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A recent preprint appeared on arXiv. It questions the appropriateness of the Black–Scholes formula for the price of a European call option in the context of already assuming the Black–Scholes model for the spot price of the underlying.

Black-Scholes Option Pricing Revisited?

An extract from the abstract:

If correct, our results invalidate the continuous-time budget equation of Merton (1971) and the hedging argument and option pricing formula of Black and Scholes (1973).

The crux is claimed to be that the self-financing condition (often assumed when rigorously deriving the formula) is inappropriate and has an unrealistic consequence, namely

His analysis therefore implicitly assumes that the portfolio rebalancing is deterministic and does not depend on changes in asset prices.


Suppose the preprint is accurate. Based on the (rather standardised) models driving derivatives pricing at modern investment banks, what kinds of derivatives are likely to be significantly mispriced (if any)? Keeping in mind that there could be "higher-order" effects by using models/results where the validity potentially also comes into question due to some degree of dependence on the Black–Scholes formula.

For example, given the way volatility is often quoted (the right number for the Black–Scholes formula to arrive at a certain price), valuation of vanilla European options is basically guaranteed to be unaffected even if the preprint is accurate.

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  • $\begingroup$ Hi Mike: I would wait until some finance whizzes comment before trying to figure out where the largest mis-pricings might be. Derivatives pricing is based on the "discount the risk adjusted payoff at the risk free rate" argument so it's a little hard to believe that that would go out the window also. In fact, if these authors were correct, it seems like the whole world of math finance would be turned on its head. Hopefully, the math finance whizzes ( some of which are on here ) will conclude that there is a mistake in the authors' argument. Thanks for pointing out an interesting paper. $\endgroup$
    – mark leeds
    Feb 15 at 2:28
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    $\begingroup$ I haven’t read the article but I think the mainstream view is that the original BS derivation was not that rigorous and you can find holes in it. But this view still maintains that you can make the derivation rigorous and the formula is correct given the key assumptions. $\endgroup$
    – fes
    Feb 15 at 10:43
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    $\begingroup$ thanks fesman. I haven't gone through it but the Gordon link that Kevin points to LOOKS REALLY USEFUL. I haven't checked out the Peter Carr one and hopefully Gordon's will be enough. I've seen enough answers from Gordon that I'm confident it will be. $\endgroup$
    – mark leeds
    Feb 15 at 13:58

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It's a common critique that the original Black-Scholes derivation is somewhat imperfect with respect to the self-financing property, as fesman mentions.

Self-financing trading strategies are a key concept in maths finance. After all, we can all replicate any payoff using a non-self-financing strategy by simply injecting funds as necessary. Thus, pricing by arbitrage typically requires identifying a self-financing portfolio which replicates the targeted payoff.

In doubt it's always useful to consult the writing of the great people in the quant finance community. The issue about self-financing in the Black-Scholes derivation has been addressed, amongst others, by Gordon and Peter Carr.

In his stellar answer, Gordon points out that the standard textbook derivation of the Black-Scholes formula is not self-financing. Whilst it gives the correct result (and yes, the final Black-Scholes formula is correct), the standard derivation is wrong.

In Carr's paper, he discusses in question V how the Black-Scholes derivation can be made rigorous. To this end, he carefully thinks what meaning the differentials need to have and how one can bypass the self-financing property.

