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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.

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  • $\begingroup$ 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. $\endgroup$
    – fesman
    Aug 8 at 17:06
  • $\begingroup$ @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. $\endgroup$ Aug 8 at 22:50
  • $\begingroup$ 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. $\endgroup$
    – fesman
    Aug 9 at 5:36
  • $\begingroup$ @fesman Ah I see thank you. But is my arbitrage strategy valid to show that the model is not free of arbitrage? $\endgroup$ Aug 9 at 6:24
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    $\begingroup$ 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. $\endgroup$ Aug 9 at 9:15
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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.

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