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Matthew Gunn
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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, but then all: the mathvalue of a basket of payoffs is justthe sum of the basket contents. An assumption that the pricing function is linear algebra and probabilityis 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.

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, but then all the math is just linear algebra and probability.

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.

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.

Source Link
Matthew Gunn
  • 7k
  • 1
  • 23
  • 32

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, but then all the math is just linear algebra and probability.

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.