Estimating Implied Volatility Without Options on an Equity
Summary
The document asks how to estimate implied volatility for a stock that has no traded vanilla options, when options are available on an index containing that stock. The questioner proposes estimating the stock’s beta to the index, potentially with a Kalman filter, but seeks an implied-volatility measure rather than a historical-volatility estimate or a forecast from an ARCH model.
The response says there is no single standard market approach and outlines two alternatives. Information-based methods transform a physical probability distribution toward a risk-neutral one using a criterion such as relative entropy. A hedging-based method defines the target Black–Scholes volatility as the value that would make expected hedged option profit and loss zero, then solves for that volatility using historical price paths over the option’s horizon. The response sketches the latter calculation but supplies no implementation details or comparative evidence. Each approach has assumptions and trade-offs, and the document does not establish that either is generally preferred.
Key ideas
- The document addresses implied-volatility estimation for an equity without listed vanilla options.
- It states that there is no universally standard method for inferring the missing implied volatility.
- Information-based approaches can adjust a physical distribution toward a risk-neutral distribution using an information criterion.
- A hedging-based approach solves for the Black–Scholes volatility associated with zero expected hedged profit and loss.
- The outlined methods have trade-offs, and the document provides no evidence that one is universally superior.
Tags
Full text
# Estimating implied volatility of an index component with no vanilla options market
# 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.
## Answer by Quantuple (score 6, accepted)
https://quant.stackexchange.com/a/31121
There is no standard approach to this problem to the best of my knowledge. Different approaches exist and each has its own pros and cons as usual. To mention a few:
- 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.
- 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.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.