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Let $\mathcal{M}$ be a one-period model with $\Omega=\{\omega_1,\omega_2\}$ and $S_t^0=1$ for $t=0,1$.

Find a $D$ such that $S^d$, $d=1,...,D$ yields a complete market without a risk-neutral measure. Why does this not contradict the second fundamental theorem of asset pricing?

So I guess this is due to the fact that it second fundamental theorem of asset pricing needs the market to be arbitrage-free. However, I was not able to construct such a model. Can someone help?

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Yes, you are right in that the second fundamental theorem of asset pricing needs the market to be arbitrage-free. Now, the model: This model is based on Nicolas Privault's Notes on Stochastic finance, chapter 1:

Consider a one period model with one risk-free asset that yields r (in your case, r=0 since $S_t^{0}=1$ for t=0,1), and one risky asset, so D=1. Now, this market is complete if and only if every contingent claim C is attainable (i.e, hedgeable). Let $C(w_1)$ and $C(w_2)$ the payoff of the Contingent claim in states $w_1$ and $w_2$, respectively. Let's call $a:=S_1^1(w_1)$ and $b:=S_1^1(w_2)$ To build a portfolio $\xi_0$, $\xi_1$ that replicates this contingent claim, we have to solve the system of equations:

$$ (1+r)\xi_0 + \xi_1a = C(w_1) $$ $$ (1+r)\xi_0 + \xi_1b = C(w_2) $$

This system has a unique solution when the determinant is not 0, i.e when $(1+r)b - (1+r)a \ne 0$, that is, when $(1+r) \ne 0$ and $b \ne a$.

Then, if $b>a$ and $(1+r) \ne 0 $, the market is complete. However, if $S_0^1 < a$ (or $S_0^1 > b$), the market has an arbitrage opportunity, by buying $S^1$ (resp, selling $S^1$), borrowing from the risk-free asset and selling at t=1.

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