# Find arbitrage opportunity in the given market model

Consider the following 3-period-market-model:

The discounted price of the risky asset $$S$$: How can I find an arbitrage opportunity in this model?

I know that there would be no arbitrage if we replace the first $$8$$ by something in $$(8,12)$$ or if we replace the second $$8$$ by something in $$(5,8)$$ but I don't know how I can explicitly state the arbitrage opportunity in the given market. So I'm looking for a portfolio which is an arbitrage opportunity.

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The arbitrage strategy is: if the stock is at 8 at $$t=1$$ buy it else do nothing. Then sell it at $$t=2$$.
Either the stock has increased to 12 and you made a profit or it is still worth $$8$$ and your PnL is 0.
So you are guaranteed not to lose any money but you have a non zero probability of making money (equal to the probability of the stock price increasing to 12 conditional on it being 8 at $$t=1$$).
• In addition, although not stated in the OP, if there is a risk-free rate $r$, you can short the asset $S$ at the first period if $S=2$ and buy it back for the same price at the next period, and the same if $S=12$ at the second period (and earn $r$ by lending the shorted amount). – Daneel Olivaw Jun 22 at 17:54