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I have been doing some research regarding options pricing (particularly using B.S) and have come across two research papers which discuss how the Black Scholes model has a tendency to overprice and underprice call options in certain scenarios.

The papers are: https://www.jstor.org/stable/2328053?seq=1#page_scan_tab_contents and: http://people.stern.nyu.edu/msubrahm/papers/wop.pdf

Particularly the first paper mentions that " B.S overprices deep in-the-money options, while it underprices deep out-the-money options." As well later mentioning "An explanation for the systematic price bias is the assumption of lognormally distributed security price, which fails to systematically capture important characteristics of the actual security price process."

I understand the fact that Black Scholes has a tendency to fall short and misprice under certain conditions (when it's assumptions do not hold true).

However I am confused by the concept of it "overpricing" and "underpricing"

The way I see it is this: if you have a pricing model based on certain assumptions and in a particular case an assumption is false then the price your model has produced is "mispriced" (i.e. it did not take an important factor into account and therefore can not reflect the true price)

But to say that an option is for example "overpriced" would you not need to know the true price of it? since "over" is a relative term.

In which case how do you get the true price of an option in order to determine whether something is over or under priced?

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Perhaps they mean that if you use the ATM implied volatity as an input to price ITM and OTM options, then some will be underpriced and some overpriced compared to the true price observed in the market. Equivalent to saying that implied volatilities exhibit a pattern and they are not constant across moneyness.

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  • $\begingroup$ so what you are saying is that if I were to look at an options chain the implied volatility of an option which is closest to being ATM best represents the true volatility and therefore true price? @Kiwiakos $\endgroup$ – Mustard Tiger Dec 29 '16 at 22:14
  • $\begingroup$ No, he is saying that there is a true option price for each strike. That is the market price. Using a black scholes model with s single volatility cannot price them all correctly. Some will be overpriced (b-s price> market price) and some underpriced (b-s price < market price). $\endgroup$ – dm63 Dec 30 '16 at 3:54
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    $\begingroup$ Then the distribution of the underlying will have fatter tails than the normal distribution, so the black scholes model will underprice the far out of the money options, for example. $\endgroup$ – dm63 Dec 30 '16 at 12:39
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Let me give you three or four of my papers. It will solve your problem. The answer "why" is simply too long to answer here. The basis of my papers is that returns are not data. Prices are data and returns are a transformation of that data. It follows then that you cannot make assumptions about the distributions of returns, but you can either derive the distribution of prices or you could make assumptions about the distributions of prices. You could actually show that it is mathematically impossible for equity prices to be either normally or lognormally distributed. It isn't possible as it would create a mathematical contradiction in an option pricing model.

It turns out that you can derive the distribution of prices by using the rules that derive the price structures and the error terms. It is also well understood in statistics how to do the transformations necessary to determine the distribution of returns for an asset class. So, for example, under Markowitz's assumption, returns on investing must be the ratio of two, independent, normal distributions centered on (0,0) in the error space. On the other hand, if you were buying and selling assets at Sotheby's, such as art, then you will encounter the ratio of two Gumbel distributions. The rules determine the distributions. Other issues, such as the budget constraint, the cost of liquidity, merger and bankruptcy risk are part of the rules and so in part determine the final distribution.

This in turn determines the rules of econometrics, which in turn determines the rules for pricing options. I also did a population test as a partial verification.

The papers at the page https://papers.ssrn.com/sol3/cf_dev/AbsByAuth.cfm?per_id=1541471 will explain why the papers see systematic mispricing. I am working with a measure theorist to extend the laws of stochastic calculus to include this situation and would like to have a fundamental extension of the rules of calculus prepared by Spring break. I have also started a paper on subjectively optimal portfolios, but I am teaching six classes so it won't be finished before summer. Just a warning, I put rough drafts out there, so the calculus paper, when it first comes out, could have poor language or be missing a boundary condition or something like that.

Please feel free to send any criticisms.

EDIT You can tell something is mispriced in two ways. First, you can do a correlation study to see if option prices are correlated with actual outcomes. Second, you can use the method of inverse probability, as is done in one of the papers referenced, to test the assumptions directly.

Informally, the findings excluded Ito calculus models from use. Because an inverse method was used, a prior probability for mean-variance finance was used, giving it 999,999:1 odds of being the true model or something like it, over the alternative that no variance existed, and it was still falsified despite prior bias.

If you have not used Bayesian or inverse methods, http://www.seaturtle.org/mtn/archives/mtn122/mtn122p1.shtml?nocount provides an informal account. A good set of youtube videos of a grad course on them are at http://www.youtube.com/user/opinionatedlessons/videos?view=0&flow=list&sort=da

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    $\begingroup$ I don't think this answers the question. The over/underpricing in question comes from the assumption of a flat volatility surface, as in Kiwiakos's answer. $\endgroup$ – Chris Taylor Dec 29 '16 at 23:26
  • $\begingroup$ Actually it does. The second paper in particular is on the kernel. A mathematical consequence of getting the wrong distribution is that the volatility surface will become flat due to the mismatch of the calculation method and the distribution. There is more than one way to a flat surface. There is a 1960's paper by Rao generalizing this to all AR processes and their continuous time cousins and a paper demonstrating this is true by Sen, also in the 60's. Don't have bibliography with me. These papers never filtered over to finance. continued $\endgroup$ – Dave Harris Dec 29 '16 at 23:42
  • $\begingroup$ Following work by ET Jaynes, you can show that models like Black-Scholes are inadmissible, or more formally, are not in the complete class of solutions under Wald's Complete Class Theorem. I found 3800 articles just on the volatility smile alone. If it were admissible, there would be zero. $\endgroup$ – Dave Harris Dec 29 '16 at 23:44
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    $\begingroup$ Let me rephrase, then - this may tangentially answer the question, but focuses on an unhelpful part of the problem, and appears to be promotional material for the author's personal research. The source of "mispricing" is the assumption of a flat vol surface in BS. I use scare quotes because in practice no one assumes a flat vol surface. There are many modifications to BS, or alternative models (e.g. SABR) which deal with a non-flat vol surface. Also see "Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula" by Haug and Taleb for an interesting historical perspective. $\endgroup$ – Chris Taylor Dec 30 '16 at 9:00
  • $\begingroup$ No self promotion intended. Taleb has done good work on the topic, but I haven't seen that article. I didn't see it as tangential. I did a population study on Black Scholes, not mentioned here. The systematic mispricing is terrible. I assumed perfect foreknowledge and adjusted for dividends. The only unknown was the closing price. The problem is some people use the algorithm or tools like it. I submitted the papers to a finance journal and was declined as obviously false. I submitted them to a stats journal and was declined as obviously true and adding nothing to the field of stats. $\endgroup$ – Dave Harris Dec 30 '16 at 10:26

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