ASSET PRICING MODEL: LAPM 6

In words, the volatility of the Treasury bond price is state contingent, and the higher the volatility, the worse the prospects for the economy. The simple logic is that volatility is high when the option is in the money and low when it is out of the money.
Remark: In the example with nonverifiable second-period income, the Black-Sholes formula yields an explicit formula for the volatility of the bond price if the process is a continuous time geometric Brownian motion.

General LAPM

Let us investigate more generally the impact of news about the state of nature in the LAPM framework of section 3. As in the example, we assume that there are subdates n = 1…../V between dates 0 and 1, at which informative signals accrue about the date-1 state of nature ш. Thus, the market’s information about the state of nature at date 1 gets refined over time. Let an denote the market’s information at subdate n with адг = ш. We let E [• | c„]denote the subdate-n expectation of a variable conditional on the information available at time n. The corporate sector purchases quantity L£ of liquid asset к at date 0, and can afterwards reconfigure its portfolio so that it holds (information contingent) quantity Lk(an) at subdate n. Asset fc’s equilibrium price given information an is denoted %((%)•

Consider the problem of maximizing the corporate sector’s expected payoff subject to the investors’ date-0 break-even condition and date-1 decisions being feasible:
w6673-23
w6673-24
A few comments are in order. First, a. is measurable with respect to ш, and so the set of feasible decisions is indeed a well-defined function of the state of nature. Second, the date-0 contract with investors specifies some portfolio adjustment at each date. We ignore the possibility that contemplated portfolio adjustments may require a net contribution by investors at subdate n %((%) — -kfc (crn_i)] > 0). While such a contribution could occur if the portfolio adjustment raised the investors’ wealth conditional on crn, it would not occur if the adjustment reduced it, since the investors would be unwilling ex post to bring in new funds, and they cannot ex ante commit to do so. However, if in equilibrium > 1, then Lk{on) = I-к for all an is an optimal policy, and so investors do not have to contribute at intermediate dates. The fictitious subdate-n reshuffling of liquid assets between the corporate sector and the rest of the economy is, as in Lucas (1978), only used to price financial assets at an intermediate date.

As before, we let ц be the multiplier of the break- even constraint in (16) and define the marginal liquidity service т(ш) as in (9). Taking first-order conditions we find that for each liquid asset к € {1,…, К} and for each subdate n €E {1,…, N — 1}:
w6673-25

Tags: , ,