# How to prove that a market is incomplete using the concept of EMMs?

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 (Equivalent Martingale Measures) 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!

Regarding your first problem, you are correct that constructing two different EMM's is sufficient to show that the market is incomplete. For a candidate measure $$\mathbb{Q}$$ to be an EMM, we require that:

1. $$q_d, q_m, q_u\in[0,1]$$, and $$q_u = 1-q_d-q_m$$, for $$\mathbb{Q}$$ to be a probability measure.
2. $$q_d, q_m, q_u>0$$, for the equivalence part.
3. $$S_0 = \mathrm{e}^{-rT}\mathbb{E}^{\mathbb{Q}}\left[S_T \right]$$, for the Martingale part.

Computing the expectation we get $$$$S_0 = \mathrm{e}^{-rT}\left(0.5S_0q_d + S_0q_m + 1.5S_0(1-q_d-q_m)\right).$$$$ You can show that this equation has multiple solutions for $$q_d, q_m \in [0,1]$$, and thus the market is incomplete.

This changes if we add a (non-degenerate) call option to the market. Take your example, of a call option $$C$$ with strike $$K=0.9S_0$$. By the Martingale property, \begin{align} C_0 &= \mathrm{e}^{-rT}\left(C_dq_d + C_mq_m + C_u(1-q_d-q_m)\right) \\ &= \mathrm{e}^{-rT}\left( S_0(1-0.9)q_m + S_0(1.5-0.9)(1-q_d-q_m)\right). \end{align} Combining this with our first equation, we get a system of equations with a unique solution, and thus only one EMM exists.

As a side note, this should make it clear why we need the new asset to be linearly independent. For example, adding a call option with strike $$K=0.4S_0$$ would not lead to a complete market, since this is a linear combination of $$S_0$$ and the bank account.