A Stochastic discount factor (SDF) is a concept in financial economics and mathematical finance.
Contents
If there are n assets with initial prices
This definition is of fundamental importance in asset pricing. The name "stochastic discount factor" reflects the fact that the price of an asset can be computed by "discounting" the future cash flow
Properties
If each
and this implies
Also, if there is a portfolio made up of the assets, then the SDF satisfies
Notice the definition of covariance, it can also be written as
Suppose there is a risk-free asset. Then
This shows that risk premiums are determined by covariances with any SDF.
The existence of an SDF is equivalent to the law of one price.
The existence of a strictly positive SDF is equivalent to the absence of arbitrage opportunities.
Other names
The stochastic discount factor is sometimes referred to as the pricing kernel . This name comes from the fact that if the expectation
is written as an integral, then
Other names for the SDF sometimes encountered are the marginal rate of substitution (the ratio of utility of states, when utility is separable and additive, though discounted by the risk-neutral rate), a change of measure,state-price deflator or a state-price density.