Editing Black–Scholes model (section)

==Criticism and comments==
Espen Gaarder Haug and [[Nassim Nicholas Taleb]] argue that the Black–Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream [[neoclassical economics|neoclassical economic]] theory.<ref>Espen Gaarder Haug and [[Nassim Nicholas Taleb]] (2011). [https://ssrn.com/abstract=1012075 Option Traders Use (very) Sophisticated Heuristics, Never the Black–Scholes–Merton Formula]. ''Journal of Economic Behavior and Organization'', Vol. 77, No. 2, 2011</ref> They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation.<ref>Boness, A James, 1964, Elements of a theory of stock-option value, Journal of Political Economy, 72, 163–175.</ref> [[Edward O. Thorp|Edward Thorp]] also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors.<ref name="thorpe">[http://edwardothorp.com/sitebuildercontent/sitebuilderfiles/thorpwilmottqfinrev2003.pdf A Perspective on Quantitative Finance: Models for Beating the Market], ''Quantitative Finance Review'', 2003. Also see [https://web.archive.org/web/20110710172106/http://www.edwardothorp.com/sitebuildercontent/sitebuilderfiles/optiontheory.doc Option Theory Part 1] by Edward Thorpe</ref> [[Emanuel Derman]] and Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes.<ref>[[Emanuel Derman]] and [[Nassim Taleb]] (2005). [http://www.ederman.com/new/docs/qf-Illusions-dynamic.pdf The illusions of dynamic replication] {{Webarchive|url=https://web.archive.org/web/20080703175403/http://www.ederman.com/new/docs/qf-Illusions-dynamic.pdf |date=2008-07-03 }}, ''Quantitative Finance'', Vol. 5, No. 4, August 2005, 323–326</ref> In response, [[Paul Wilmott]] has defended the model.<ref name="Wilmott Defence">{{cite web |last=Wilmott |first=Paul |author-link=Paul Wilmott |title=Science in Finance IX: In defence of Black, Scholes and Merton |url=http://www.wilmott.com/blogs/paul/index.cfm/2008/4/29/Science-in-Finance-IX-In-defence-of-Black-Scholes-and-Merton |date=2008-04-29 |archive-url=https://web.archive.org/web/20080724100130/http://www.wilmott.com/blogs/paul/index.cfm/2008/4/29/Science-in-Finance-IX-In-defence-of-Black-Scholes-and-Merton |archive-date=2008-07-24}}; And the subsequent article: <br/>{{cite web |last=Wilmott |first=Paul |author-link=Paul Wilmott |title=Science in Finance X: Dynamic hedging and further defence of Black-Scholes |url=http://www.wilmott.com/blogs/paul/index.cfm/2008/7/23/Science-in-Finance-X-Dynamic-hedging-and-further-defence-of-BlackScholes |date=2008-07-23 |archive-url=https://web.archive.org/web/20081120003133/http://www.wilmott.com/blogs/paul/index.cfm/2008/7/23/Science-in-Finance-X-Dynamic-hedging-and-further-defence-of-BlackScholes |archive-date=2008-11-20}}</ref><ref>See also: Doriana Ruffinno and Jonathan Treussard (2006). [https://web.archive.org/web/*/http://www.bu.edu/econ/workingpapers/papers/RuffinoTreussardDT.pdf  ''Derman and Taleb's The Illusions of Dynamic Replication: A Comment''], WP2006-019, [[Boston University]] - Department of Economics.</ref>

In his 2008 letter to the shareholders of [[Berkshire Hathaway]], [[Warren Buffett]] wrote: "I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the long-term variety are being valued... The Black–Scholes formula has approached the status of holy writ in finance ... If the formula is applied to extended time periods, however, it can produce absurd results. In fairness, Black and Scholes almost certainly understood this point well. But their devoted followers may be ignoring whatever caveats the two men attached when they first unveiled the formula."<ref>{{cite web |publisher=[[Berkshire Hathaway]] |last=Buffett |first=Warren E. |author-link=Warren Buffett |title=2008 Letter to the Shareholders of Berkshire Hathaway Inc. |language=en |date=2009-02-27 |url=https://www.berkshirehathaway.com/letters/2008ltr.pdf |access-date=2024-02-29}}</ref>

British mathematician [[Ian Stewart (mathematician)|Ian Stewart]], author of the 2012 book entitled ''[[In Pursuit of the Unknown: 17 Equations That Changed the World]]'',<ref name="Stewart_17Equations_2012">{{cite book |url=https://books.google.com/books?id=ezzWkITecN8C |title=In Pursuit of the Unknown: 17 Equations That Changed the World |date=13 March 2012 |isbn=978-1-84668-531-6 |publisher=Basic Books |location=New York}}</ref><ref name="Nahin2012">{{cite journal|last1=Nahin|first1=Paul J.|author-link=Paul J. Nahin|series=Review|title=In Pursuit of the Unknown: 17 Equations That Changed the World|journal=Physics Today|volume=65|issue=9|year=2012|pages=52–53|issn=0031-9228|doi=10.1063/PT.3.1720|bibcode=2012PhT....65i..52N}}</ref> said that Black–Scholes had "underpinned massive economic growth" and the "international financial system was trading derivatives valued at one quadrillion dollars per year" by 2007. He said that the Black–Scholes equation was the "mathematical justification for the trading"—and therefore—"one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" that contributed to the [[2008 financial crisis]].<ref name="theguardian_Stewart_2012">{{Cite news| issn = 0029-7712| last = Stewart| first = Ian| title = The mathematical equation that caused the banks to crash| work = The Guardian |series=The Observer| access-date = April 29, 2020| date = February 12, 2012| url = https://www.theguardian.com/science/2012/feb/12/black-scholes-equation-credit-crunch}}</ref> He clarified that "the equation itself wasn't the real problem", but its abuse in the financial industry.<ref name="theguardian_Stewart_2012"/>

The Black–Scholes model assumes positive underlying prices; if the underlying has a [[negative price]], the model does not work directly.<ref>{{cite web |last1=Duncan |first1=Felicity |title=The Great Switch – Negative Prices Are Forcing Traders To Change Their Derivatives Pricing Models |url=https://www.intuition.com/the-great-switch-negative-prices-are-forcing-traders-to-change-their-derivatives-pricing-models/ |website=Intuition |access-date=2 April 2021 |date=22 July 2020}}</ref><ref name="bloomberg Traders Rewriting Risk Models After">{{cite news |title=Traders Rewriting Risk Models After Oil's Plunge Below Zero |url=https://www.bloomberg.com/news/articles/2020-04-21/negative-oil-prices-are-literally-breaking-traders-risk-models |access-date=3 April 2021 |work=Bloomberg.com |date=21 April 2020 |language=en}}</ref> When dealing with options whose underlying can go negative, practitioners may use a different model such as the [[Bachelier model]]<ref name="bloomberg Traders Rewriting Risk Models After"/><ref>{{cite news |title=Switch to Bachelier Options Pricing Model - Effective April 22, 2020 - CME Group |url=https://www.cmegroup.com/notices/clearing/2020/04/Chadv20-171.html |access-date=3 April 2021 |publisher=[[CME Group]] |language=en}}</ref> or simply add a constant offset to the prices.