Checking for Arbitrage with State-Contingent Asset Payoffs

Summary

The document illustrates a basic arbitrage test using a portfolio formed from two assets. It proposes buying one unit of Asset 3 and shorting two units of Asset 2, then compares the portfolio’s terminal payoff across five states. The listed outcomes are nonnegative in every state and positive in two, which would indicate an arbitrage if the initial cost were truly zero and the payoffs were correctly specified.

The example’s arithmetic needs careful checking: it says two units of Asset 2 are shorted at $200 each but records only $200 of proceeds, then claims the initial net cash flow is zero against a $200 purchase. Thus the stated zero-investment condition does not follow from the described trades. The payoff comparison is a useful template, but the prices, short-sale proceeds, and initial cash flow must be reconciled before concluding that an arbitrage exists.

Key ideas

• An arbitrage candidate can be checked by comparing its payoff in every possible state.
• A zero-cost portfolio with nonnegative payoffs and a positive payoff in at least one state meets the basic payoff criterion for arbitrage.
• The example proposes buying Asset 3 and shorting two units of Asset 2.
• The stated short-sale proceeds conflict with the per-unit sale price, so the initial-cost claim requires correction.

Tags

# Is there an arbitrage ? (Solve question 1c)


# Is there an arbitrage ? (Solve question 1c)












How do we solve question 1 part c?

## Answer by carry_and_pray (score 1)

https://quant.stackexchange.com/a/81150

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.