# Martingales and Arbitrage in Multiperiod Securities Markets

I have been reading the paper "Martingales and Arbitrage in Multiperiod Securities Markets".

The paper works in the probability space $$(\Omega, F, \mathbf{P})$$. $$X$$ is defined as the set of all random variables on $$(\Omega, F)$$. $$M$$ is a subspace of $$X$$.

The paper defines a consumption bundle $$(r, x) \in (\mathbb{R}, X)$$ where $$r$$ is consumed today and $$x$$ at a later time $$T$$ based on a random state of the world ($$\omega \in \Omega$$).

A price system is a pair $$(M, \pi)$$ where $$\pi$$ is a linear functional on $$M$$. The agents can purchase a bundle $$(r,m)$$ for a time $$0$$ units of date zero consumption of $$r + \pi(m)$$.

A viable price system $$(M, \pi)$$ is viable if there exists an agent with preference $$\succsim$$ and a bundle $$(r^*,m^*) \in \mathbb{R} \times M$$ such that,

$$r^* + \pi(m^*) \le 0$$ and $$(r^*,m^*) \succsim (r,m)$$ for all $$(r, m) \in \mathbb{R} \times M$$ such that $$r + \pi(m) \le 0$$.

Note that $$\succsim$$ is a preference relation that is transitive, continuous and convex. The continuity is based on a topology defined later.

My Question:

why is $$r^* + \pi(m^*)$$ less than equal to zero? The author states that it is a budget constraint.

Also is the preference across all agents? Or is it specific to an agent. The fact that he uses there exists seems to imply $$(r^*, m^*)$$ is preferred by all agents?

It is assumed that the agent's trades cannot have positive cost. In other words, the agent cannot promise to spend more than they make.

(cf. this slides, pg. 9)

For the second part, it should be for a specific agent only. It is mentioned in page 5 of the paper (after equation 2.4):

This says that there is some agent from the class $$\mathbf{A}$$ who, when choosing a best net trade subject to his budget constraint $$r+\pi(m)\leq 0$$, is able to find an optimal trade.

• Thank you for your response. That still did not resolve my doubts. These are two consumptions. It is easy to understand a budget equation where I have a specific $X$ dollars to spend on goods and $X >= \prod q_i p_i$ where $q_i$ and $p_i$ are quantities and prices. If I am exchanging consumption, $r$ today and $\pi(m)$ at some time $T$, I do not understand the meaning of positive cost. Is it stating, I need to be paid to consume tomorrow instead of today? Jul 14, 2022 at 10:19
• Thanks again. Continuing the reasoning, So I may choose to sell short today $r$ and buy it back at $T$ for $m < 0$. The constraint is that $r + \pi(m) \le 0$? So the market is essentially sustained by short selling and buying? Jul 14, 2022 at 11:11
• @RameshKadambi I think your second comment is correct. The paper did mention (pg 1) that it is assuming frictionless markets (no transaction costs and unrestricted short sales). Further on in pg 20, it is mentioned that "short sales (of the bond) give you the necessary funds". Jul 14, 2022 at 12:34
• Thank you for all your responses. That seems reasonable. Jul 14, 2022 at 23:08
• Thank you William. Jul 14, 2022 at 23:09