Testing Arbitrage in a Two-Period Stock and Cash Model

Summary

The document poses an arbitrage-free condition for a finite-state, two-period market with a nonnegative interest rate, a specified initial stock price, and stock prices at each future state. It asks which interest rates rule out arbitrage and reports the questioner’s preliminary one-period analysis, which produces a candidate interval. No answer or proof is included in the supplied text, so that interval is not established as correct.

The setup illustrates why a multi-period arbitrage question must account for the joint paths of asset prices, rather than checking each period in isolation. A complete solution would need to examine self-financing trading strategies over both periods, including the possibility of adjusting holdings after the first-period outcome, and determine when a nonnegative terminal payoff can be made positive in at least one state at no initial cost. The document supplies the model inputs but no probability values beyond positivity for every state, and it gives no derivation or caveats beyond the unanswered status of the question.

Key ideas

• The model has four possible states and stock prices specified at two future dates.
• Arbitrage-free rates must be assessed across complete trading paths.
• Checking each period separately may miss strategies that change after the first date.
• The proposed rate interval is only a preliminary result because no solution is provided.

Tags

# For which interest rates r is the model arbitrage-free?


# For which interest rates r is the model arbitrage-free?












> Given $\Omega=\{\omega_1,...,\omega_4\}$ and a probability measure $\mathbb{P}$ on $(\Omega, \mathcal{P}(\Omega))$ where $\mathbb{P}(\{\omega_i\})>0$ for all $i$. Let, furthermore, $r\geq 0$, $S_0=5$ and $S_1(\omega_1)=S_1(\omega_2)=8$, $S_1(\omega_3)=S_1(\omega_4)=4$, $S_2(\omega_1)=9$, $S_2(\omega_2)=S_2(\omega_3)=6$, $S_2(\omega_4)=3$. For which interest rates $r$ is the model arbitrage-free?

I looked at the one-period arbitrage possibilities and got $r\in [0,1/3)$. I'd appreciate it if someone could double-check it since my textbook gives no solutions.

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