Stochastic discount factor

The stochastic discount factor (SDF) is a concept in financial economics and mathematical finance. The name 'stochastic discount factor' reflects the fact that the price of an asset can be computed by 'discounting' the future cash flow x ~ i {displaystyle { ilde {x}}_{i}} by the stochastic factor m ~ {displaystyle { ilde {m}}} and then taking the expectation. This definition is of fundamental importance in asset pricing. The stochastic discount factor (SDF) is a concept in financial economics and mathematical finance. The name 'stochastic discount factor' reflects the fact that the price of an asset can be computed by 'discounting' the future cash flow x ~ i {displaystyle { ilde {x}}_{i}} by the stochastic factor m ~ {displaystyle { ilde {m}}} and then taking the expectation. This definition is of fundamental importance in asset pricing. If there are n assets with initial prices p 1 , … , p n {displaystyle p_{1},ldots ,p_{n}} at the beginning of a period and payoffs x ~ 1 , … , x ~ n {displaystyle { ilde {x}}_{1},ldots ,{ ilde {x}}_{n}} at the end of the period (all xs are random variables), then SDF is any random variable m ~ {displaystyle { ilde {m}}} satisfying The stochastic discount factor is sometimes referred to as the pricing kernel. This name comes from the fact that if the expectation E ( m ~ x ~ i ) {displaystyle E({ ilde {m}},{ ilde {x}}_{i})} is written as an integral, then m ~ {displaystyle { ilde {m}}} can be interpreted as the kernel function in an integral transform. 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'. 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 p i {displaystyle p_{i}} is positive, by using R ~ i = x ~ i / p i {displaystyle { ilde {R}}_{i}={ ilde {x}}_{i}/p_{i}} to denote the return, we can rewrite the definition as