Complete Markets Without a Risk-Neutral Measure
Summary
The document constructs a one-period market with two possible states, a risk-free asset, and one risky asset. When the risky asset has different payoffs in the two states, the two payoff equations can be solved uniquely for any contingent claim, so the market is complete. In the example, the risk-free asset has zero return, and completeness requires the risky asset’s state payoffs to differ.
The initial risky-asset price can nevertheless lie outside the range of its possible payoffs. That creates an arbitrage opportunity, such as borrowing to buy the asset when its future payoff is always higher than its initial price. The example shows why completeness alone does not guarantee a risk-neutral measure: the second fundamental theorem’s connection between completeness and uniqueness of that measure applies under the no-arbitrage condition. The discussion is limited to a simple one-period, two-state model and does not develop the general theorem.
Key ideas
- With two states and a risk-free asset, one risky asset can complete the market if its payoffs differ across states.
- The replication portfolio is found by solving one payoff equation for each state.
- An initial price outside the range of possible future payoffs can create arbitrage.
- The second fundamental theorem relies on no-arbitrage assumptions.
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Full text
# Complete market without risk-neutral measure
# Complete market without risk-neutral measure
> 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?
## Answer by Confused Quant (score 1)
https://quant.stackexchange.com/a/77662
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.