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## Saturday, June 4, 2011

### constant-vol assumption ^ varying daily realized vols

As stated in http://bigblog.tanbin.com/2012/06/var-swap-daily-mark-to-market.html, people expect daily realized vol (DRVol) to fluctuate during next 3 months (or any given period). Monday 9%, Tue 17%, Wed 8.9%... However, we also know that BS diffusion equation assumes a constant vol. Any contradiction?

Well, BS is not so naive as to assume every single day's ln(PriceRelative) == the same value throughout the life of an option. That would be a deterministic asset price model. Such a model would absolutely predict the exact IBM closing price tomorrow. It's clearly unrealistic -- no one can predict the exact IBM closing price tomorrow.

No, BS is all about diffusion/randomness, so the exact price on any (future) day is random (even though price now is realized and known). That means BS can't predict the exact value of ln(PriceRelative), which is the to-be-realized vol. Even in the constant-vol model, this ln() could be 10% tomorrow, and 20% next day (annualized vols). Such varying vol on a daily basis is perfectly legitimate I a constant-vol model.

Q: So What is the constant-vol assumption by BS?
A: the sigma for a given stock, once calibrated using historical data, is assumed to permanently characterize the diffusion or the random walk or the geometric Brownian motion. So even though BS can't predict the exact value of ln(PR) today vs yesterday closing, BS does predict the Distribution of that ln() value. It treats the ln() as a random variable following a precise Normal distribution. Consequently, today closing price is another random variable following a precise LOGnormal distribution.

To a layman, this is revolutionary thinking. A layman like me tries to predict today's closing price. Knowing how hard it is, we try to draw a narrow band of the closing. BS is smarter in treating that unknown closing just like a temperature, and predicting its probability distribution instead of its exact value.