Viable Price Systems and Budget Constraints in Securities Markets

# Martingales and Arbitrage in Multiperiod Securities Markets


# 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?

Thank you in advance.

## Answer by William Wu (score 1, accepted)

https://quant.stackexchange.com/a/71564

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.