## ASSET PRICING MODEL: LAPM 7

Asset prices (or liquidity premia) form a martingale because there is no liquidity service within the periods where news arrives. Only at the last subdate (date 1) will the liquidity premium disappear and hence the martingale property fail. Note that the martingale condition reflects the fact that firms are indifferent regarding the timing of the purchase of liquidity, as long as the purchase is made before the final date. The martingale condition stems from arbitrage within the investors’ budget constraint.

Define a “generalized fixed-income security,” as one with expected return unchanged as news accrues, that is,

Condition (19) rules out volatility stemming from news about the assets’ dividends. Such volatility must be added (with a correction depending on the covariance with the innovations about liquidity needs) to our price formulae when expected payoffs change over time.

Asset volatility in the case of a single state of liquidity shortage.

Assuming that there is a single state (state шн) of liquidity shortage, let /я(о’п) denote the posterior probability of the bad state of nature at subdate n, conditional on the information available at that subdate. Let

denote the relative variance of the posterior probability. ^s(an) is a measure of the informativeness of the signal accruing at subdate n + 1.

Note, from (20), that the ratio formula (12) continues to apply with дк{сгп) in place of qk So at subdate n and for any two assets к and I,

Under the (strong) assumption of just one liquidity constrained state, and with a government bond on the market, all assets with constant expected dividend are priced according to a linear formula involving the liquidity premium on that bond. This is a close analog to the С АРМ.

Contingent volatilities and clustering.

Let Vk and v^ denote the absolute and relative volatilities of the price of asset к conditional on information an: