Edit: I'm a dumbass. The thing below is supposed to be just the motivation of asking. I want to ask for below and in general, hehe.
Assume that we have a general one-period market model consisting of d+1 assets and N states.
Using a replicating portfolio $\phi$, determine $\Pi(0;X)$, the price of a European call option, with payoff $X$, on the asset $S_1^2$ with strike price $K = 1$ given that
$$S_0 =\begin{bmatrix} 2 \\ 3\\ 1 \end{bmatrix}, S_1 = \begin{bmatrix} S_1^0\\ S_1^1\\ S_1^2 \end{bmatrix}, D = \begin{bmatrix} 1 & 2 & 3\\ 2 & 2 & 4\\ 0.8 & 1.2 & 1.6 \end{bmatrix}$$
where the columns of D represent the states for each asset and the rows of D represent the assets for each state
What I tried:
We compute that:
$$X = \begin{bmatrix} 0\\ 0.2\\ 0.6 \end{bmatrix}$$
If we solve $D'\phi = X$, we get:
$$\phi = \begin{bmatrix} 0.6\\ 0.1\\ -1 \end{bmatrix}$$
It would seem that the price of the European call option $\Pi(0;X)$ is given by the value of the replicating portfolio
$$S_0'\phi = 0.5$$
On one hand, if we were to try to see if there is arbitrage in this market by seeing if a state price vector $\psi$ exists by solving $S_0 = D \psi$, we get
$$\psi = \begin{bmatrix} 0\\ -0.5\\ 1 \end{bmatrix}$$
Hence there is no strictly positive state price vector $\psi$ s.t. $S_0 = D \psi$. By 'the fundamental theorem of asset pricing' (or 'the fundamental theorem of finance' or '1.3.1' here), there exists arbitrage in this market.
On the other hand the price of 0.5 seems to be confirmed by:
$$\Pi(0;X) = \beta E^{\mathbb Q}[X]$$
where $\beta = \sum_{i=1}^{3} \psi_i = 0.5$ (sum of elements of $\psi$) and $\mathbb Q$ is supposed to be the equivalent martingale measure given by $q_i = \frac{\psi_i}{\beta}$.
Thus we have
$$E^{\mathbb Q}[X] = q_1X(\omega_1) + q_2X(\omega_2) + q_3X(\omega_3)$$
$$ = 0 + \color{red}{-1} \times 0.2 + 2 \times 0.6 = 1$$
$$\to \Pi(0;X) = 0.5$$
I guess $\therefore$ that we cannot determine the price of the European call using $\Pi(0;X) = \beta E^{Q}[X]$ because there is no equivalent martingale measure $\mathbb Q$
I noticed that one of the probabilities, in what was attempted to be the equivalent martingale measure, is negative. I remember reading about negative probabilities in Wiki and here
Half of a Coin: Negative Probabilities
However the following links
Negative Probabilities in Financial Modeling
Why so Negative to Negative Probabilities?
mentioned by Wiki seem to assume absence of arbitrage so I think they are not applicable. Or are they?
Is it perhaps that this market can be considered to be arbitrage-free under some quasiprobability measure that allows negative probabilities?
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