Talk:Black–Scholes model

Model vs. Equation?
As currently written, the second sentence is "From the model, one can deduce the Black–Scholes formula". Hmmm... "deduce"? Really? That phrasing seems muddled, to put it nicely. Quantitative models often *consist of* formulas but rarely "generate" formulas. Isn't it better to say that the B-S model consists of the B-S equation? Hugantik (talk) 07:30, 9 March 2013 (UTC)


 * Most people would say a mathematical model consists of assumptions and sometimes deductions, some of which are given in equations. According to mathematical model, a mathematical model is a "description of a system using mathematical concepts and language".  In any case, the Black-Scholes equation is not a description of the Black-Scholes "universe" per se, but part of it, and in fact a deduction one makes from more basic parts of the model, which are not in fact originally due to Black and Scholes, but people like Samuelson.  In fact, people do often speak of models leading to formulae.   --C S (talk) 01:33, 13 March 2013 (UTC)

Why no mention of American Style options?
Since the vast majority of options traded today are American-style, why is there no mention of how the BS model needs to be changed to accommodate them? — Preceding unsigned comment added by Hsfrey (talk • contribs) 04:36, 5 May 2012 (UTC)
 * Good point, I added in a small section discussing the difference (mainly partial differential inequality instead of equality). Zfeinst (talk) 15:57, 5 May 2012 (UTC)

Black-Scholes calculators
The section on Black-Scholes calculators written by User:Novolucidus, which refers to his/her webpage, is based on a misconception. S/he seems unaware that the rate put into Matlab's price function is not the same as the one put into his/her function. Matlab's function requires the rate be continuously compounded while his/her does not. In other words, if Matlab's rate is r, and Novolucidus' rate is R, 1+R = e^r. The use of a continuously compounded interest rate is fairly common in this area, and I expect that's why "the majority of such calculators" do not agree with Novolucidus'. --C S (talk) 17:45, 7 August 2012 (UTC)

The material is also a violation of the Wikipedia policy on original research. Since Novolucidus is new to WIkipedia, I doubt s/he is aware of this, which is why I am focusing on showing the flaw in the material. But the policy violation should be pointed out, nonetheless. C S (talk) 17:55, 7 August 2012 (UTC)

Leibniz rule in derivation of Black-Scholes equation
In The Black–Scholes equation, we read


 * The value of these holdings is


 * $$\Pi= -V +\frac{\partial V}{\partial S}S.$$


 * Over the time period $$[t,t+\Delta t]$$, the total profit or loss from changes in the values of the holdings is:


 * $$\Delta \Pi= - \Delta V +\frac{\partial V}{\partial S}\,\Delta S.$$

What happened to the term $$ S\Delta\frac{\partial V}{\partial S}$$? If this can be ignored, we should explain why.

S.racaniere (talk) 14:24, 15 November 2012 (UTC)


 * Why would there be such a term? :) After all, we're holding a certain amount of stock over a discrete time period.


 * But yes, I have an idea of why you are raising this question. You'd like to skip this whole discretization bit and immediately apply Ito's formula (which will of course introduce a "Leibnizian" term).  But that's missing the whole point of the discretization, which was to avoid that.  This is actually something that was missed, or not clearly outlined, in the original Black--Scholes paper.


 * The essential thing is to have a self-financing condition on the portfolio. The usual way of stating this is to assume the portfolio satisfies $$d\Pi = -dV + \frac{\partial V}{\partial S}\ dS. $$  This is the continuous version of what pops up in the discrete case.  --C S (talk) 09:51, 19 January 2013 (UTC)


 * To clarify, it takes a bit of work to show that the self-financing condition I stated right above holds. But it does.  And that's what justifies the derivation in the end. --C S (talk) 08:29, 25 April 2013 (UTC)

Article is getting longer
For example, we now have some mammoth, garbled explanation in the "interpretation" subsection. It just kept getting longer, and now it is a mash of several different expository threads. The problem with this article is that everyone keeps adding their favorite bit, but nobody wants to clean it up and neatly organize the material.

In the hopes that some of these people read the talk page before adding their material, please STOP. At least give an eye toward cleaning up the article before adding what you'd like. --C S (talk) 09:52, 19 January 2013 (UTC)
 * I agree that this article is fairly unwieldy as it is. I think it would probably be best if we came up with an outline to follow in this talk page. And of course the more input on an outline would be best.  Then cleanup can occur with more consensus and hopefully won't result in parts being deleted and added back in shortly after. Zfeinst (talk) 15:18, 19 January 2013 (UTC)


 * Yes, an outline is a great idea! --C S (talk) 20:06, 20 January 2013 (UTC)

As the beginnings of an outline, I suggest:


 * Black--Scholes formula
 * Basic interpretation -- explain the essential features of the formula (asymptotic behavior, significance of the N(d1) term (the delta), and so on)
 * some history


 * most basic Black--Scholes model -- This would get into the basic assumptions of the model. This should help simplify the presentation of the Black-Scholes formula.  There should also be discussion of the no-arbitrage concept, as even those studying Black-Scholes often find surprising the idea that the expected future value plays no role in the pricing.


