# From small markets to big markets

## TL;DR

This paper investigates utility maximization in a large financial market modeled by the Arbitrage Pricing Model, demonstrating the existence of optimizers under weaker assumptions and showing convergence of solutions from small to large markets.

## Contribution

It establishes the existence of optimal investment strategies under weaker conditions and proves convergence of solutions from finite to infinite market models.

## Key findings

- Existence of optimal strategies under weaker assumptions.
- Convergence of maximal satisfaction and reservation prices from small to large markets.
- Continuity rules for optimal investments in large markets.

## Abstract

We study the most famous example of a large financial market: the Arbitrage Pricing Model, where investors can trade in a one-period setting with countably many assets admitting a factor structure. We consider the problem of maximising expected utility in this setting. Besides establishing the existence of optimizers under weaker assumptions than previous papers, we go on studying the relationship between optimal investments in finite market segments and those in the whole market. We show that certain natural (but nontrivial) continuity rules hold: maximal satisfaction, reservation prices and (convex combinations of) optimizers computed in small markets converge to their respective counterparts in the big market.