"Valuation and Martingale Properties of Shadow Prices - An Exposition"

Lucien Foldes
London School of Economics

ABSTRACT

Concepts of asset valuation based on the martingale properties of shadow (or marginal utility) prices in continuous-time, infinite-horizon stochastic models of optimal saving and portfolio choice are reviewed and compared with their antecedents in static or deterministic economic theory. Applications of shadow pricing to valuation are described, including a new derivation of the Black-Scholes formula and a generalised net present value formula for valuing an indivisible project yielding a random income. Some new results are presented concerning (i) the characterisation of an optimum in a model of saving with an exogenous random income and (ii) the use of random time transforms to replace local by true martingales in the martingale and transversality conditions for optimal saving and portfolio choice.

Keywords: Valuation, Investment, Optimisation, Continuous Time, Martingales, Transversality, Time Change

JEL Classification: D81; D9; C61; D46
AMS (1991) Classification: 93E20; 90A09; 90A16; 60G44; 49K45


Economics Department, London School of Economics, Houghton Street, London WC2A 2AE, UK

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