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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?

Richard Hardy
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Andrew Dabrowski
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    Given your question is not really about the mathematics, but perhaps the philosophy of implying volatility, you might enjoy http://www.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
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    "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

4 Answers4

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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

Community
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Matt Wolf
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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 04 '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 05 '13 at 00:52
  • 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 05 '13 at 00:59
  • So what exactly do you mean by IV? How do define, calculate, or estimate it? – Andrew Dabrowski Nov 06 '13 at 01:15
  • 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 06 '13 at 01:31
  • You say that IV is the input to any vanilla option pricing model i.e. it doesn't matter about higher moments as long as we know the IV. I don't see how something like skewness wouldn't also affect the price. I.e. a normal distribution and chi-square distribution can have the same mean and variance but would result in very different option prices if the stochastic process governing the movement of the underlying stock followed either of them. – rwolst Nov 06 '13 at 10:42
  • @MattWolf: So what's the difference betweeb your notion of IV and the ordinary notion estimated volatility? Since the actual volatility of a stock cannot be measured directly it must always be estimated or appraised by individual investors. – Andrew Dabrowski Nov 06 '13 at 13:48
  • @rwolst, I did not make any claims about the distributional assumption of any of the pricing models. What I mean to say is that IV is generally the variable to which vanilla options are most sensitive to (assuming delta-hedged option positions mostly eliminate exposure to the underlying price). I tried to disagree with the notion that IV is "just" a "fudge factor" as implied by the OP. – Matt Wolf Nov 06 '13 at 14:18
  • @AndrewDabrowski, I am not sure I understand what you are trying to say. I suggest to look at the big picture. Single plain vanilla options represent views on implied volatility (for most part). If you think implied volatility should be lower than the market prices then you sell the option and vice versa. Simple as that. How you arrive at your own implied volatility estimate is limited only by your imagination. – Matt Wolf Nov 06 '13 at 14:21
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    @MattWolf: I think you're confusing IV with estimated volatility. The former has a precise meaning for quants in the context of the BS formula, it's what you get when you solve for sigma in terms of option price. Estimated volatility is each trader's estimate of the volatility of the stock price. Of course IV may be useful in estimating volatility, but it's a distinct concept. – Andrew Dabrowski Nov 06 '13 at 16:04
  • IV varies with the terms of the option: strike price, maturity date. Estimated volatility varies with the trader but not with strike price or maturity. When different IVs are used as inputs into the BS formula to adjust for different option terms they are being used as fudge-factors: they do not represent anyone's estimate of the underlying stock's volatility. That doesn't mean they're not very useful fudge-factors. – Andrew Dabrowski Nov 06 '13 at 16:05
  • I just want to say that your reference article is a life changer. Thanks a bunch for linking it! – htd Nov 20 '13 at 20:31
  • @AndrewDabrowski, I stand by my explanation. Implied volatility is never a "fudge factor" but rather an actively traded "asset class". Maybe you want to write up an answer to your own question if you think all other answers are incorrect, which of course begs the question why you would have asked in the first place. – Matt Wolf Nov 21 '13 at 02:29
  • @MattWolf: I'd appreciate it if you would address some of my specific points. In particular how do you account for the fact that a single stock at a single moment can have several different IVs? – Andrew Dabrowski Nov 21 '13 at 14:09
  • I addressed many of your comments and questions and you all debunked them. I also added references for you to read up on. What else do you need? There are different IVs across moneyness and the reason is explained in detail in many answers on this board. Please search for one of my past answers to this particular question as I am a little short of time right now. – Matt Wolf Nov 21 '13 at 22:43
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    @AndrewDabrowski..."particular how do you account for the fact that a single stock at a single moment can have several different IVs?"...this has been covered in the answer...IV is a traded asset! The dollar price of IV is the option price! – Don Shanil May 18 '14 at 13:52
  • sorry the above is not clear -- what I mean is that the different IV correspond to differences in 'moneyness' -- if this was not the case there would be a clear arbitrage situation. – Don Shanil May 18 '14 at 14:06
  • @DonShanil, are you addressing the above to me or Andrew? – Matt Wolf May 18 '14 at 14:26
  • @MattWolf I am addressing this to Andrew. Cheers! – Don Shanil May 19 '14 at 00:23
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Implied Black-Scholes volatility is much more than just a parameter in a formula that can be fudged to produce a reasonable price. When an option position is hedged in Black-Scholes, the daily P&L is proportional to the realized minus implied variance. It follows that implied volatility corresponds to the consensual prediction of realized volatility by market participants committing monetary stakes on their prediction.

Nicole El Karoui calls this 'Robstness of Black-Scholes' in her 1998 paper, and Rolf Poulsen calls it 'Fundamental Theorem of Derivatives Trading' in his 2015 paper. Short term option traders (also called 'gamma traders') constantly compare implied volatility to their own predictions and take (delta-hedged) option positions to realize their views on volatility. Bruno Dupire produced a number of important results based on these considerations.

For the occasion of Bruno's 60th birthday last month, I made a short presentation in RiO on this topic and some important applications. The presentation was recorded and is found on YouTube: https://www.youtube.com/watch?v=-YiAMxjOKHg. The presentation slides are found on SlideShare: https://www.slideshare.net/AntoineSavine/60-years-birthday-30-years-of-ground-breaking-innovation-a-tribute-to-bruno-dupire-by-antoine-savine. I also develop these ideas in the first part of my volatility lectures at Copenhagen University, which slides are also on SlideShare: https://www.slideshare.net/AntoineSavine/lecture-notes-from-volatility-modelling-lectures-at-copenhagen-university

Antoine Savine
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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.

