1$, $p_2$ and $p_3$, such that $p_1 + p_2 + p_3 = 1$ and such that $$ 8 = \frac{1}{1 + r} \left(9p_1 + 11p_2 + 12p_3\right) $$ Solving for $r$, and substituting $p_3 = 1 - p_1 - p_2$, one obtains that $r$ is an arbitrage-free fixed interest rate iff $$ r \in \left\{\frac{1}{2} - \frac{3}{8}p_1 - \frac{1}{8}p_2 :\mid p_1, p_2 \geq 0, p_1 + p_2 \leq 1\right\} = \left[\frac{1}{8}, \frac{1}{2}\right] $$

Note, in particular, that when $r \in \{1/8, 1/2\}$ the market is arbitrage free, contrary to what the official solution states.

What am I doing wrong? Can you please give me a concrete example for an arbitrage-yielding portfolio when $r \in \{1/8, 1/2\}$?

Answer 1

When looking for an equivalent martingale measure it is important to not forget the "equivalent" part. In order for a martingale measure $\mathbb Q$ to be equivalent to $\mathbb P$ we need $\mathbb Q(\omega_i)>0 \Leftrightarrow \mathbb P(\omega_i)>0 $. This means they must agree on what events are possible.

You did everything right except that we need $p_1, p_2>0$ with strict inequality. we see that if $r=1/2$ then this implies $\mathbb Q(\omega_1)=\mathbb Q(\omega_2)=0$ so the only martingale measure we can devise is not equivalent. An arbitrage in that case is simple: Short the stock and put everything in the bank. Then, in the worst case (i.e. $\omega_3$) we lose $4$ units while always getting $4$ from the interest.

Answer 2

Edit:

Equilibrium Measure: $\pi_i$= π ( ω i ) on the set Ω of possible market scenarios is said to be an equilibrium measure (or risk-neutral measure) if, for an asset A , the share price of A at time t = 0 is the discounted expectation, under π , of the share price at time t = 1,

Fundamental Theorem of Arbitrage pricing: There exists an equilibrium measure if and only if arbitrages do not exist.

Given these two definitions, I would say that for every equilibrium measure, there happens to be a discounting interest rate if and only if arbitrages do not exist. In this context, my below reasoning is quite a good fit.

For it to be market arbitrage free, there must exist $p_1$ and $p_2$ such that the equality for r holds true in the range.

You have nicely derived the expression for r. Let us take that expression and play around with inequalities. For any range outside of the interval $(\frac{1}{8},\frac{1}{2})$, there should not be a combination of probabilities that will obey the simple laws of probability.

So, $r = \frac{1}{2} - \frac{3}{8}p_1 -\frac{1}{8}p_2$

Let us analyse the region $r<\frac{1}{8}$

$\frac{1}{2} - \frac{3}{8}p_1 -\frac{1}{8}p_2 <\frac{1}{8}$

$\frac{3}{8}p_1 +\frac{1}{8}p_2 >\frac{3}{8}$

$ 3p_1 +p_2 > 3$. Extreme values of $p_1$ taking the value of 1 shall not maet the inequality hold.

Similarly $\frac{1}{2} - \frac{3}{8}p_1 -\frac{1}{8}p_2 >\frac{1}{2}$

$\frac{3}{8}p_1 +\frac{1}{8}p_2 <0$

This does not hold true too for any non negative probabilities. Hence outside of the interval there do not exist market arbitrage free interest rate.

For a concrete example of a market arbitrage free interest rate in the give interval, we shall be able to find $p_1,p_2,p_3$ where $p_1 = \frac{2}{5}, p_2 = \frac{1}{10} and p_3 = \frac{1}{2}$ that shall make the discounted expected payoff equal to the today's market price.