Stochastic discount factor
The concept of the stochastic discount factor (SDF) is used in financial economics and mathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow formula_1 by the stochastic factor formula_2, and then taking the expectation. This definition is of fundamental importance in asset pricing. If there are "n" assets with initial prices formula_3 at the beginning of a period and payoffs formula_4 at the end of the period (all "x"s are random (stochastic) variables), then SDF is any random variable formula_2 satisfying The stochastic discount factor is sometimes  referred to as the pricing kernel as, if the expectation formula_7 is written as an integral, then formula_2 can be interpreted as the kernel function in an integral transform. Other names sometimes used for the SDF 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". Properties. The existence of an SDF is equivalent to the law of one price; similarly, the existence of a strictly positive SDF is equivalent to the absence  of arbitrage opportunities (see Fundamental theorem of asset pricing). This being the case, then if formula_9 is positive, by using formula_10 to denote the return, we can rewrite the definition as and this implies Also, if there is a portfolio made up of the assets, then the SDF satisfies By a simple standard identity on covariances, we have Suppose there is a risk-free asset. Then formula_15 implies formula_16. Substituting this into the last expression and rearranging gives the following formula for the risk premium of any asset or portfolio with return formula_17: This shows that risk premiums are determined  by covariances with any SDF. See also. Hansen–Jagannathan bound