Viable prices in financial markets with solvency constraints

Abstract

We study the viability of a securities market model with continuous trading in which agents are required to remain solvent at all times and cannot add funds to their portfolio of securities in excess of an exogenous endowment. A financial security is characterized by its dividends and its usefulness as a collateral for borrowing. We show that viability, a notion introduced by Harrison and Kreps (1979), is equivalent to the existence of a pair of continuous linear functionals, defined on the consumption space and “risk-adjusted reserve” space, such that their sum represents a linear pricing rule for cash flows consistent with security prices. We also show that viability is equivalent to the existence of a state price process such that the adjusted total gains from securities are local martingales. In particular, one can construct a probability measure under which the discounted total gains of securities with the same collateral quality as the riskless asset are martingales. The discounted total gains of lower quality securities are submartingales. The state price process is always equal to the process of cash flow “shadow” price times an increasing process. Finally, we characterize the riskless rate and local risk premia on securities.