Completeness and Arbitrage in a Deterministic Market with a Rising Asset
Summary
The example considers a deterministic market with a constant asset priced at one and a second asset whose value rises over time. With no uncertainty, the market is complete in the stated one-period setup because any terminal payoff can be obtained by holding the corresponding amount of the constant-price asset.
The increasing asset creates an arbitrage: buy one unit and short one hundred units of the constant asset at the initial time, then close both positions later. The initial net cost is zero, while the later net payoff is positive. This illustrates that completeness and absence of arbitrage are separate properties. The explanation relies on the simplified setup, including zero interest rates and unrestricted trading; it does not discuss transaction costs or market constraints.
Key ideas
- Completeness means that every attainable terminal payoff can be replicated by trading in the market.
- A constant-price asset can replicate any payoff in the deterministic one-period setup described.
- Buying the rising asset while shorting an equal initial value in the constant asset produces a zero-cost positive payoff.
- A market can be complete and still contain arbitrage opportunities.
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Full text
# Show a model is complete but not free of arbitrage
# Show a model is complete but not free of arbitrage
Let $\mathcal{F}=\{\Omega, \emptyset\}$ be the trivial $\sigma$ -algebra, and consider the deterministic financial market model with zero interest rates, $S_{0} \equiv 1$, and $n=1$ additional asset $S_{1}(t)=100+t$. Show that this model is complete but not free of arbitrage.
Edit: can someone point me in the right direction. I am quite lost as it seems quite trivial to me.
For example to show that the model is not free of arbitrage, I can construct an arbitrage strategy as below:
At time t: $\text{buy } K * S_{1}(t) = K * (100+t) \\ \text{cash} = - K * (100+t)$
At time T: $\text{sell } K * S_{1}(T) = K * (100+T) \\ \text{cash} = (K * (100+t)) - ( K * (100+t)) > 0 $
Hence arbitrage. But I am not sure if I am right or how to show that the model is complete.
## Answer by fes (score 4, accepted)
https://quant.stackexchange.com/a/66360
There is no uncertainty. Assume at $t=0$ I buy one unit of asset $1$ and sell $100$ units of asset $0$. Moreover, at $t=1$, I close both positions. At $t=0$ my payoff is $100-100=0$ and at $t=1$, it is $101-100=1$. Hence there is arbitrage.
Assume I want to attain a payoff of $\xi(T)$ at period $T$. I can obtain this e.g. by investing $\xi$ in asset $0$ at $t=0$ and selling it at $t=T$. Because $\xi(T)$ is arbitrary the market is complete.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.