Wsrtjtyk

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
$$bar{S}_1(omega_1) = begin{pmatrix} 1\3\9end{pmatrix}, quad bar{S}_1(omega_2) = begin{pmatrix} 1\1\5end{pmatrix}, quad bar{S}_1(omega_3) = begin{pmatrix} 1\5\10 end{pmatrix}$$
(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 :

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

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:
$$begin{align}
& t=0: && xibar{S}_0=xi^0S_0^0+xi^1S_0^1+xi^2S_0^2 = 1.5+2-3.5=0
\
& t=1: && xibar{S}_1(omega_1)=1.5+3-4.5=0
\
&&& xibar{S}_1(omega_2)=1.5+1-2.5=0
\
&&& xibar{S}_1(omega_3)=1.5+5-5=1.5>0
end{align}$$
Thus:
$$xibar{S}_0=0, quad mathbb{P}(xibar{S}_1geq0)=1, quad mathbb{P}(xibar{S}_1>0)>0$$

Hence the market has arbitrage.

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).