## ASSET PRICING MODEL: LAPM 5

Information filtering and volatility

As we noted in the introduction, many recent advances in empirical finance were motivated by the observation that conditional variances and covariances change over time. It is well-known, for instance, that volatility is clustered, that asset volatilities (stock volatilities, bond volatilities across maturities) move together, and that stock volatility increases with bad news.16 This section does not attempt to provide a general theory of the impact of liquidity premia on volatility. Its only goal is to suggest that a liquidity-based asset pricing model has the potential to deliver interesting insights into state-contingent volatilities.

Example

Let us first return to the example of section 2. In this example with nonverifiable second-period income, the liquidity benefit of the Treasury bond is a put option, since m(x) decreases linearly with first-period income x until it hits zero. We also observed that with verifiable second-period income and under some mild regularity conditions, m(x) decreases and is convex until it hits zero.

Suppose now that news arrives intermittently between dates 0 and 1 containing information about the realization of x at date 1. Specifically, suppose that there are N news dates between 0 and 1 (the first distinct from date 0 and the Nth equal to date 1). Assume further that the realization of x is given by either an additive or a multiplicative process

where the increments r]m are independently distributed.

The early accrual of information about the state ш will have no impact on the optimal decision rule d(-) and hence no retrading of financial contracts occurs between dates 0 and 1. Yet, we can price Treasury bonds by arbitrage at each subdate n. Contingent on the available information at subdate n, summarized by xn,

where En[-1-] denotes the conditional expectation given information at subdate n. (Formula (14) is derived formally for the general framework in section 4.2.)

It can be shown that, provided m! < 0 and m" > 0, for either the additive (13a) or the multiplicative process (13b), we have