Let $(\Omega, \mathcal{F}, \mathbb{P})$ a probability space. Consider a random variable $R \in \mathbb{L}^1$ such that $\mathbb{E}[R] > 0$ and $\mathbb{P}( R < 0) >0$. Define a 1-period market with riskless asset $S^0_0 = 1$, $S^0_1 = 1$ and risky asset $S^1_0 = 1$, $S^1_1 = 1 + R$. In this model an arbitrage opportunity is a vector $(\alpha, \beta) \in \mathbb{R}^2$ such that:
- $\alpha + \beta \leq 0$;
- $\alpha + \beta + \beta R \geq 0$ $\mathbb{P}$-a.s.;
- $\mathbb{P}(\alpha + \beta + \beta R > 0) > 0$.
It seems to me that taking as $\alpha = -1$ and $\beta = 1$ we get a (possible) arbitrage opportunity (to rule it out we would need more hypothesis on $R$. Hence, we can not conclude a priori that this market is arbitrage-free.
Is there anything wrong in this argument? If so how can I prove that this market is arbitrage free?
If we add the standard assumptions that $$ 1+R\ge 0\quad \mathbb P\text{-a.s. ( asset prices are nonnegative )} $$ $$ \mathbb E[1+R]=1\quad \text{( $S^1$ is a martingale )} $$ Then for any $\alpha,\beta$ with $\alpha+\beta\le 0$ and $\alpha+\beta+\beta R\ge 0$ $\mathbb P$-a.s. we have $$ \mathbb E[\alpha+\beta+\beta R]=\alpha+\beta\,\mathbb E[1+R]=\alpha+\beta\le 0\,, $$ and $$ \mathbb E[\alpha+\beta+\beta R]\ge 0\,. $$ Therefore, $$ \mathbb E[\alpha+\beta+\beta R]= 0\,. $$ Due to $\alpha+\beta +\beta R\ge 0$ it is not possible that $$ \mathbb P\{\alpha+\beta+\beta R>0\}>0\,. $$ Therefore, not arbitrage opportunity exists.