This paper is based on a commissioned research study by the Casualty Actuarial Society with a focus on a theoretical framework for a liquidity risk premium and the interaction of illiquidity with credit effects on the valuation of assets and liabilities. The problem addressed is the development of a theory of liability valuation distinct from a theory of asset pricing or valuation.
Our proposed solution is to formulate a risk theory for two price economies when markets fail to converge to the law of one price. The traditional one price economy prices the credit, market and some components of liquidity risk using traditional methods that include exponential tilting in the presence of constant absolute risk aversion utilities. The two price economy on the other hand explicitly models illiquidity issues by developing expressions for bid and ask prices that ensure the acceptability of residual risks. The presence of residual risk and its associated market incompleteness is central to the market's inability to converge to the law of one price in two price economies. The resulting two prices are differentiated from classical linear pricing rules as they are nonlinear functions of the cash flows being priced.
Specifically bid prices are concave functions of the claims being priced while ask prices are convex functions. Asset and liability valuation then part company as we propose to employ ask prices for evaluating liabilities while assets are to be priced at bid. Two price economies also provide new hedging objectives for corporations with hedging strategies being designed to economize on the commitment of capital needed to cover residual risks. The static two price theory is extended to its dynamic counterpart using recent developments in the theory of non linear expectations. The dynamic model is in discrete time with a tenor reáecting a time horizon at which it is anticipated that genuine counterparties normally arrive. The longer the tenor, the greater the illiquidity and the greater is the spread between bid and ask prices.
The dynamic theory is illustrated on pricing a simple compound Poisson gamma insurance loss process. In this context capital minimization as a hedging objective illustrates the construction of optimal reinsurance points. Financial hedging using the securitization of catastrophic loss exceedances and mortality contingent securities for life risk further illustrate the construction of two prices in the context of capital minimizing hedges. Further it is observed that bid prices are sensitive to changes in credit risk but this is not the case for the ask or liability pricing counterpart. A final empirical section addresses issues of measuring the size of cones of acceptability using data on daily high, low and close prices of publicly traded equity.
