Game-Theoretic Foundations for Probability and Finance

15Numeraires in Market Spaces

The theory of martingale spaces developed in the preceding chapter implies that a constant is a capital process. This can be acceptable when the betting picture is used as a metaphor in scientific applications, but it is unrealistic for actual applications to finance, as it implies that the trader can borrow money without interest.

The standard way to avoid this unrealistic assumption is to insist that the trader's strategies be self‐financing; each time the trader rebalances his portfolio, he must preserve its total value. In Section 15.1, we use this idea to define the concept of market space, a self‐financing analog of the concept of martingale space. Market spaces can be considered more general than martingale spaces, because trading strategies in a martingale space become self‐financing if we add cash as one of its basic securities, and the addition makes no difference in what the trader can accomplish.

The continuous martingales in martingale spaces have an analog in market spaces; we call them continuous gales. If a market space has a positive continuous gale , then, as we explain in Section 15.2, we can turn the market space into a martingale space by dividing all the prices at time by . The positive continuous gale we choose for this purpose ...