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    $\begingroup$ Thanks Kevin. I'll check those out. I'm surprised then that the authors either don't mention or don't know about the work of Gordon and Peter Carr. It seems like they've tried to involve a good amount of people as seen in the footnote on the bottom of the first page. Thanks again. $\endgroup$
    – mark leeds
    Feb 15 at 13:50
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    $\begingroup$ On Gordon's answer, note the reason why the original Black-Scholes derivation does still yield the correct result is explained under Comments: the bank account's holding gets cancelled when derivative and portfolio dynamics get equated. I'd also draw attention to a somewhat forgotten paper, "A Characterization of Self-Financing Portfolio Strategies" by Bergman (1981) which provides conditions for a replicating portfolio to be self-financing by construction, amounting to the well-known trick of investing the difference between derivative and market hedging instruments in the bank account. $\endgroup$ Feb 15 at 14:14
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    $\begingroup$ On my latter point above, see also this intuitive explanation by the late Mark Joshi: answer to "Dynamic Delta Hedging And a Self Financing Portfolio". $\endgroup$ Feb 15 at 14:17
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    $\begingroup$ My answer to the aforementioned question also goes through this. $\endgroup$ Feb 15 at 14:26
  • $\begingroup$ @DaneelOlivaw Thanks very much for all the additional links to other answers and Bergman's paper as well as summarising the underlying intuition. Greatly appreciate that :) $\endgroup$
    – Kevin
    Feb 15 at 18:43
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Let me begin by noting that about three paragraphs from the end of the Black-Scholes paper is the disclosure that the authors attempted to validate it and it failed to be significantly different from chance and it has never passed a validation test.

His argument is probably correct, but also for a different reason. Ito calculus violates the converse of the Dutch Book Theorem when the assumption that all parameters are known does not hold. As nobody knows the true parameters, the converse of the Dutch Book Theorem would clearly support the idea that some hedges are not hedges and it is possible to assure a sure win over all prices.

It is likely the author has noticed one of the mathematical structures that allow you to arbitrage an Ito calculus user. I actually have five examples that I use for training that I have worked out.

Without reading the paper, my guess is that the author may have found a sixth way to attack Ito based solutions, or it is a variant on the five I have already found. Dubbins and Savage wrote an entire book on it, a book on stochastic inequalities. I think it was in the 70s.

The short answer is that for any game with Frequentist pricing or point estimation, there exist a subset of contracts greater than zero, in some cases 100% of all contracts, where it is possible to construct a pure arbitrage where someone can win in all states of nature. I have proposed an alternate calculus that drops Ito's assumption that parameters are known but I cannot get it published. It complies with both the Dutch Book Theorem and its converse. I also propose a conjecture that a Frequentist calculus exists, but it may not.

I had a measure theorist help me either to prove or disprove the conjecture and we could not. It isn't clear at this time whether a non-Bayesian stochastic calculus that does not assume the parameters are known can exist.

One aspect of Frequentist decisions is that they are mechanistic algorithms. That is the construction by design. A decision rule is intentionally an algorithm. I can see potential attacks on the paper having gone down this road for a decade now, but, having not read it, I don't know. Nonetheless, they may be arguing against axiomatic design and that would be a non-starter.

By axiomatic design, I don't mean the economic assumptions, I mean things like Kolmogorov's axioms of probability or Wald's axiomatic construction of mathematical decisions.

They are correct on the continuous time budget being an intrinsic problem. It is already known in probability theory that a sure-thing arbitrage can be set up if that is your construction methodology.

It is not a problem if the parameters do not have to be estimated but fatal if they do. As I said, Dubbins and Savage wrote a whole book on that and the first chapter is mostly dedicated as to why that cannot be used. At least if my memory holds correctly, that is the part of the book it is in.

Without reading the paper, it should be possible to show definitively as it has been known this is the case since 1955, that the continuous time budget guarantees an arbitrage opportunity at least some percentage of the time.

If I were writing the article and didn't know anything about the Dutch Book argument, then I would proceed thusly.

First, unless actors are careful, there could exist a convex combination of contracts where pure arbitrage would exist if all offered contracts were accepted.

Second, if some actors have the mathematical equivalence of color blindness and so cannot use a method of detecting the presence of arbitrage, then some of those combinations will actually fill.

My third argument would be to find one and publish it. As Frequentist methods are "color blind" to arbitrage, by theorem, they are not that hard to find.

See for example:

de Finetti, Bruno (1937), “Foresight: Its Logical Laws, Its Subjective Sources”, in Henry E. Kyburg and Howard E.K Smokler (eds.), Studies in Subjective Probability, Huntington, NY: Robert E. Kreiger Publishing Co.

––– (1972), Probability, Induction and Statistics, New York: Wiley.

Kemeny, John (1955), “Fair Bets and Inductive Probabilities ”, Journal of Symbolic Logic, 20 (3): 263–273.

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