How do investors' expectations drive asset prices?

thus, the current price F_t has to be lower, since the terminal value F_r of the
forward price is given by lʃ (remember Fp = Ip).¹°

With equation (16) we can give an explanation for the well documented
μ_t = σ_t (√1 — p²ʌɑ∖t) + λ^²∖t)p_t) for 0 ≤ t ≤ T, defined as usual) :

μ := √1 — pλ<¹ >(«) + λ<²>(t)p, , 0 ≤ t ≤ T.           (17)

^σ

This equation shows that the sharpe ratio ɪ may be time-varying and sto-
chastic even if the risk premia are constant, i.e. ʌɑ∖t) = const., λ^²∖t) =
const, in equation (17). Hence, equation (17) suggests that seemingly un-
explainable variations of the risk premia λ and the sharpe ratio -μ may be
due to the assumption of a constant correlation p between W^f and W^v.

To illustrate our results we now consider the case when the volatility of
the information process depends on time only.

Example Let σ( depend on time only. Then the volatility of the information
process satisfies

Then, by Theorem 1, the forward price of the asset satisfies the sto-
chastic differential equation

dF_t = F_tλ^α( (t) σ¹_t dt + F_tσ¹_t dW^F ,   0 ≤ t ≤ T.

With this simplified model it is easily seen, that the properties of the
volatility of the information process transfer to the properties of the
drift and the volatility of the asset price process. The drift of the as-
set price process is equal to the volatility of the information process
multiplied by the market price of risk. The volatility of the informa-
tion process and the volatility of the asset price process are identical.

¹⁰The described effect is known as the volatility feedback effect. For similar results in
discrete time see Campbell, Hentschel [3] and Wu [33].

19

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