How do we solve question 1 part c?
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1$\begingroup$ Hint:notice that the third asset is >= the 2nd asset in all scenarios. $\endgroup$– dm63Commented Nov 13 at 11:44
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$\begingroup$ Though the question is clear enough, it would be nice if you also cited the source in detail. $\endgroup$– AlperCommented Nov 22 at 2:19
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1 Answer
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At time $t_0$
- buy 1 unit of Asset 3 at $\\\$200$
- short sell 2 units of Asset 2 at $\\\$200$ each receiving $\\\$200$
- Our net cash flow at $t_0$ is $\\\$200$ (from short selling asset 2) - $\\\$200$ (from buying asset 3) $= 0$
At time $t_2$, we can calulate the net payoff in each state by considering the payoffs from asset 3 and the obligations from the short position in asset 2
| State $(\omega)$ | Asset 3 Value at $ t_2 $ | Short Position in Asset 2 (2 units) | Net Payoff |
|---|---|---|---|
| $\omega_1$ | \$360 | $-2 \times \\\$180 = -\\\$360$ | \$0 |
| $\omega_2$ | \$120 | $-2 \times \\\$60 = -\\\$120$ | \$0 |
| $\omega_3$ | \$260 | $-2 \times \\\$126 = -\\\$252$ | \$8 |
| $\omega_4$ | \$200 | $-2 \times \\\$100 = -\\\$200$ | \$0 |
| $\omega_5$ | \$150 | $-2 \times \\\$72 = -\\\$144$ | \$6 |
Notice, we have non-negative payoffs in all states, and positive payoffs in states $\omega_3$ and $\omega_5$.
Thus, this strategy yields riskless profit with zero initial investment. The presence of this strategy indicates that the market is not free of arbitrage opportunities when the third asset is included.
