Testing Market Completeness with Equivalent Martingale Measures

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


# 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 $T = 1, r = 0.05, S_0 = 1, u =1.5, m = 1, d =\frac{1}{u}.$

- By considering the set of EMMs (Equivalent Martingale Measures) or otherwise, show that the market is incomplete.

- 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

- 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 $$\mathbb{P}_1 \neq \mathbb{P}_2.$$

- 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 by Achrbot (score 2, accepted)

https://quant.stackexchange.com/a/78632

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:

- $q_d, q_m, q_u\in[0,1]$, and $q_u = 1-q_d-q_m$, for $\mathbb{Q}$ to be a probability measure.

- $q_d, q_m, q_u>0$, for the equivalence part.

- $S_0 = \mathrm{e}^{-rT}\mathbb{E}^{\mathbb{Q}}\left[S_T \right]$, for the Martingale part.

Computing the expectation we get \begin{equation} S_0 = \mathrm{e}^{-rT}\left(0.5S_0q_d + S_0q_m + 1.5S_0(1-q_d-q_m)\right). \end{equation} 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.