# Estimating implied volatility of an index component with no vanilla options market

There are liquid vanilla options trading on an index of 20 equity components.

The question is how to price an option on one of the index components, knowing that there are no options trading on that particular equity, hence no implied volatility available.

Is there any market practice on how to estimate that implied volatility?

My way of doing it would be to estimate beta of the stock to that index, probably with the help of kalman filter, and manipulate beta to get "implied" volatility for the underlying. But maybe there is some better way? I don't want to use any metrics such as historical standard deviation etc. Arch model would be useful to predict "true" future volatility, but I would like to have implied.

• Information-based methods: these aim at "risk-neutralising" the distribution observed under the physical measure relying on some information criterion e.g. minimising the KL divergence, or relative entropy, between $\Bbb{P}$ and $\Bbb{Q}$. See Derman & Zou's method for instance.
• Econometric methods: postulate and calibrate some ARCH/GARCH dynamics under $\Bbb{P}$ and translate it to some "risk-neutral" world $\Bbb{Q}$ (should be equivalent to directly modelling/calibrating the dynamics of the stochastic discount factor). See NGARCH model proposed by Duan in 1995 (can't put my hand on the paper) or the one by Barone-Adesi, Engle & Mancini.
• Hedging-based approach: Define the target implied volatility $\sigma_T$ as the BS volatility which would have allowed you to obtain a zero expected P&L if you had hedged a vanilla option of maturity $T$ using that precise vol figure. This amounts to solving some non-linear equation of the form: $$\Bbb{E}^\Bbb{P}[ \text{P&L}_{[0,T]}] = \frac{1}{2M} \sum_{m=1}^M \left[ \sum_{i=1}^N \Gamma(\sigma_T,S_i^{(m)}) \left(S_i^{(m)}\right)^2 \left( \left(\frac{S_i^{(m)}-S_{i-1}^{(m)}}{S_i^{(m)}}\right)^2 - \sigma_T^2 \delta t \right) \right] = 0$$ with $m=1,...,M$ representing past realisations of the price process $(S_t)$ over a uniform time partition $t_0,...,t_N$ spanning over $T$ years. See this slide deck by Dupire.