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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.

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What is "alpha value"? –  Alexey Kalmykov Jul 12 '13 at 8:24
    
Does your question imply that you do not have access to option market data you can derive implied volatilities from? (i.e. you only have the underlying market?) –  AdAbsurdum Jul 12 '13 at 10:38
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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

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Yeah, I quite liked using the drift-independent volatility estimator outlined here: atmif.com/papers/range.pdf But as others have commented, I don't think this is quite what is being asked. –  experquisite Jul 15 '13 at 14:58
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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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