 * derivation of the formula for a European call
 * dynamic delta-hedging argument
 * solution of the PDE
 * relation of the PDE to the martingale approach (Feynman-Kac)
 * martingale approach
 * option replication via a portfolio of a share digital and regular digital option


 * An overview section, that leads into many more articles, explaining how the Black-Scholes framework has been extended and how pricing works in that general framework
 * Some comments about closed-formula versus numerical solutions to PDEs and Monte Carlo, especially discussing American options and exotics like Asians and barriers


 * Limitations of the Black-Scholes framework -- this might be a good place to discuss misconceptions perpetuated by the popular press (e.g. Black-Scholes was behind the LCTM collapse or the recent financial crisis).

The reason I list the formula first is that i think that's what most layman want to see first. Right now the article starts like a textbook, listing assumptions, then a derivation and so forth. That's not so helpful for most people, and I daresay even those studying the subject for the first time. We should start by showing how the formula makes sense and satisfies consistency checks. I think also emphasizing how the two terms represent a long position in the stock and a short position in the money market is very useful and should help later with the delta-hedging argument. --C S (talk) 09:21, 26 February 2013 (UTC)


 * I agree with CS on this. The article is academic in style and doesn't give a good summary of applications. Statoman71 (talk) 05:34, 27 February 2013 (UTC)


 * Ok, so many moons later, here we are. I've moved a lot of the actual derivations that were in the article to Black-Scholes equation.  I think there should also be a separate article for  non-PDE derivations of the formula, which we can call Black-Scholes formula.  Then this article can be called Black-Scholes model and be more of an overview of Black-Scholes theory, which is probably what the layman wants anyway. Of course, we should show the formula and give some intuition behind the formula and so forth too.  The hedging idea is too nice to leave out entirely, but I suggest just describing it loosely and leaving the actual derivation to the equation article.  Hedging then leads to risk-neutral evaluation, which can also be described intuitively.  --C S (talk) 23:41, 2 October 2013 (UTC)

Here is a proposal for a more concise interpretation section:


 * The Black–Scholes formula can be interpreted fairly handily, with the main subtlety the interpretation of the $$N(d_\pm)$$ (and a fortiori $$d_\pm$$) terms, particularly $$d_+$$ and why there are two different terms.


 * The formula can be interpreted by first decomposing a call option into the difference of two binary options: an asset-or-nothing call minus a cash-or-nothing call (long an asset-or-nothing call, short a cash-or-nothing call). A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms equal the value of the binary call options. These binary options are much less frequently traded than vanilla call options, but are easier to analyze.


 * Thus the formula:
 * $$C = D \left[ N(d_+) F - N(d_-) K \right]$$


 * breaks up as:
 * $$C = D N(d_+) F - D N(d_-) K$$,


 * where $$D N(d_+) F$$ is the present value of an asset-or-nothing call and $$D N(d_-) K$$ is the present value of a cash-or-nothing call. The D factor is for discounting, because the expiration date is in future, and removing it changes present value to future value (value at expiry). Thus $$N(d_+) ~ F$$ is the future value of an asset-or-nothing call and $$N(d_-) ~ K$$ is the future value of a cash-or-nothing call. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure.


 * Adopting the risk-neutral probability measure is equivalent to setting $$\mu = r$$. This results in a log-normal probability distribution for $$S_T/S_t$$, the increase in the stock price, with parameters $$\mu' = \left(r - \frac{1}{2} \sigma^2\right)\tau$$ and $$\sigma' = \sigma\sqrt{\tau}$$, where the term $$\frac{1}{2}\sigma^2$$ in $$\mu'$$ results from Itō's lemma applied to geometric Brownian motion. Under this log-normal distribution, $$N(d_-)$$ is the probability that the option expires in the money, $$P(S_T>K)$$. Similarly, $$N(d_+) ~ F$$ is the partial expectation of the log normal distribution with threshold $$K$$, giving the expected value of the stock for cases in which the option expires in the money.


 * Note that, while the risk-neutral probability measure is a probability in a measure theoretic sense, it does not give the true probability of expiring in-the-money under the real probability measure. To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk.

The version above connects the first term of the BS formula to the partial expectation of the log-normal distribution, and removes discussion of the numeraire. I think that the concept of numeraire is not particularly intuitive to beginners, while the explanation in terms of the partial expectation of the log-normal does explain the difference between $$d_-$$ and $$d_+$$ and allows the reader to easily derive them, provided one accepts that $$\mu = r$$. If we do keep the interpretation in terms of numeraire, we should cite sources that derive it.

Non-smooth solution
Why isn't C(S, t) = max{S − D K, 0} a solution? u(x, τ) = K [exp(max{x + σ2τ/2, 0}) − 1]. Granted, the first derivatives have a jump at S = D K. Do the sources just assume that the solution has continuous first derivatives or do they require it for some reason?—pivovarov (talk) 09:07, 12 April 2013 (UTC)


 * Ito's lemma doesn't apply unless the 'C' has continuous first derivatives. --C S (talk) 08:27, 25 April 2013 (UTC)

Requested move

 * The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section. 

