Overview
Stochastic cash flows are cash flows where the amount (and possibly timing) of the cash flow is not known ahead of time. As a general rule, the discount rate can be taken to be the expected return of the cash flow. (see below)
Expected Return and the Discount Rate
Starting with todays price {% P_0 %}, we form the following equation. (using the expectation)
{% \mathbb{E}(P_t) = P_0 e^{rt} %}
Note, at this point this equation is a tautology. That is, we have not yet defined the rate {% r %}
found in the equation. It just asserts that such a {% r %} exists. The following computes
what {% r %} has to be equal to.
{% e^{rt} = \frac{\mathbb{E}(P_t)}{P_0} %}
In order to quote the rate of return in annual terms, we assume that the time period is equal to 1 year,
or {% t=1 %}.
Then we get
{% r = log(\frac{\mathbb{E}(P_t)}{P_0}) \approx \frac{\mathbb{E}(P_t)}{P_0} %}
{% r %} is termed the risk adjusted discount rate. (Note, this is not the same as the
stochastic discount)
At this point, there is no theory here, this is just a definition.
Pricing Theories
Pricing theories turn the discount equation around in order to determine the fair price of an asset today. That is, re-arranging terms you get
{% P_0 = e^{-rt} \mathbb{E}(P_t) %}
That is, if we have a forecast of expected future prices (or cash flows) and we have an appropriate discount
rate to apply, we can determine what a fair price today would be for the asset in question.
Most pricing theories then take the forecast as given, and then try to formulate what the appropriate
discount rate should be.
As a general rule, it is thought that assets that are subject to the same types of risks, will have roughly the same risk adjusted discount. This assumption is generally intuitive and can be somewhat justified by appealing to the stochastic discount. It can be used to price an asset, by forecasting and expected cash flow from the asset, and by approximating what the risk adjusted discount rate should be by looking at similar assets, then apply the discount to the forecast to arrive at a price.
Additional pricing theories are listed here: