Seeking clarity on the concept of equivalent martingale measures

Question

Question

Consider a one-step trinomial tree, where there are two traded assets, a bond with risk-free rate, $r$, a stock with initial price, $S_0$, and terminal price

$$S_T = \begin{cases} S_0u,& \text{with probability} \ p_u \\ S_0m,& \text{with probability} \ p_m \\ S_0d,& \text{with probability} \ p_d \end{cases}$$ where $p_u, p_m, p_d >0.$ Suppose that $T = 1, r = 0.05, S_0 = 1, u =1.5, m = 1, d =\frac{1}{u}.$

1. By considering the set of EMMs or otherwise, show that the market is incomplete.

2. Suppose a European call on the stock with strike price $0.9$ and maturity time T is an asset traded in the market. Explain whether or not including this option as a traded asset in the market has made the market complete.

My attempt

1. From my understanding, a market is complete if there exists a unique EMM. Also, if there exists infinitely many EMMs, then the market is incomplete. Thus, I think it is sufficient to prove this by establishing the existence of two EMMs, $\mathbb{P}_1$ and $\mathbb{P}_2$ and conclude that $$\mathbb{P}_1 \neq \mathbb{P}_2$$ as we can then construct infinitely many EMMs using a linear combination of $\mathbb{P}_1$ and $\mathbb{P}_2$. However, I am stuck as I am unsure how to construct $\mathbb{P}_1$ and $\mathbb{P}_2$ and prove that $$\mathbb{P}_1 \neq \mathbb{P}_2.$$

2. From my understanding, a market is complete if there are at least as many tradable assets as risk sources. In a discrete setting, we will need at least as many traded assets with linear independent payoffs as there are states of nature. However, I am not entirely sure how to interpret the preceding sentence and as a result, I am unsure whether the inclusion of the call makes the market complete or not.

Any intuitive explanations will be highly appreciated!

Answer 1

As you stated a market is complete if there exists a unique EMM (Second Fundamental Theorem of Asset Pricing).

To show that the standard trinomial model is incomplete we can proceed as follows.

The martingale condition requires that

$$ S_{n-1} = \frac{1}{1+r}\mathbb{E}^{\mathbb{Q}}[S_n|\mathcal{F}_{n-1}] $$

we then couple this with the requirement that the probabilities sum to $1$. In our case:

$$ \begin{equation} \begin{cases} 1+r = uq_u + mq_m+dq_d \\ q_u+q_m+q_d = 1 \end{cases} \end{equation} $$

as you can see this system does not admit a unique solution, hence the EMM is not unique and the model in therefore incomplete. (try and solve it)

On the other hand, if we include an option (or in general a second asset), we need to include the martingality condition for that asset as well. this adds an equation in the previous system which makes the solution unique and hence the model complete.