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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

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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)

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