# Showing that a market model has arbitrage and describing martingales

This is an exercise which I came upon while studying an introduction to financial mathematics.

Exercise :

Consider the finite sample space $$\Omega = \{\omega_1,\omega_2,\omega_3\}$$ and let $$\mathbb P$$ be a probability measure such that $$\mathbb P[\{\omega_1\}] > 0$$ for all $$i=1,2,3$$. We define a financial market of one period which is consisted by the probability space $$(\Omega,\mathcal{F},\mathbb P)$$ with $$\mathcal{F} := 2^\Omega$$ and the securities $$\bar{S} = (S^0,S^1,S^2)$$ which are consisted of the zero-risk security $$S^0$$ and two securities $$S^1,S^2$$ which have risk. Their values at the time $$t=0$$ are given by the vector $$\bar{S}_0 = \begin{pmatrix} 1\\2\\7 \end{pmatrix}$$ while their values at time $$t=1$$, depending whether the scenario $$\omega_1,\omega_2$$ or $$\omega_3$$ happens, are given by the vectors $$\bar{S}_1(\omega_1) = \begin{pmatrix} 1\\3\\9\end{pmatrix}, \quad \bar{S}_1(\omega_2) = \begin{pmatrix} 1\\1\\5\end{pmatrix}, \quad \bar{S}_1(\omega_3) = \begin{pmatrix} 1\\5\\10 \end{pmatrix}$$ (a) Show that this financial market has arbitrage.

(b) Let $$S_1^2(\omega_3) = 13$$ while the other values remain the same as before. Show that this financial market does not have arbitrage and describe all the equivalent martingale measures.

Attempt :

(a) We have that a value process is defined as :

$$V_t = V_t^\bar{\xi} = \bar{\xi}\cdot \bar{S}_t = \sum_{i=0}^d \xi_t^i\cdot \bar{S}_t^i, \quad t \in \{0,1\}$$

where $$\xi = (\xi^0, \xi) \in \mathbb R^{d+1}$$ is an investment strategy where the number $$\xi^i$$ is equal to the number of pieces from the security $$S^i$$ which are contained in the portfolio at the time period $$[0,1], i \in \{0,1,\dots,d\}$$.

Now, I also know that to show that a market has arbitrage, I need to show the following :

$$V_0 \leq 0, \quad \mathbb P(V1 \geq 0) = 1, \quad \mathbb P(V_1 > 0) > 0$$

I understand that the different $$S$$ vectors will be plugged in to calculate $$V_t$$ but I really can't comprehend $$\xi$$. What would the $$\xi$$ vector be ?

Any help for me to understand what $$\xi$$ really is based on the problem and how to complete my attempt will be much appreciated.

For (b), showing that it does not have arbitrage is similar to (a) as I will just show that one of these conditions will not hold. What about the martingale stuff though ? It's a mathematical substance we really haven't been into so, if possible, I would really appreciate an elaborations.

The parameter $$\xi$$ represents your strategy, namely the quantity you hold in your portfolio of each security $$S^0$$, $$S^1$$ and $$S^2$$. Consider the following strategy: $${\xi}=(\xi^1,\xi^2,\xi^3)=(1.5,1,-0.5)$$ Then: \begin{align} & t=0: && \xi\bar{S}_0=\xi^0S_0^0+\xi^1S_0^1+\xi^2S_0^2 = 1.5+2-3.5=0 \\ & t=1: && \xi\bar{S}_1(\omega_1)=1.5+3-4.5=0 \\ &&& \xi\bar{S}_1(\omega_2)=1.5+1-2.5=0 \\ &&& \xi\bar{S}_1(\omega_3)=1.5+5-5=1.5>0 \end{align} Thus: $$\xi\bar{S}_0=0, \quad \mathbb{P}(\xi\bar{S}_1\geq0)=1, \quad \mathbb{P}(\xi\bar{S}_1>0)>0$$
For question b), you need to generalize to prove that there is no portfolio $$\xi$$ that allows arbitrage (instead of just finding a counterexample as in a).
• Hello, why is it legit to just take a random $\xi$ and show the conditions ? Wouldn't the $\xi$ need to be derived from the exercise ? I'm asking because I'm a true beginner at this lesson. Also, for the martingale stuff, what is needed ? – Rebellos Nov 11 '18 at 20:51
• It is not a random $\xi$, it is the product of careful thought by someone who knew what he was looking for. You have to construct $\xi$, and there is a logic to it. – Alex C Nov 11 '18 at 22:48
• @Rebellos I Iooked at the relative prices $S^1/S^2$ and $S^2/S^1$ and noticed that $S_1^1(\omega_1)/S_1^2(\omega_1), S_1^1(\omega_3)/S_1^2(\omega_3)>S_0^1/S_0^2$ whereas only $S_1^2(\omega_2)/S_1^1(\omega_2)>S_0^2/S_0^1$ so I tried to come up with an arbitrage portfolio long the security most expected to increase relatively, namely $S^1$. – Daneel Olivaw Nov 11 '18 at 23:02