Does anybody know the formule which transforms Black vol into normal Vol?

To a first approximation,(*) [$]\sigma_{normal} = \sigma_{Black} \times S_0[$] If you mean you want to convert implied vols from one model to another exactly, then (*) is only a first approximation.To do it exactly, just convert the Black implied vol into a price, and then solve [$]\text{price} = c_{\text{normal}}(\sigma)[$],where [$]c(\cdot)[$] is the option value function, suppressing all parameters but [$]\sigma[$]. Of course, the 'normal' model value function is ill-defined until you say exactly what happens when the asset price reaches 0, but I suppose you must already have a formula in mind.

In Managing Smile Risk, Hagan gives a more accurate formula (in Appendix formula A.64)

and what does happen if market price of the option does not always follow BS pricing model?

QuoteOriginally posted by: list1and what does happen if market price of the option does not always follow BS pricing model?Out of curiosity, what would that have to do with anything?

list1 - you don't really believe that people sit in front of their screens trading options believe in BS dynamics/assumptions do you?BS is nothing more than a translation mechanism

Once I asked a question about how much institutions use BS methodology for their trades. The question was directed to a person who knows the situation in whole market. It was the answerThe usual view on options markets is that there are 3 types of players1) Retail investors eg you or me2) Institutional investors eg Aegon an insurance company3) Market makers eg Morgan Stanley or CitadelAll 3 types can either buy or sell optionsFor example, a buy write strategy is popular for 1) and 2)1) and 2) never do anything resembling BS hedging. In contrast 3) does do something close to BS hedging. Then it was asked a question If 1), 2) never do something like BS how do then they interpret option price?The answer wasI would say 1) and 2) look at option valuation from an absolute or fundamental perspective whereas 3) look at it from a relative perspectiveThen the question was " if 1), 2) never do something like BS how do then they interpret option price?"and the answer was"So eg if a terrorist is planning on attacking Paris, he might buy puts on a French hotel stock beforehand because he thinks the put price is cheap on an absolute basis.He does not consider combining the long one put with long half a French hotel stock so as to create a portfolio with riskless gains under some strong assumptions like no jumps or only one possible jump size"I do not think that my question regarding "what does happen if market price of the option does not always follow BS pricing model?" completely irrelevant. Even if option pricing follows BSE with calibration or other innovation it makes a theoretical sense to imagine that market pricing can sometimes deviate from BS methodology.

Of course DocToc is correct about the translation idea. For vanilla options, traders and market makers are certainly free to translatemarket option prices into Black or Black-Scholes implied vols. After all, the models span all the market possibilities, certainly for equity options with [$]S_0 > 0[$].There is an issue when traders try to ram *everything* through the funnel of GBM. For example, for some simple barrier options, it's easy to cook up arbitrage-free option prices that are simply impossible to represent as the value of the corresponding GBM model with some parameter [$]\sigma[$].Please don't ask for an example, as it would take a lot of digging I don't have time for. But the idea is, you use some jump-diffusion model that can produce prices outside of "robust" bounds that are known to hold for any and all pure diffusion processes. You don't need bizarre parameters, so the resulting model prices are easily potential market prices, esp. when it is quite possible that theparticle may jump over the barrier.So, in this sense, list1 is correct that his question is not completely irrelevant.

I can comprehend the practice of "traders and market makers are certainly free to translate market option prices into Black or Black-Scholes implied vols" but how it can be interpreted on theoretical level of comprehension? For illustrative example let takes second Newton law F = ma. We want illustrate translation between observed data F for fixed m = 1 and a which plays the same role as input volatility. Though such analogy may not a good example.

It's not that deep. It just means many traders and especially market makers think about their markets and manage their lifein terms of Black-Scholes implied volatilities. They don't have to believe in GBM dynamics to do this -- mostly -- with some exceptions I pointed out.

