# The Information Process

• Erik Lüders
Part of the ZEW Economic Studies book series (ZEW, volume 24)

## Abstract

Up to here we have not discussed the distribution of FT. 1 It follows from (2.4) that to characterise the forward price Ft all we need besIDes the pricing kernel is a characterization of the conditional distribution of the terminal asset price FT More precisely, we need for every time t the distribution of FT conditioned on the information at time t. Loosely stated this means for our framework with homogeneous expectations that we need to know what is known when Such a characterization of information is called an information structure Technically the information structure is characterised by the filtration Ft In every model explicitly or implicitly some assumptions about the information structure have to be made How to model the relevant information, however, is somewhat arbitrary For the purpose of asset pricing the relevant information is a characterization of the distribution of future cash flows In our model, where no divIDends are paID until the terminal date T, the relevant information is the (exogenous) distribution of the terminal value This information can be modeled by the so-called information process. Generally, the information process I is defined as
$$I_t = E^P (X_T)|F_t),0 \leqslant t \leqslant T,$$
(3.1)
where XT = FT = IT is the exogenous terminal value of the asset Hence, this process may be interpreted as the representative investor’s conditional expectation about the terminal value of the asset. 2 Since the information process characterises conditional expectations, it is a martingale, $$I_t = E^P (I_T|F_t)\forall _T \in [t,T],$$, its drift is zero Intuitively, it is also clear that the expectations of a rational investor have to follow a martingale Otherwise, the investor could improve his forecasts by anticipating the expected change in his forecasts Note also that in the case of risk neutral investors the information process is equal to the forward price process F. Using an information process as defined in (3.1)has several advantages First, it is a parsimonious way to characterise all the relevant information, i.e it characterises all the conditional distributions of the terminal value Ft- Secondly, since it characterises investors’ expectations, it has an economic interpretation.

## Keywords

Information Process Cash Flow Stochastic Volatility Price Process Deterministic Function
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## Notes

1. 1.
This Chapter is partly based on Lüders [131] and Sect 3.4 is based on Lüders and Peisl [133].Google Scholar
2. 2.
A similar approach to characterise investors’ information is already proposed in Brennan [22].Google Scholar
3. 3.
This analysis is closely related to the work of, for example, Brennan and Xia [24]. See also the discussion on p 45.Google Scholar
4. 4.
The relationship between asset price processes and information processes will be discussed later in this monograph That the geometric Brownian motion as a model for the behaviour of asset prices implies an information process which is also governed by a geometric Brownian motion was shown in Pranke, Stapleton and Subrahmanyam [74] This relationship can be proved analogously to Proposition 1 Investors can observe the asset price process and build their expectations based on these observations Assuming that the asset price is governed by a geometric Brownian motion and applying the Theorem of Feynman-Kac yields that the information process is also governed by a geometric Brownian motion.Google Scholar
5. 5.
See for example Schwartz and Moon [172] and Schwartz and Moon [173] or Dechow, Hutton and Sloan [52].Google Scholar
6. 6.
See again, for example, Schwartz and Moon [172] and Schwartz and Moon [173] or Dechow, Hutton and Sloan [52].Google Scholar
7. 7.
This derived information process is a slight variation of the information process assumed in Lüders and Peisl [132], see also Chap 7 In Chap 7 we assume that the information process is governed by a stochastic process similar to (3.8).Google Scholar
8. 8.
Similar problems are discussed in the context of learning The articles of Brennan and Xia [24], Johnson [106], Veronesi [184] and Ziegler [189] are closely related.Google Scholar
9. 9.