No Free lunches and price uniqueness relation problem

Question

In the article Martingales and arbitrage in multiperiod securities markets from J Michael Harrison and David M. Kreps, they begin considering a market with time interval $[0, T]$ with only two periods $0$ and $T$. In this market they consider $(Z_{t})$ as a $k+1$ dimensional process adapted with respect to $(\Omega, \mathcal{F}_{t \in \lbrace 0, T\rbrace}, P)$ where $Z^{(k)}_{t}$ represents the value of the $k$-th asset at time $t$, and $Z_{t}^{(0)}=1$ for all $t$ and $w \in \Omega$. They assume that $\mathcal{F}_{0} = \lbrace \emptyset, \Omega \rbrace $. They define a self-financing simple strategy as a process $(\theta_{t})$ such that it satisfies the following

1. $(\theta_{t})$ is a process $\mathcal{F}_{t}$ adapted
2. $\theta_{t}\cdot Z_{t} \in L^{2}(\Omega, \mathcal{F}, P)$ for all $t$
3. $\theta_{t}$ is constant over the interval $[0,T)$ (this is the case for only two periods. This can emmediately be generalized to the scenario of more than two periods )
4. $\theta_{t_{n-1}}\cdot Z_{t_{n}} = \theta_{t_{n}}\cdot Z_{t_{n}} $ where $t_{n} = 0$ for $n=0$; and $t_{n} = T$ for $n=1$. (self-financing property)

They say that a self-financing strategy $\theta$, is a free lunch if $\theta_{0} \cdot Z_{0} \leq 0$ and $\theta_{T} \cdot Z_{T} \in (L^{2}(\Omega, \mathcal{F}, P))^{+}$

The authors say that in order to price a claim $x \in L^{2}(\Omega, \mathcal{F}, P) $ in only one way, we must ensure that if there exist two strategies $(\theta_{t})$ and $(\theta^{*}_{t})$ such that $\theta_{T}\cdot S_{T} = x = \theta^{*}_{T} \cdot S_{T} $, then $\theta_{0}\cdot S_{0} = \theta^{*}_{0} \cdot S_{0} $.

The author claimed this proposition:

If the there is not free lunches, then for all self-financing strategies $(\theta_{t})$ and $(\theta^{*}_{t})$ such that $\theta_{T}\cdot S_{T} = x = \theta^{*}_{T} \cdot S_{T} $, we get $\theta_{0}\cdot S_{0} = \theta^{*}_{0} \cdot S_{0} $.

Quesiton I wonder if there is something else that is not mentioned in this article, because this proposition is not true.

Consider this counter example

$\theta_{t} =(a_{t}, b_{t}, c_{t})_{t \in \lbrace 0, T \rbrace}, \ \ \ \ \theta^{*}_{t} =(a^{*}_{t}, b^{*}_{t}, c^{*}_{t})_{t \in \lbrace 0, T \rbrace}, \ \ \ \ Z_{t} =(1, Z^{(1)}_{t}, Z^{(2)}_{t})_{t \in \lbrace 0, T \rbrace}$

with $Z^{(1)}_{t}, Z^{(2)}_{t} > 0$ for all $t$

Now consider the following set of equations

1. $a_{T} + b_{T}Z^{(1)}_{T} + c_{T}Z^{(2)}_{T} = \gamma $ with $\gamma >0$
2. $a^{*}_{T} + b^{*}_{T}Z^{(1)}_{T} + c^{*}_{T}Z^{(2)}_{T} = \gamma $
3. $a_{0} + b_{0}Z^{(1)}_{T} + c_{0}Z^{(2)}_{T} = \gamma $
4. $a^{*}_{0} + b^{*}_{0}Z^{(1)}_{T} + c^{*}_{0}Z^{(2)}_{T} = \gamma $
5. $a_{0} + b_{0}Z^{(1)}_{0} + c_{0}Z^{(2)}_{0} = \alpha $
6. $a^{*}_{0} + b^{*}_{0}Z^{(1)}_{0} + c^{*}_{0}Z^{(2)}_{0} = \alpha^{*} $

with $\alpha, \alpha^{*} > 0$ and $\alpha \neq \alpha^{*} $

If $\begin{vmatrix} Z^{(1)}_{T} & Z^{(2)}_{T} \\ Z^{(1)}_{0} & Z^{(2)}_{0} \end{vmatrix} \neq 0 $ we get that

$\left[ \begin{matrix} Z^{(1)}_{T} & Z^{(2)}_{T} \\ Z^{(1)}_{0} & Z^{(2)}_{0} \end{matrix} \right] \left[ \begin{matrix} b_{0}^{*} \\ c_{0}^{*} \end{matrix} \right] = \left[ \begin{matrix} \gamma - a_{0}^{*} \\ \alpha^{*} - a_{0}^{*} \end{matrix} \right] (\text{equation 4 and equation 6})$

and

$\left[ \begin{matrix} Z^{(1)}_{T} & Z^{(2)}_{T} \\ Z^{(1)}_{0} & Z^{(2)}_{0} \end{matrix} \right] \left[ \begin{matrix} b_{0} \\ c_{0} \end{matrix} \right] = \left[ \begin{matrix} \gamma - a_{0} \\ \alpha - a_{0} \end{matrix} \right] (\text{equation 3 and equation 5})$

have a solution for $Z_{t} =(1, Z^{(1)}_{t}, Z^{(2)}_{t})_{t \in \lbrace 0, T \rbrace}$; $\gamma > 0$, and $\alpha, \alpha^{*} > 0$ and $\alpha \neq \alpha^{*} $.

Notice that

1. Equations 1) and 2) are equivalent to say that $\theta_{T} \cdot Z_{T} = \gamma = \theta^{*}_{T} \cdot Z^{*}_{T}$
2. Equations 3) and 4) are equivalent to say that $\theta_{T} \cdot Z_{T} = \theta_{0} \cdot Z_{T}$ and $\theta^{*}_{T} \cdot Z_{T} = \theta^{*}_{0} \cdot Z_{T}$
3. Equations 5) and 6) are equiavalent to say that $\theta_{0} \cdot Z_{0} = \alpha > 0 $ and $\theta^{*}_{0} \cdot Z_{0} = \alpha > 0 $, but $ \theta_{0} \cdot Z_{0} \neq \theta^{*}_{0} \cdot Z_{0}$ . In other word strategies that are not free lunch and compute different prices.

I choosed this example because this is pretty easy to work with random variables equal to constants. This also can be done with random variable differnt from constants using the same method but multiplying each one of the terms by their respective probability. The inequalities hold using convex combinations and changing the probabilities (the convex coefficients) in order to get the inequalities.

The proposition holds if we assume that $(Z_t)$ is supermartingale, but the authors do not say anything else about the $Z$ in the article.

Thanks in advance.