Too many academics on this site. Hesitant to say anything lest I get downgraded en masse.

The statement above is bit far from correct.

  1. Directional traders, even in the OTC space, who trade options do so if their expectation of their “mu” (actual trend) is different from the risk free rate. They are the guys providing the main liquidity to the options market across both liquid and illiquid option underlyings.

For these traders, who happen to be the majority of option traders, the IV part is often insignificant compared to their views on the difference between their expected trend and the risk free rate.

  1. Volatility traders who are a relatively limited lot and focus mostly on the most liquid option underlying, trade options that are delta hedged to get an exposure to IV minus RV differences. Without the directional traders, there would be really limited or no market for the volatility traders to play in.

Yes, to a large extent IV can be a fudge factor. I have had 3 occasions in the past, where I had to negotiate some 5 year options on illiquid instruments where the bank quoted 30% and I had to negotiate it down to 20%, and the main objective was to limit the option premium.

In the OTC space, exotics like Barriers, worst of options etc, all exist to lower the price of options. Price (demand, supply) is definitely more important.

For super liquid markets like S&P on the other spectrum, the IV is completely driven by supply and demand of the underlying options.

For example, even “IV” indices like VIX, OVX, TYVIX etc are not averages of any implied IV —— they are instead weighted averages of actual traded option prices.

It’s only the FX space, where you have banks quoting IV directly — and that’s a characteristic of that market , because of needing to quote OTC exotics like barriers and any random strike if needed , where it’s more continent for the banks to deal with IV rather than standardized strikes.

In general,

  1. the more listed the market, the more supply and demand (option prices) drive the IV. Not uncommon to see illiquid stocks with huge bid and ask IV where the market maker has limited clue on the true IV and just throwing out quotes that wide enough to not get it.

  2. the more OTC the market , and hence non-standardized, the more IV will drive the option prices

uday
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  • I no longer follow the Black-Scholes world, because I have trouble remembering the details. But here is a paper I wrote to summarize what I learned. https://www.academia.edu/6642079/A_Quick_and_Dirty_Overview_of_Black-Scholes-Merton – Andrew Dabrowski Dec 29 '18 at 18:50
  • @AndrewDabrowski thanks for sharing. I could only access the first 5 pages in the link that you sent. Did your paper have only 5 pages or more? – uday Dec 30 '18 at 19:16
  • Right, just 5 pages. I don't pretend to have a lot to say. :) – Andrew Dabrowski Dec 31 '18 at 02:20
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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.

Brian B
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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 04 '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 04 '13 at 18:17
  • Au contraire: assumptions 2 and 6 are the main drivers for cancellation of $\mu$. – Brian B Nov 04 '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 04 '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: http://quant.stackexchange.com/questions/8247/why-drifts-are-not-in-the-black-scholes-formula – Matt Wolf Nov 05 '13 at 04:33
  • Of course I'm mostly incorrect, that's why I'm asking questions! Thanks for the link to the other discussion. Multiple arguments for the cancellation are discussed there. It seems that your (@MattWolf) favorite approach is the risk-neutral martingale, which is the one I haven't yet studied. Do you think the other approaches (continuous self-financing replicating portfolios, CAPM, put-call parity) are lacking in some way, i.e. do you think risk-neutral martingales provide the only completely valid answer to the question of how the $\mu$ cancels? – Andrew Dabrowski Nov 05 '13 at 14:56
  • @AndrewDabrowski, no, there are many approaches, pick whatever suits your needs. I find the approach through the risk neutral probability measure very intuitive, thats all. – Matt Wolf Nov 05 '13 at 15:12
  • Using martingales is the most sophisticated method mathematically, although risk-neutral probabilities per se can be approached through binary tree models. In the latter case risk-neutrality arises as a way of forcing the efficient-market hypothesis via arbitrage: if market prices don't conform to the risk-neutral probabilities then a guaranteed profit can be made. Is that true in the continuous case, martingales, also? – Andrew Dabrowski Nov 05 '13 at 16:25
  • I believe solving PDEs is a more sophisticated and also the more popular approach at exotic trading desks. – Matt Wolf Nov 06 '13 at 01:36
  • I'm talking about the method of deriving the pde. – Andrew Dabrowski Nov 06 '13 at 02:31
  • Like Matt, I favor the martingale approach. It is, by the way, that approach that combines with Black's assumptions 2 and 6 to achieve cancellation of $\mu$. There are mathematical equivalences between all approaches, though the PDE has a special twist that is often ignored: a well-known and tricky "quant" interview question is: if you're buying $\Delta$ shares, then why don't you differentiate the $\Delta$ in applying the PDE? – Brian B Nov 06 '13 at 13:48
  • @BrianB: Is the correct answer: because we're approximating a self-financing portfolio we look only at capital gains? – Andrew Dabrowski Nov 06 '13 at 16:10
  • @BrianB: Ah, I see your point now: by assuming the processes are continuous you can move to differentials where a linear hedge formula is possible. But in that article Fisher is explicitly referring only to the original argument of his and Schole's (not even Merton's) which was using CAPM. Would you agree with me that none of the 10 assumptions Fisher addresses in the article touch on the way $\mu$ cancels in the CAPM argument? – Andrew Dabrowski Nov 06 '13 at 16:16