${\displaystyle Price_{j}=\sum _{s}(p_{s}Y_{s}X_{sj})/r}$

${\displaystyle =\sum _{s}(q_{s}X_{sj})/r}$

${\displaystyle =\sum _{s}p_{s}X_{sj}{\tilde {m}}_{s}=E}$

${\displaystyle =\sum _{s}\pi _{s}X_{sj}}$

Four equivalent formulations,^[6] where:

${\displaystyle j}$ is the asset or security

${\displaystyle s}$ are the various states

${\displaystyle r}$ is the risk-free return

${\displaystyle X_{sj}}$ dollar payoffs in each state

${\displaystyle p_{s}}$ a subjective, personal probability assigned to the state; ${\displaystyle \sum _{s}p_{s}=1}$

${\displaystyle Y_{s}}$ risk aversion factors by state, normalized s.t. ${\displaystyle \sum _{s}q_{s}=1}$

${\displaystyle q_{s}\equiv p_{s}Y_{s}}$, risk neutral probabilities

${\displaystyle {\tilde {m}}\equiv Y/r}$ the stochastic discount factor

${\displaystyle \pi _{s}=q_{s}/r}$ state prices; ${\displaystyle \sum _{s}\pi _{s}=1/r}$