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I have a questions on arbitrage. These questions are from my lecture notes. And I don’t understand their some points.

Ex1: there is a single step market as follows enter image description here

here, there are two securities and security prices are (1, 1.2), And 1 is bond, 1.2 is a risky asset.

We have a bond with its price 1 and interest rate 10%. So bonds next time prices are (1.1, 1.1, 1.1) and $(\alpha, \beta, \gamma)$ is the next time prices for risky asset. Question: what is the conditions on the given parameters such that there is an arbitrage opportunity?

Such type of questions are solved by two ways:

First way:

Arbitrage: At first, we have a portfolio, and the value of this initial portfolio is zero. So, The portfolio $\phi = (\phi_0, \phi_1)$=(bond, asset)

$$\phi=1*\phi_0+1.2\phi_1=0$$

From here, $$\phi_1 = -\frac{\phi_0}{1.2}$$

And in the next step, we have must non-negative value of portfolio and at least one of them is strictly positive such that we can say that there is an arbitrage in the market.

So the next period values are as follows:

$$1.1\phi_0+\alpha\phi_1$$

$$1.1\phi_0+\beta\phi_1$$

$$1.1\phi_0+\gamma\phi_1$$

These three values must be non-negative and at least one of them must be strictly positive such that there is an arbitrage opportunity in the market. After that, I guess, I need to substitute $\phi_1 = -\frac{\phi_0}{1.2}$ into three equations.

But, I am stack at this point.

Second way:

Second way is related to martingale.

$P_1, P_2, P_3$ are probability.

$E[S_1^{(1)}]= P_1*\alpha + P_2*\beta+P_3*\gamma$

And $$\frac{E[S_1^{(1)}]}{1.1}=1.2$$

I need to check the existence of $(p_1, p_2, p_3)$. We have 3 unknowns, 2 equations and it’s solution must satisfy the positivity constraint $P_i^*\ge 0$ such that there is an arbitrage opportunity.

What I understand from the lecture is that. But I could not proceed these two solutions. Please help me at this point. Thanks a lot.

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