Aligning Option Prices and Index Data for Pricing Kernel Estimation
Summary
The note outlines a procedure for estimating an empirical pricing kernel, also called a stochastic discount factor, for an index. It estimates the physical density from index observations over a historical interval and the risk-neutral density from option prices. Its central recommendation is to take the option prices at the final date of the historical interval, when the physical-density estimate incorporates the available observations up to that point.
The proposed workflow uses closing index prices over the interval, then gathers options observed at its endpoint with a common expiration horizon and varied strikes. After estimating both densities, it forms their ratio as the pricing kernel. The explanation is intuitive rather than empirically demonstrated, and does not discuss estimation techniques, data quality, or how uncertainty in either density affects the ratio. The suggested date alignment is therefore a practical framing of the question, not a validated universal rule.
Key ideas
- Estimate the physical density from index data observed over a historical interval.
- Use option prices from the interval’s final observation date to estimate the risk-neutral density.
- Keep option expiration horizons equal while varying strikes in the described procedure.
- The empirical pricing kernel is estimated as the ratio of the risk-neutral density to the physical density.
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# Data Selection for Empirical Pricing Kernel Estimation (Stochastic Discount Factor)
# Data Selection for Empirical Pricing Kernel Estimation (Stochastic Discount Factor)
I want to estimate an empirical pricing kernel for an index. Hence, I need to estimate a physical and risk neutral density. For estimating the physical density, only the index data in an observed time interval is needed. Moreover, I know for estimating the risk neutral density I can use option prices with different strike prices and time to maturity. However, I think the observation date for the option prices should somehow correspondent to the time interval used for estimating the physical density.
Therefore, my question is how the observation date for the option prices has to correspond to the used time interval.
Note: The empirical pricing kernel equal is also referred to as the stochastic discount factor.
## Answer by Finance_Newbie (score 6)
https://quant.stackexchange.com/a/15718
I found and answer to my own question. So, I post it here for people who maybe have the same problem. The answer, however, is quite intuitive. The last observation used for the estimation of the physical density is also the time point where the investors know the most about the physical density because at this point the most possible historical observations are used. Hence, they evaluate the future rationally based on this pseudo-true measure. Therefore, the option prices at the last observed time point need to be used to estimate the risk neutral density.
So, the estimation procedure for a empirical pricing kernel could be as follows:
- Get the closing prices of the considered index in a reasonable time interval $[t,T]$
- Get the option prices for the index at time $T$ with equal time to expiration but different strike prices
- Estimate the physical density $\hat p$ with data obtained in the first step
- Estimate the risk neutral density $\hat q$ with date obtained in the second step
- Calculate the empirical pricing kernel as $\hat k = \frac{\hat q}{\hat p}$
Note: The empirical pricing kernel equal is also referred to as the stochastic discount factor.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.