Noise, Risk Premium, and Bubble

Abstract

The existence of the pricing kernel is shown to imply the existence of an ambient information process that generates market filtration. This information process consists of a signal component concerning the value of the random variable X that can be interpreted as the timing of future cash demand, and an independent noise component. The conditional expectation of the signal, in particular, determines the market risk premium vector. An addition to the signal of any term that is independent of X, which generates a drift in the noise, is shown to change the drifts of price processes in the physical measure, without affecting the current asset price levels. Such a drift in the noise term can induce anomalous price dynamics, and can be seen to explain the mechanism of observed phenomena of equity premium and financial bubbles.

Lévy information and the aggregation of risk aversion

“…Furthermore, if we make use of the market price of risk to effect a change of measure, a calculation shows that (i) the random variables ξ t and X are independent under Q, and (ii) the probability law for X under Q is given by p(dx); that is to say, it is the same as it is under the physical measure P. Therefore, an 'observer' in the risk-neutral frame of reference (Ω, F , Q) detects the 'message' {ξ t }, or equivalently the price {S t }, but finds that it contains no information about the level of risk aversion-this is the sense in which the level of risk aversion cannot be inferred a priori from derivative prices in the context of Brownian-motionbased models. If stronger modelling assumptions are made about the structure of the pricing kernel in a Brownian model, then in some contexts it is possible to infer information about the risk aversion level from derivative prices [20,[26][27][28]. The approach that we are taking is, perhaps, more practically oriented, inasmuch as an explicit estimation formula for the risk premium, such as that given by (4.7), can be obtained in a direct and transparent manner without any reference to the risk-neutral measure.…”