Pricing Kernels, Stochastic Discount Factors, and Their Processes

Summary

The document distinguishes a pricing kernel, a stochastic discount factor (SDF), and an SDF process. A pricing kernel is described as a time-indexed stochastic process that makes the price-weighted asset value satisfy a conditional martingale relation. Dividing adjacent values of that kernel yields the one-period SDF used to price a future payoff by taking its expected discounted value.

The response also relates the SDF to the change in marginal utility of wealth between periods. The term “SDF process” refers to the time series of these discount factors. This is a concise conceptual explanation, not an empirical pricing model: it does not discuss how to estimate the kernel or SDF, assumptions about investor preferences, or applications to particular assets.

Key ideas

• A pricing kernel is a stochastic process that supports a martingale pricing relation.\nThe one-period SDF is the ratio of consecutive pricing-kernel values.\nExpected future payoffs weighted by the SDF determine current prices.\nThe SDF is linked to the change in marginal utility of wealth across periods.\nAn SDF process is the sequence of period-specific discount factors.

Tags

# What is the difference between stochastic discount factor and stochastic discount factor process?


# What is the difference between stochastic discount factor and stochastic discount factor process?












What is the difference between stochastic discount factor and stochastic discount factor process and how are they both related?

## Answer by fni (score 4)

https://quant.stackexchange.com/a/31413

You should consider the following different, but related, concepts:

-Pricing kernel: It is a stochastic process $\{M_t\}_{t=1}^\infty$ such that the pricing process is a martingale, i.e., $M_t S_t =E_t[M_{t+1}S_{t+1}] \quad \forall t$

-Stochastic discount factor: you can divide both sides of the previous equation by $M_t$ to obtain that $P_t=E_t\left[\frac{M_{t+1}}{M_t}S_{t+1}\right]=E[SDF_{t+1}\times Payoff_{t+1}]$ . The SDF tells you about the ratio between marginal utility of wealth at time t+1 and at time t, i.e., $SDF_{t+1}=\left(\frac{\partial U( W)}{\partial W_{t+1}}\right)/\left(\frac{\partial U( W)}{\partial W_{t}}\right)$

-Stochastic discount factor process: it is the stochastic process of $\left\{SDF_t\right\}_{t=1}^\infty$

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