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We know Black-Scholes is an imperfect model for options pricing. Why is so much of the analysis of its defects focused on implied volatility? The fact that IV varies for the same stock at the same time proves that BS is faulty (and cannot be fixed by making volatility time-dependent). It also proves that $\sigma$ is not actually the standard deviation of the stock price, but just some fudge factor that is used to tweak results.

Why all the attention on this fudge-factor? Once you know the model is defective I would think you should go back to the drawing board and see how the derivations of the BS pde could be modified to get a more sophisticated model. But instead most of the attention in the literature seems to be on how to control the fudge-factor (Derman has 12 lectures on The Smile), as if the BS equations had descended from heaven and our job as mortals was merely to provide midrashim on them.

My own opinion is that people are too bemused by the famous cancellation of $\mu$ in the BS pde. After all, that's what impressed the Nobel Prize committee, so it must be right. But both Black & Schole's original derivation using asset pricing theory and Merton's later self-financing portfolio argument are enormous over-simplifications of the real world, and contra the Sveriges Rijksbank, Black, Scholes, & Merton did not show that "... it is in fact not necessary to use any risk premium when valuing an option".

Merton's argument was the more sophisticated mathematically but also the less robust and flexible. Continuously self-financing portfolios will never be approached in reality: even if IT makes the continuous aspect feasible, the self-financing aspect will always be consumed in transaction fees. But the asset pricing argument could easily be modified to allow greater suppleness.

The logical first pass at explaining the volatility smiles ought to be in going back and not making the facile assumptions that led to the cancellation of $\mu$ in the asset pricing argument. This immediately introduces an extra parameter into the model that makes more sense than the IV fudge-factor and might in fact be able to explain much of the smile. For example you might get a modified BS pde like $$ rV=\mu S {\partial V \over \partial S} + {\partial V \over \partial t} + \frac{1}{2}\sigma^2S^2{\partial^2 V \over \partial S^2}, $$ which leads to a similarly modest change in the pricing formula.

As you have no doubt inferred by now I'm not an expert. Have these ideas already been explored and found wanting?

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Given your question is not really about the mathematics, but perhaps the philosophy of implying volatility, you might enjoy ito33.com/sites/default/files/articles/0601_ayache.pdf –  experquisite Nov 10 '13 at 22:05
    
@experquisite: That article turns out to exemplify the confusion I was complaining about. I read only 3 pages of it. That would have been enough time for any decent writer to have set out the basic issues, but Ayache had still not addressed a simple very pertinent fact: IV varies with the terms of the option used to calculate it. That immediately makes it clear that the BS $\sigma$ is a fudge-factor covering up for holes in the model. But Ayache wants to wax profound on metaphysics and use some of the French post-structuralist lingo he learned in college. –  Andrew Dabrowski Nov 11 '13 at 22:11
    
"I shall argue that the step towards implied volatility is in fact so radical that it ought to change the whole attitude towards statistical inference." I get the impression he never studied any science at Ecole Polytechnique de Paris, which ought not to be possible. But his attitude toward statistics is so innocent as to invite that surmise. However I see that my hero NN Taleb is an admirer, so I will have a look at "The Blank Swan". –  Andrew Dabrowski Nov 11 '13 at 22:19

2 Answers 2

First of all, may I point out two big misperceptions that you may have:

  • Implied Volatility (IV) is the input to any vanilla option pricing model (not just Black Scholes (BS) that impacts the pricing the most. You can verify this by flipping through the different risk exposures (greeks and higher order sensitivities) and study mean volatilities in such risk factors and their impact on the pricing of such options.

  • Traders who price and buy/sell options in effect trade future realized volatility/expected variation in the underlying asset returns. Hence, option traders express views on such future asset price variation and thus buy and sell volatility. The term "implied" volatility is in my opinion a bit of a misnomer because the trade starts with an agreed level of volatility and not an option price.

    (In fact, you hardly ever hear any professional traders agreeing on an option price, they most often agree on the exact implied vols they trade at, and often times also trade the delta alongside the option (at least in equity space) in order to have the option delta-hedged at initiation.)

Option pricing models are used to translate the expressed IV -> Price. When you see option prices on your trading screen then those are the outputs of automated pricing applications which as input take among couple other mostly statics, IV.

Hence, your view of IV being a "fudge factor" is very simplistic. Most everything that is traded in an option in fact is IV. (Of course you have other option inputs but you would trade specific dividend swaps or interest rate derivatives, for example, if you wanted to express a view on such inputs). The option price is just a translation in order to pay for the implied volatility that is traded.

And: Just because an option pricing model is imperfect does not make it worthless. In fact, I challenge you to come up with an alternative model that is equally simple (computationally as well as intuitively) and more accurate than B-S, I am sure the market will embrace it and thank you for your efforts.

