On the Consistency of the Lucas Pricing Formula

Abstract

Finally, the Gordon growth formula is extended from its deterministic origin, to the present model of uncertainty, and it is indicated how this can be used to to possibly shed some light on the volatility puzzle. In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance. In this paper, we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version. While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included. In order to find the real market value of an asset in an exchange economy, one would typically apply the Lucas formula, developed in a discrete time framework. This theory has also been extended to continuous time models, in which case the same pricing formula has been universally applied. While the discrete time theory is rather transparent, there has been some confusion regarding the continuous time analogue. In particular, the continuous time pricing formula must contain a certain type of a square covariance term that does not readily follow from the discrete time formulation. As a result, this term has sometimes been missing in situations where it should have been included. In this paper, we reformulate the discrete time theory in such a way that this covariance term does not come as a mystery in the continuous time version. In most real life situations dividends are paid out in lump sums, not in rates. This leads to a discontinuous model, and adding a continuous time framework, it appears that our framework is a most natural one in finance. Finally, the Gordon growth formula is extended from its deterministic origin, to the present model of uncertainty, and it is indicated how this can be used to to possibly shed some light on the volatility puzzle.