Black-Scholes: Why the focus on volatility?

Question

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?

Answer 1

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

Answer 2

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

Answer 3

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.

2. 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

Answer 4

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.