EDIT:

I highly recommend to go through the following short paper: Option Pricing Q&A

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My understanding was that IV is the value obtained by plugging the market price into the BS equation and solving for sigma. Thus IV is by definition an output, not an input. True volatility, if there were any way to measure it accurately, would be the input you'd want to use. Are you using IV to mean estimated volatility? That's something different. –  Andrew Dabrowski Nov 4 '13 at 17:22
    
IV = implied volatility and no, it is not an output but an input. Any reputable option dealer/trader/sales person should have a keen understanding at exactly which implied vol levels their products trade whereas hardly anyone knows the quoted prices. It is a huge misperception even within the quant community to believe that option prices are plugged in and what comes out is an implied vol level. A good comparison are bond price vs yields on the fixed income side: Hardly anyone quotes bond prices but everyone has a keen understanding (so I hope) of yields. –  Matt Wolf Nov 5 '13 at 0:52
    
But just to make sure, this was not the center of my answer, my answer attempts to make the point that implied volatility is the corner stone of option pricing. In fact, I claim that different implied vol levels cause more variation and potential error in pricing an option than the choice of pricing model (I cannot prove that but its a conclusion drawn from many years in the "war zone"). The choice of model in fact makes zero difference when dealing options as long as both counterparties agree on the same model at the time they translate IV-> Invoice Price. What they trade is IV, nothing else. –  Matt Wolf Nov 5 '13 at 0:59
    
So what exactly do you mean by IV? How do define, calculate, or estimate it? –  Andrew Dabrowski Nov 6 '13 at 1:15
    
Implied volatility is the expected future return volatility of the underlying asset over the lifetime of the option. It is not a value that is calculated but it is in itself something that is bid and offered in the market as a function of each trader's view on future volatility. Participants express a view on implied volatility and trade it, paid for through the translated option price. Implied volatility can be modeled and forecast and there are various models that accomplish such. –  Matt Wolf Nov 6 '13 at 1:31

The focus on volatility comes about because all price changes "look like" volatility, no matter their source. Improvements in volatility treatment are therefore conflated with improvements in the model, and typically when people consider altered models, they first look to how well the alterations do in providing prices that explain skew for the classical model.

Thus, though there are indeed many problems with the model I wouldn't necessarily agree with you that the research focus is on "volatility". It's more on the "form of price changes", albeit only tractable ones. Some of the flaws in the model lack clear treatment. My favorite such problem is corporate actions, like spinoffs. How would one propose to model that?

Fischer Black himself wrote a clear-eyed paper on the model's limitations, and how to use them. If you want to ponder improvements, it's the right place to start.

You happen to focus on drift in your question, so I'll specifically address that. Drift is closely related (observationally) to using a mis-estimated discount rate in the classical formula. This sort of issue can be perceived by examining implied vols for vanilla puts and calls at the same strike, which enjoy a pricing relation derived from the put-call parity formula (which is, I stress, model-free)

$$ C_K - P_K = e^{-rT}(F-K) $$

When those implied vols differ, either the model or its parameterization is in error. As a matter of fact, many underlyings have this property. Often the explanation is mundane: the carry cost is wrong. Other times, the problem is more in the model, as it would be with a bad drift.

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Thanks for the link to Black's article, I'll read it shortly. In the meantime let me just clarify that I was thinking specifically about the focus on the implied volatility smile, which is purely an artifact of the limitations of the BS formula and has nothing to do with changes in the underlying stock price. As for drift, surely it's possible that different investors simply have different estimates of a stock's future performance? –  Andrew Dabrowski Nov 4 '13 at 17:37
    
OK, I've had a look at the Black article "Holes" and it was disappointing. He addresses 10 unrealistic assumptions that BS is based on, but does not even mention the ones that have me worked up - the ones that lead to the cancellation of $\mu$. Has this aspect really never been explored? In fact Black says "I don't know of any variation of the formula where the stock's expected return affects the option's value..." Maybe there's a simple arbitrage argument that shows my modified BS pde is unworkable? –  Andrew Dabrowski Nov 4 '13 at 18:17
    
Au contraire: assumptions 2 and 6 are the main drivers for cancellation of $\mu$. –  Brian B Nov 4 '13 at 19:39
    
No, I think the most important reason for the cancellation (in the asset price argument) is the assumption that the option price follows exactly the same capital asset price model as does the stock. Options buyers and stock buyers are two significantly different populations, with the former on average more sophisticated, and they may have different (not necessarily better) ideas about how the stock will behave. Assumptions 2 and 6 are standard idealizations which have no particular bearing on the cancellation of $\mu$. –  Andrew Dabrowski Nov 4 '13 at 20:39
    
Andrew Dabrowski, please take a look at this question, (so far) you are incorrect in most of your claims regarding the subject matter of option pricing: quant.stackexchange.com/questions/8247/… –  Matt Wolf Nov 5 '13 at 4:33

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