Pricing Cash Flows Through Linear Pricing and Discount Factors
Summary
The document explains a geometric and economic interpretation of a pricing cash flow using linear pricing. Under the law of one price, the price of a payoff basket is the corresponding linear combination of its component prices. A linear pricing function can be represented as an expectation against a stochastic discount factor. Projecting that discount factor onto the market's payoff space gives a pricing cash flow that prices payoffs in that space equivalently.
The explanation then decomposes price into a covariance term and a product of expected values, connecting the definition to regression beta models and the mean variance frontier. It suggests the textbook's definition may include scalar multiples of the projected discount factor. The note is a conceptual explanation rather than a full derivation: it does not establish the precise conditions for the representation or projection, and it leaves the scalar-multiple interpretation tentative.
Key ideas
- Linear pricing expresses the price of a payoff basket as the sum of the prices of its components.
- A linear pricing function can be represented using a stochastic discount factor.
- Projecting the discount factor onto the traded payoff space can preserve prices for payoffs in that space.
- The covariance decomposition links pricing to regression beta models and the mean variance frontier.
- The proposed interpretation of a pricing cash flow as a scalar multiple is presented tentatively.
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# Economic intuition behind pricing cash flow
# Economic intuition behind pricing cash flow
I read the book of Skiadas Asset Pricing Theory 2009. I don't quite understand what does mean pricing cash flow. In the book it's written:
$\textbf{Definition 2.9}$ A cash flow $x^*$ is a pricing cash flow if $x^* \in X$ and there exist constants $\rho$ and $q$ such that $q \neq 0$ and:
$x(0) + \rho\mathbb{E}[x(1)] + q\mathbb{Cov}[x^*(1),x(1)] = 0 \qquad \forall x \in X$
$X$ - market of cash flows
$x(0)$ price of cash flow
$x(1)$ payoff at different states of nature $K$
I'm looking for some geometric and/or economic interpretation. Thanks in advance!
## Answer by Matthew Gunn (score 4, accepted)
https://quant.stackexchange.com/a/37081
All that's going on here are essentially consequences of a linear pricing function.
That asset prices should be linear in their payoffs makes intuitive economic sense: the value of a basket of payoffs is the sum of the basket contents. An assumption that the pricing function is linear is sometimes referred to as the law of one price.
#### Quick review
Let $f$ be a pricing function which gives you the current price $X_0$ of a future, stochastic payoff $X_1$. If $f$ is linear, $f(a X_1 + b Y_1) = a f(X_1) + b f(Y_1)$, then $f$ can be written as the inner product with some stochastic discount factor.
$$ f(X_1) = \mathbb{E}[MX_1]$$
Let $X^*$ be the projection of $M$ onto the space of payoffs $\underline{X}$. $X^* \in \underline{X}$ will also work as the discount factor for $X_1 \in \underline{X}$.
$$ f(X_1) = \mathbb{E}[ X^* X_1] $$
Now we can just do some algebra:
$$ X_0 = \mathbb{E}[X^*X_1]$$ $$ X_0 = \mathbb{Cov}[X^*, X_1] + \mathbb{E}[X^*] \mathbb[X_1]$$
I'm following how John Cochrane defines $X^*$ in his book Asset Pricing. The book here appears to call any scalar multiple of $X^*$ a pricing cash flow?
Anyway, you can manipulate these equations to bring out classic regression beta models and the mean variance frontier.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.