Weighting Option Prices When Fitting a Risk-Neutral Density
Summary
The document describes fitting a parametric risk-neutral density to observed call prices. For each strike, the model price is the discounted expected payoff under the proposed density. The density parameters are estimated by minimizing a weighted sum of squared differences between model and market prices. Weights are intended to reduce the influence of less reliable or illiquid observations; suggested liquidity measures include trading activity and bid-ask spreads.
The question asks what to do when those measures are unavailable, including whether to downweight deep out-of-the-money options, but the answer offers only limited guidance. It reports a practitioner approach of weighting each option by its share of total volume and mentions an alternative of excluding options below a volume threshold, citing prior work on index options. The response does not establish that either approach is generally validated, nor does it supply a substitute weighting rule when volume and spread data are missing.
Key ideas
- Parametric risk-neutral density estimation can minimize weighted squared errors between observed and theoretical call prices.
- Weights can reflect option liquidity, with trade activity and bid-ask spreads offered as possible measures.
- One practitioner approach assigns each option a weight based on its share of total volume.
- An alternative mentioned is to omit options whose volume falls below a chosen threshold.
- The response does not provide a validated weighting function for settings without liquidity data.
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Full text
# Weighting function for parametric estimation of the Risk-neutral density function
# Weighting function for parametric estimation of the Risk-neutral density function
I would like to estimate the Risk-neutral density function (RND) implicit in financial Call option prices by a parametric approach where the parameters of the RND (for instance mean and variance for a log-normal distribution) are obtained by minimizing the weighted squared deviations of the theoretical and empirical prices.
According to standard textbook theory, the theoretical Call option price is equal to the expected discounted payoff at maturity, hence:
\begin{align} C^{\textit{th}}_K &= e^{-r \tau} \int_{K}^{\infty} \! (S - K)f_{RND}(S\vert\vartheta) \mathrm{d}S \nonumber \\ \end{align}
where $r$ is the risk-free interest rate, $\tau$ the option maturity, $K$ the strike price, $S$ the price of the option's underlying at maturity and $f_{RND}$ the risk-neutral density function with parameters $\vartheta$.
As described above, I consider $N$ empirical option prices with different strike rates, $C_K^{emp}$, and estimate $\vartheta$ by a least squares optimization procedure:
\begin{align} \hat{\vartheta} = \operatorname*{arg\,min}_{\vartheta} \sum_{K}^{N} \omega_K \cdot (C_K^{emp} - C^{th}_K(\cdot \vert \vartheta))^2 \end{align}
Following Jondeau et al. (2007, p. 388, "Financial Modeling Under Non-Gaussian Distributions"), I include weights $\omega_K$ associated with each option. The idea is that illiquid options have a relatively little weight in the estimation procedure. Jondeau et al. (2007) suggest using a measure of liquidity such as the number of trades or based on the bid-ask-spread.
My question: Does anyone know how I can define weights if no such data is available? I would probably have to define a (admittedly more or less arbitrary) weighting function where I assign little weight to (deeply) out-of-the-money options. Are there papers about it that suggest such weighting functions?
I appreciate any comment. Many thanks in advance
## Answer by DomingoBrown (score 1, accepted)
https://quant.stackexchange.com/a/51016
A colleague of mine used to simply weight the options by its volume over the total of the volumes. Not sure if this is backed by academic papers tho.
In other approaches such as in Figlewski (2012), the authors simply discards the option with a volume under a certain threshold (40 contracts for S&P500 options)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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