# 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.

• The market is complete as there is no uncertainty but two riskless assets. For a more formal proof you could first state the definition of completeness you are referring to.
– fes
Commented Aug 8, 2021 at 17:06
• @fesman hanks for your reply! The definition of completeness I am referring to is: - The market model is complete if, for any finite $T>0$, every $T$ -claim $\xi$ with bounded discounted payoff $\xi / S_{0}(T)$ is attainable. - If the market is complete, then every $T$ -claim $\xi$ with $\mathbb{E}_{\mathbb{Q}}\left[\frac{|\xi|}{S_{0}(T)}\right]<\infty$ is attainable. Commented Aug 8, 2021 at 22:50
• You can attain payoff $\xi$ at $T$ e.g. using the first riskless asset by saving $\xi$ in the first period and then selling it at $T$. Hence the market is complete.
– fes
Commented Aug 9, 2021 at 5:36
• @fesman Ah I see thank you. But is my arbitrage strategy valid to show that the model is not free of arbitrage? Commented Aug 9, 2021 at 6:24
• For no-arbitrage, note that for $k\in\{0,1\}$, we have in your model: $\text{d}S_k(t)=r_k\text{d}t$, with $r_0=0$ and $r_1=1$. Then per Proposition 7.9 in Arbitrage Theory in Continuous Time (Björk, 2020, 4th Ed.), the model admits arbitrage, namely there can be only one riskless/deterministic asset/return in a market model to avoid arbitrage. Commented Aug 9, 2021 at 9:15

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.