Thanks Alan, I understand your point.These days my vision is changed and I will appreciate your comments or corrections. For me pricing role model is coupon bond. If coupon is defined by a floating rate the benchmark can be adjusted. Price at t = 0 is defined as PV or EPV of the future cash flow.Let us take a look at option and assume that dt is 1 day or 30 min it does not matter. An we chose dt such that the values of the order (dt)[$]^{1+h}[$] can be ignored at least in the theory. BS price I interpret as following bank sells not a single option it sells Bs portfolio that is defined at t[$]\,_0[$]. Buyer can sell or by depending on sign of the delta of the [$] \delta\,( t_0 ) [$] stocks and we arrive at BS price. Here we assumed that stock can be sell or by immediately and we ignore bid-ask gap. At t[$]\,_0[$] we know that at t[$]\,_1[$] the portion of stock in BS portfolio should be changed and that should happen next during lifetime of the option. Hence the value of the BS portfolio at t[$]\,_0[$] should be cumulative value of the EPV of the all future adjustments and not only the value as it was defined by B&S. The EPV of all adjustments from t[$]\,_k[$] , k = 1, 2, ... n should be added to the value of the BS portfolio which is defined at t[$]\,_0[$] to cover [$][\,t\,_0 \, , t\,_1[$]] period.Doing this we will arrive at another formula for the option price at initiation. Of course it can be close or far from BS price depending whether the cumulative adjustments are biased or not. This ideas are also relate to connection between volatilities. If market somewhat takes into account all future transactions then implied volatility and BS theoretical volatility would be different

If I gloss over a lot of ambiguities, you are describing valuation by one-factor dynamic hedging, aka dynamic replication. It 'works' if you clean up the ambiguities and the stock price dynamics are 1D diffusions, which are complete markets. Personally I don't find it especially useful to think too much in terms of dynamic replication.That's because my mental picture of the markets, within the dubious paradigm of stochastic process theory, is that of an essentially incomplete market with jumps and stuff.

If market somewhat takes into account all future transactions then implied volatility and BS theoretical volatility would be different ofcourse, I think you would really benefit from a few days in a real trading environment, I think you will grasp some fundamental issues which you don't seem to get.in practical terms there are three prices for any trade (at a high level).1. Mid price - this is what I think is your "fair/theoretical BS or whatever model volatility" - apart from Marking your book and doing analysis a mid is practically a useless concept in my opinion2. Bid price - the price at which some one is willing to buy the asset.3. Offer price - the price at which some one is willing to sell the asset.the difference between 1 and 2. or 1 and 3. is the price difference/volatility difference that the trader is charging you for whatever transaction costs + other costs he is going to accrue over the life of the trade to completely hedge it or not. He could do this in infinitely many different ways. EDIT: further, he might charge you even more than just transaction costs +... as at the time of trading the transaction costs are not fully known/forecastable.

QuoteOriginally posted by: DocTocIf market somewhat takes into account all future transactions then implied volatility and BS theoretical volatility would be different ofcourse, I think you would really benefit from a few days in a real trading environment, I think you will grasp some fundamental issues which you don't seem to get.in practical terms there are three prices for any trade (at a high level).1. Mid price - this is what I think is your "fair/theoretical BS or whatever model volatility" - apart from Marking your book and doing analysis a mid is practically a useless concept in my opinion2. Bid price - the price at which some one is willing to buy the asset.3. Offer price - the price at which some one is willing to sell the asset.the difference between 1 and 2. or 1 and 3. is the price difference/volatility difference that the trader is charging you for whatever transaction costs + other costs he is going to accrue over the life of the trade to completely hedge it or not. He could do this in infinitely many different ways. EDIT: further, he might charge you even more than just transaction costs +... as at the time of trading the transaction costs are not fully known/forecastable.DT, I agree with your point about importance to get Bid-Mid-Offer prices in real trading. It seems that they are used also throughout all market instruments including simple ones such as for example variable coupon bonds. I chose a variable rate coupon to highlight the fact that coupon value will be known only in future moments of its delivery. Its functionality in pricing bonds similar to adjustments of the BS portfolio.My point which I expressed earlier is concerning Mid-price. Though I actually understand importance Bid-Offer prices in trading but I do not quite understand in what degrees they are subjective/objective based on trader experience or somewhat theoretical. Concerning Mid price based on BS model I would to note that based on variable coupon bond valuation its play the same role in valuation as zero coupon bond component as far as the PV of future coupon value do not represented in the spot price which is EPV of all future payments.

how subjective/objective bid/offers are depends on many different things. I would argue that nearly all the time they are subjective as there are too many unknowns to be 100% certain and therefore there is always a degree of finger in the air-ness about making prices. This is what makes making a mkt hard and risky.regarding mid prices are you trying to ask how a mid price in the mkt is established? if so, like I said - mids are irrelevant and only a theoretical construct to make marking a book and analysis easier. Therefore mid can have countless definitions, the most common is probably mid = (bid+ ask) /2, but that is just because it is simple definitely not correct. the only thing that you do know for sure is that the fair value of the asset is some where between the bid and the offer.