Volatility Estimation Choices for Black–Scholes Option Pricing
Summary
The document discusses whether an alternative parameter, possibly GARCH alpha, can replace historical volatility in a Black–Scholes option pricing project. It distinguishes realized volatility, estimated from past returns, from implied volatility, which represents the market’s forward-looking volatility input. Black–Scholes pricing uses an assumption about future variability, so historical estimates are not automatically the appropriate input; they may instead help inform an implied volatility estimate.
For realized volatility estimation, the answer notes that the usual close-to-close estimator relies on a constant-volatility geometric Brownian motion model. It suggests more frequent regularly spaced observations or high, low, open, and close data as alternatives, citing the Garman–Klass estimator as having lower variance in the stated comparison. For departures from Black–Scholes, it suggests specifying another formal process or treating volatility as locally stable over a shorter estimation window. These approaches involve assumptions, and the discussion does not provide a GARCH estimation procedure or a tested option-pricing modification.
Key ideas
- Close-to-close realized volatility is an estimator under a constant-volatility geometric Brownian motion assumption.
- Black–Scholes requires a view of future volatility, while historical realized volatility measures past variation.
- More frequent observations or high-low-open-close data can improve realized volatility estimation.
- The answer describes Garman–Klass as a lower-variance estimator than close-to-close volatility in its cited comparison.
- A different volatility model or a locally stable volatility assumption can be considered when extending the Black–Scholes framework.
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# how to calculate more efficient volatility figure than historical volatility? # how to calculate more efficient volatility figure than historical volatility? can we use alpha value to calculate option price instead of historical volatility. And if we can please explain how. I am doing my MMS in Finance and this for a project i am doing. the project is about tweaking the black scholes formula on option pricing. ## Answer by Mark Higgins (score 4) https://quant.stackexchange.com/a/8455 The standard realized volatility calculation assumes an underlying model: geometric Brownian motion with constant drift and volatility. Then realized vol squared is an unbiased estimator of the process volatility squared. If you want to move beyond Black Scholes then you have two possibilities: look at a different formal model and the estimators for its values; or assume that volatility is stochastic but varies more slowly than your estimation window, so you can (roughly) assume it's constant for that window. I think by "alpha value" you're referring to the alpha parameter in the GARCH model? There are lots of references on estimators for GARCH process parameters, though I don't have any at hand right now. Trying googling for GARCH estimation. In practice, though, people do the second more than the first, even though it's difficult to justify formally. One way you can give yourself more confidence there is to use a more accurate estimator of realized vol than the usual close-to-close one, since then you can use a shorter historical window for the same uncertainty on your volatility estimate. Two usual ways there: use regularly-spaced data that's more frequent than close-to-close (eg sampled every 10 mins from realtime data); or use high-low-open-close (HLOC). There's a really cool paper by Garman and Klass on estimating vol using HLOC data which I love, and gives an estimator with a variance about 9x smaller than the variance on the close-to-close realized vol estimator (so std dev of the estimate is ~3x more accurate): http://www.cims.nyu.edu/~ra1221/riskPortfolio/garmanKlass.pdf ## Answer by Matt Wolf (score 2) https://quant.stackexchange.com/a/8450 Black Scholes makes the assumption of deterministic (time varying) volatility of the underlying asset. Also, the volatility input to the option pricing model is implied by nature and does not rest on realized historical volatilities. Think about it, the whole notion of being able to price a derivative with contingent future payoff rests to large degree on the assumed future variability of the underlying asset. Therefore if you look to make changes to the Black Scholes framework then you can start with modeling implied volatility, not historical realized volatility. The next step would be to relax more assumptions of the BS framework to account for discontinues jumps, varying interest rates, different dividend models, and such forth. But try to remove yourself from considering realized volatility metrics as they are of no use to the BS framework unless of course you utilize historical volatility to model your implied volatility which you then plug into BS.
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