Let $\left(S_{n}^{0}, S_{n}\right)_{0 \leqslant n \leqslant T}$ be a discrete-time asset price model in $\mathbb{R}^{d+1}$ with numéraire.

(i) What is meant by an arbitrage for such a model?

(ii) What does it mean to say that the model is complete?

Consider now the case where $d=1$ and where

$$S_{n}^{0}=(1+r)^{n}, \quad S_{n}=S_{0} \prod_{k=1}^{n} Z_{k}$$

for some $r>0$ and some independent positive random variables $Z_{1}, \ldots, Z_{T}$ with $\log Z_{n} \sim N\left(\mu, \sigma^{2}\right)$ for all $n$.

(iii) Find an equivalent probability measure $\mathbb{P}^{*}$ such that the discounted asset price $\left(S_{n} / S_{n}^{0}\right)_{0 \leqslant n \leqslant T}$ is a martingale.

(iv) Does this model have an arbitrage? Justify your answer.

(v) By considering the contingent claim $\left(S_{1}\right)^{2}$ or otherwise, show that this model is not complete.