Suppose that we are in one period model, where assets are bought (in positive and/or negative values) at $t=0$ and are redeemed for their value at time $t=1$. Furthermore, suppose that there are three assets: $A$, $S$, and $R$.

Asset $A$ is a riskless investment with initial cost of $90$ and final payoff of $100$. Asset $S$ is a risky asset with an initial cost of $20$, that returns $25$ if state $\omega_1$ realizes at $t=1$, and returns $15$ if state $\omega_2$ realizes at $t=1$.

All that we know about Asset $R$ is that it if $\omega_1$ occurs, it returns $30$, and if $\omega_2$ occurs, it returns $28$. Given that there are no arbitrage opportunities, what is the strongest statement we can make about the initial price of Asset $R$?

Obviously, the price for one unit of $R$ must be bigger than 0. My next inclination was to say that the price of $R$ must also be bigger than $28$, but when I attempted to prove this via contradiction, I was unable to. Is there an effective strategy for solving this type of question?


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