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First I'll describe the way I understood things so far from the literature, feel free to correct me here, and then I formulate some questions. I'd search through QSE, but haven't found so far similar question.

The BS model assumes the stock to follow an SDE with a linear diffusion term, hence constant volatility: $$ \mathrm dS_t = \mu S_t \mathrm dt + \sigma S_t\mathrm dW_t. $$ Although model showed reasonable match with market prices back in early 70s, in a decade it was noticed that when assuming constant volatilities calibrated e.g. with ATM options, options on the same stock with different maturities/strikes differ in value from what BS model suggests.

Implied volatility $\sigma_{\mathrm{imp}}(K,T)$ hence was defined as a value of constant volatility which is when put in BS formula returns the market price for the option with maturity $T$ and strike $K$. Since the plot of $\sigma_{\mathrm{imp}}$ appeared far from being flat and showed infamous skew/smile effects, it was clear that assuming any constant level of volatility does not lead to observed market prices. Due to this reason, Derman and others in mid-90s introduced the local volatility model $$ \mathrm dS_t = \mu S_t\mathrm dt + \sigma_{\mathrm{loc}}(S_t,t)\mathrm dW_t $$ and suggested calibrating $\sigma_{\mathrm{loc}}$ with $\sigma_{\mathrm{imp}}$ rather than e.g. with historical volatilities. Further, there were even models of dynamics for the entire volatility surface - I think there are some people doing this now with SPDEs.

  1. Now, the implied volatility is a model-dependent concept. Even if that would be a very good estimate of market's expectation about the future volatility (which some people don't think to be true), that means assuming that everybody on the market is using the same model. E.g. classical implied volatility is computed via BS formulae, so taking it into account requires assuming most people on the market are using BS to price e.g. vanillas. Does it make sense?

  2. Implied volatility necessary contains some noisy information - e.g. demand/supply of OTM options make their prices more volatile than those of ATM options, which is not an intrinsic property of the stock (which volatility is supposed to be). Even if one decides to use local deterministic/stochastic volatility model, why don't calibrate this on the historical data? I do know the criticism of the latter method, but since most of the models are Markovian anyways, that seem to justify historical approach way better than the implied one, doesn't it?

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4 Answers 4

My try to answer this question with some other questions:

  1. Is the BS model right? No. Is it useful: yes. Taking a traded price and the BS Model there is only one input factor that is not given by the market: the implied volatility. It is a measure to compare options across time and strike.
  2. Are there better models? yes. Those that you mention: The local vol models. There are stochastic vol models, Lévy models and if calibrated correctly they perform much better than the BS model. In fact the BS model often is a trivial special case of the more complicated models. How are those models evaluated? One way is to see if the BS-implied vol of the prices calculated by these models fits the BS-implied vol that can be observed on the market. Again the BS-implied vol can be used to do comparisons.
  3. Why not use historical data? The volatility that you use for an option that expires in the future has only limited connection to past/observed volatility. E.g. if I trade an option on NIKKEI and the Fokushima catastrophy happens then implied vol will rise immediately. Historical vol will rise only gradually. In short: implied vol is forward looking and reactive whereas historical vol is backward looking and slow.
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+1 as always ;) –  Probilitator Apr 3 at 14:21
Thanks for the answer. Can you clarify a couple of points? 1. It is useful: yes. Do you mean here, that people in fact use BS to price simple contracts often enough? Cause in that case, I'd agree that it is interesting to take a look at implied volatility. 2. BS-implied vol of the prices calculated by these models fits the BS-implied vol that can be observed on the market. Isn't that equivalent to saying that model prices coincide with market ones? –  Ilya Apr 4 at 9:37
@Ilya addressing your points: with useful I mean that BS is in any case useful as a tool to compare various options by their implied volatility (IV). High IV means rather expensive, small IV means rather cheap. BS with the correct IV (taken from the market) gives the price which is a tautology. If you use BS with the wrong IV then the price is wrong. For OTM options with short time to maturity BS is only able to get the price right with extremely large IVs. AD 2: yes it is equivalent. Again the IV smile/smirk is equivalent to the prices but on another scale. –  Richard Apr 7 at 6:21

we use implied vol for similar reasons why we use duration. we know that security prices are not linear functions of rates, yet we look at the duration, because it gives us an idea of sensitivity to a rate. implied vol gives you a measure of volatility, it doesn't perfectly describe it, but as long as we know this, it's still a valuable metric.

