# Black Scholes: How does it help to transform uncertainty and still not be able to calculate a fair price?

Recapitulating the history of Black-Scholes:

1. Nobody knows the fair price of options.
2. Revolution: BS! You put in all the parameters and get a price -> A Nobel Prize for that one!
3. Wait: Nobody knows the true value of future volatility, so we cannot calculate the fair price after all and are back to square one.
4. But ok: Assuming Mr. Market is always right we now have it backwards: putting in real market prices of options and calculating their implied volatility via BS.

So what we are basically doing is exchanging the uncertainty about the fair price of an option for the uncertainty about future volatility - or just a mathematically sophisticated transformation of uncertainty per se. We could have had the "Mr. Market is always right"-thing without the complex mathematical machinery, couldn't we?

I know, one of the accomplishments of BS is that the option price is independent of any drift of the underlying but still my question is:

Is my short characterisation above correct and if yes, how does it really help to transform one form of uncertainty into another and still not be able to calculate a fair price (why you started the whole endeavour in the first place)?

Your characterisation is correct but incomplete.

1) The most important part of Black-Scholes is not the model but the more general framework of dynamic hedging: you can replicate your payoff by continuously trading the underlying and the amount (delta) you should hold is the derivative of the current premium with respect to the current spot. This is a much more general fact than the actual BS model $dS_t = rS_tdt + SdW_t$.

2) Now how does it help to transform prices in volatilities using the BS formula?

Prices are hard to compare. Fix the underlying and the maturity and try to compare premiums for calls and puts with different strikes. It is not obvious to see which is "cheap" because they do not have the same scale. If you plot them, it will be hard to say anything based on the curve.

By looking at the BS implied volatility instead, you only have one parameter with uniform scale (the standard deviation of the log-returns divided by the square-root of maturity). This makes data much more tractable. Calls and puts correspond to a unique value of volatility and if you plot the smile you get a better representation of the market data. You can list stylized facts and interpret them e.g. on equity markets, vol is higher for low strikes (negative skew), one reason is that lots of risk averse investors buy out of the money puts as insurance against a crach, so the implied distibution for the spot has a fatter left tail than a lognormal one etc...

Might not be the answer you're looking for, but just some thoughts that immediately come to mind...

As you've alluded to, the BS model is much more than just a tool to price options. But there's no need to get into this here.

Prior to publication of the BS model, option prices already traded at more-or-less the price implied by B.S. (part of the justification for B.S. was that its predicted prices matched closely with those of the market at the time).

Supply and demand forces of the market will always dictate the prices of assets, in spite of what any theory says; as such, Black Scholes can be regarded (and is used in practice) as a guiding tool to this price discovery process and a useful interpolation method between option prices quoted in the market. Furthermore, option trading is a constantly evolving market; for instance, prior to the late 1980's there were no volatility smiles/skews observed, and these came about due to events unrelated to any theoretical advances in option price modeling (by that time the BS model was already in active use).

Keep in mind that just because the BS model is used to back-out volatilities from market prices, these implied volatilities aren't really a measures of the asset's volatility in the BS model; they are reflections of supply, demand, and expectations about future price movements. This can be seen from the existence of volatility smiles/skews and the divergence of implied volatility of put and call options (when they should be the same!) And when other volatility models are used (stochastic volatility models, historical volatility, etc.), we actually do need a pricing model that can explicitly provide prices.