The concept of thestochastic discount factor (SDF) is used infinancial economics andmathematical finance. The name derives from the price of an asset being computable by "discounting" the future cash flow${\displaystyle {\tilde{x}}_i}$ by the stochastic factor${\displaystyle \tilde{m}}$, and then taking the expectation.^[1] This definition is of fundamental importance inasset pricing.

If there aren assets with initial prices${\displaystyle p_1,\dots ,p_n}$ at the beginning of a period and payoffs${\displaystyle {\tilde{x}}_1,\dots ,{\tilde{x}}_n}$ at the end of the period (allxs arerandom (stochastic) variables), then SDF is any random variable${\displaystyle \tilde{m}}$ satisfying

${\displaystyle E(\tilde{m}{\tilde{x}}_i)=p_i,\text{for }i=1,\dots ,n.}$

The stochastic discount factor is sometimes referred to as thepricing kernel as, if the expectation${\displaystyle E(\tilde{m}{\tilde{x}}_i)}$ is written as an integral, then${\displaystyle \tilde{m}}$ can be interpreted as the kernel function in anintegral transform.^[2] Other names sometimes used for the SDF are the "marginal rate of substitution" (the ratio ofutility ofstates, when utility is separable and additive, though discounted by the risk-neutral rate), a (discounted) "change of measure", "state-price deflator" or a "state-price density".^[2]

In a dynamic setting, let${\displaystyle \mathbb{F}=({\mathcal{F}}_t{)}_{t\ge 0}}$ denote the collection of information sets at each time step (filtration), then the SDF is similarly defined as,

${\displaystyle E_t[\tilde{m}(t+s)\tilde{x}(t+s)]=p(t),s>0}$

where${\displaystyle E_t[\cdot ]=E[\cdot |{\mathcal{F}}_t]}$ denotes expectation conditional on the information set at time${\displaystyle t\ge 0}$,${\displaystyle \tilde{x}=({\tilde{x}}_1,\dots ,{\tilde{x}}_n{)}^{\prime }}$ is the payoff vector process, and${\displaystyle \tilde{p}=({\tilde{p}}_1,\dots ,{\tilde{p}}_n{)}^{\prime }}$ is the price vector process.^[3]

The existence of an SDF is equivalent to thelaw of one price;^[1] similarly, the existence of a strictly positive SDF is equivalent to the absence of arbitrage opportunities (seeFundamental theorem of asset pricing). This being the case, then if${\displaystyle p_i}$ is positive, by using${\displaystyle {\tilde{R}}_i={\tilde{x}}_i/p_i}$ to denote the return, we can rewrite the definition as

${\displaystyle E(\tilde{m}{\tilde{R}}_i)=1,\mathrm{\forall }i,}$

and this implies

${\displaystyle E\left[\tilde{m}({\tilde{R}}_i-{\tilde{R}}_j)\right]=0,\mathrm{\forall }i,j.}$

Also, if there is aportfolio made up of the assets, then the SDF satisfies

${\displaystyle E(\tilde{m}\tilde{x})=p,E(\tilde{m}\tilde{R})=1.}$

By a simple standard identity oncovariances, we have

${\displaystyle 1=\mathrm{cov}(\tilde{m},\tilde{R})+E(\tilde{m})E(\tilde{R}).}$

Suppose there is a risk-free asset. Then${\displaystyle \tilde{R}=R_f}$ implies${\displaystyle E(\tilde{m})=1/R_f}$. Substituting this into the last expression and rearranging gives the following formula for therisk premium of any asset or portfolio with return${\displaystyle \tilde{R}}$:

${\displaystyle E(\tilde{R})-R_f=-R_f\mathrm{cov}(\tilde{m},\tilde{R}).}$

This shows that risk premiums are determined by covariances with any SDF.^[1]

In thestandard consumption-based model with additive preferences, the stochastic discount factor is given by,

${\displaystyle M_{t+s}=\frac{\beta u^{\prime }(c_{t+s})}{u^{\prime }(c_t)}}$

where${\displaystyle (c_t{)}_{t=0}^T}$ denotes an agent's consumption path, and${\displaystyle \beta }$ is their subjective discount factor.

In theBlack–Scholes model, the stochastic discount factor is the stochastic process${\displaystyle \xi =({\xi }_t{)}_{t\ge 0}}$ defined by,

${\displaystyle {\xi }_t=e^{-rt}{Z}_t=e^{-rt}{e}^{-\lambda W_t-\frac{1}{2}{\lambda }^{2}t}}$

where${\displaystyle W=(W_t{)}_{t\ge 0}}$ denotes a standard Brownian motion,${\displaystyle \lambda =\frac{\mu -r}{\sigma }}$ is a given market price of risk, and${\displaystyle Z=(Z_t{)}_{t\ge 0}}$ is the Radon-Nikodym process of the risk-neutral measure with respect to the physical measure.

Hansen–Jagannathan bound

• Stochastic calculus
• Financial economics
• Mathematical finance

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