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I am not a trader and will pass on question 1 for now.

To answer your second question. You want you model to be able to reproduce market prices of certain vanilla instruments. This way you achieve consistency with the market. Thus if you want to price an exotict call option you will calibrate your model to liquid call option prices. If the option has more optionality than a standard call it should be more expensive than one. If you calibrated your model to market quotes this will be the case. For historical data this relationship might not hold.

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In that case, don't we completely exclude from our glance the situation when "market prices derivatives incorrectly" which we may think of taking advantage of? –  Ilya Apr 4 at 9:40
A good point :) This is why one is always mostly calibrating to liquid instruments. For these the price is a result of supply and demand and thus assumed to be efficient under a given market setting. –  Probilitator Apr 4 at 11:47
Another aspect is model risk - e.g. when the market undervalues instruments because it is not aware of a certain price relevant factor. Two examples from the recent past are counter party risk and liquidity. Market participants weren't really considering liquidity permiums etc. thus certain instruments were overall cheaper than they should have been. –  Probilitator Apr 4 at 11:51

I'll try to anwswer too.

1) You seem to try to interpret implied volatility as having a statistical nature. In fact implied volatility is nothing but (today's) market prices except that you look at them through Black and Scholes "glasses". Why the Black-Scholes model? There are many reasons for that.

  • this is the simplest sensible model: basically you just assume that log-returns are iid normally distributed.
  • you get a clear interpretation of sensibilities and how to hedge the different risks (the most basic example being Delta = derivative of option price w/r to asset price = amount of stock needed to hedge)
  • you get closed formula for calls and puts (you can actually read the replicating portfolio on this formula)
  • you get the probabilistic and the PDE approach.
  • and you get all that with basically a single parameter to calibrate: the volatility. So all other things being fixed, you get a one to one correspondance between call (or put) prices and volatility.

As convenient as it is, this model is way too simplistic for many well known reasons (fat tails volatility clustering etc...) but a simple model beats an accurate one if the latter isn't useful for the practionner. Implied volatility allows you to import the understanding you get from Black-Scholes into more realistic models. For example, it gives you a natural and visual idea of how far your stock returns are from being normally distributed.

2) As you said calibration on historical data is bad because it simply doesn't work.

First of all there is a practical problem: you have to obtain and treat your historical data. In particular you have to chose a range, sampling frequency and estimator. A typical problem is that if your window is too small you will get a poor estimation, but if your window is too large you will adapt to market changes only days after they occur (same problem with VaR estimation).

That being said there is a more fundamental explanation based on the market efficiency hypothesis which says that all available information is incorporated in the current price (One can debate about the accuracy of this hypothesis. I consider it as a good rule of thumb). This means that all the work you might do to calibrate based on historical data is pointless since all the past information is already in the current price or equivalently in the implied volatility.

Note that the fact that models are Markovians is actually an argument for the implied approach rather than the historical one. You want to make predictions about the future and by definition a markovian process's future only depends of its past through its current state so you should focus on the present (current price = implied volatility) rather than the past (historical data).

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I see a point in your first argument - but as in my comment to @Richard, isn't that argument only true given that a lot of people on the market are using BS model for vanilla, or at least using IV? It seems, that IV is of the following feature: if everybody uses it, then it is also of value to you as you are playing against others. If nobody uses it, it does not give you a lot of information, though. Am I correct? –  Ilya Apr 4 at 9:44
For your second argument: I've only traded on FX a couple of years ago, and there the frequency of data seemed quite enough to make good estimates of volatility just based on a 5-minute-wide window. Of course, the market is quite dynamical, but even for such a fast market 5 minutes did not seem to be such a big window. Although that's a historical data, it seems to be more relevant to "current volatility" given the latter is continuous, than the IV. –  Ilya Apr 4 at 9:48

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