The result of the proposal was moved. --BDD (talk) 18:42, 26 November 2013 (UTC)

Black–Scholes → Black–Scholes model – Per WP:NOUN: "Nouns and noun phrases are normally preferred over titles using other parts of speech" and "Adjective and verb forms (e.g. democratic, integrate) should redirect to articles titled with the corresponding noun". The phrase "Black–Scholes" is an attributive adjective phrase, whereas the article is about the "Black–Scholes model". Since 2009, "" has been a redirect to "". (Note that there is a separate article about the Black–Scholes equation.) BarrelProof (talk) 20:20, 17 November 2013 (UTC)


 * No objection. It might be a bit weird to do so at this stage since there's so much more in here and Black-Scholes formula doesn't have its own article yet.  --C S (talk) 06:59, 19 November 2013 (UTC)
 * When referring to the "Black-Scholes formula", are you talking about something different from the Black–Scholes equation (which has an article)? —BarrelProof (talk) 14:59, 19 November 2013 (UTC)


 * Strong support of some move. This one seems fine. Red Slash 00:33, 20 November 2013 (UTC)
 * The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Instruments paying continuous yield dividends
I think the presented price formulas are not correct, the correct ones are available here: http://www.macroption.com/black-scholes-formula/. — Preceding unsigned comment added by 217.173.8.141 (talk) 22:47, 15 May 2015 (UTC)

Shuffled terms : price, value, payoff, premium
95.132.143.157 (talk) 19:13, 22 November 2015 (UTC)


 * The exercise price is the strike price is generally written as $$K$$ but only equals $$S_0$$ if it is at the money
 * The payoff or payout can be $$S_T - K$$, though that is really a futures contract. If it is a European call option then the payoff is $$(S_T - K)^+$$ or a European put option is $$(S_T - K)^-$$.  If we truly want general then the payoff is $$V(S_T,T)$$.
 * $$V(S,T)$$ is known because for any actualized price value $$S \in \mathbb{R}$$ at the terminal time $$T$$ the value of the option is equal to its payoff which is known (otherwise the two parties who are engaged in the transaction will disagree on the deal and it is then a matter for the lawyers).
 * Does this help? Zfeinst (talk) 23:18, 22 November 2015 (UTC)

92.113.144.245 (talk) 04:05, 24 November 2015 (UTC)

Can anybody explain me how is next formule obtained $$C(S, t) = N(d_1)S - N(d_2) Ke^{-r(T - t)}$$ ? First autors use Ito's Lemma and get differential of function which is absolutely unrelated to proving. So differential of any function of 2 variables can be represented as $$ df = \left(\frac{\partial f}{\partial t} + \frac{1}{2}\frac{\partial^2 f}{\partial B_t^2}\right)dt + \frac{\partial f}{\partial B_t}\,dB_t $$ (http://www.math.tamu.edu/~stecher/425/Sp12/brownianMotion.pdf) . By substituting $$B_t$$ to $$S$$ and $$dS$$ to $$dS = \mu S dt + \sigma S dW$$ authors get $$dV = \left(\mu S \frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S \frac{\partial V}{\partial S}\,dW$$ (https://en.wikipedia.org/wiki/Black–Scholes_equation)

But what is next? How do they obtain Black–Scholes equation and $$C(S, t) = N(d_1)S - N(d_2) Ke^{-r(T - t)}$$ ??? I've heard they use Feynman-Kac formula. But that formula gives expectation value of function, not ready formula. 0.0.0.0 (talk) 05:49, 26 November 2015 (UTC)

Dr. Wu's comment on this article
Dr. Wu has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock".[5] Their dynamic hedging strategy led to a partial differential equation which governed the price of the option. Its solution is given by the Black–Scholes formula.

Add the following:

The key insight of the model is that one can completely eliminate the risk of the option by dynamically and continuously trading the underlying stock. In practice, one can only trade discretely due to transaction cost. Even so, the risk of an option position can be drastically reduced through hedging with the underlying stock. This insight contributed in a large part to the booming of the derivatives industry."

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Wu has published scholarly research which seems to be relevant to this Wikipedia article:


 * Reference : Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, EconWPA.

ExpertIdeasBot (talk) 09:12, 16 June 2016 (UTC)


 * Indeed just noticed the article fails to stress that Black-Scholes is the dynamically obtained risk-neutral argument. The article promotes the misconception that it was a mere pricing formula. Will incorporate such a central idea. Limit-theorem (talk) 12:18, 5 November 2017 (UTC)


 * I think I fixed the article and addressed Dr. Wu's very, very valid concerns. In the past I have given it to students not noticing the article had such inaccuracies and incomplete description. Limit-theorem (talk) 01:07, 15 November 2017 (UTC)

So everything is for the best in the best of all possible worlds? So folk don't lose their homes, their jobs, their security, and their hope because of abuse of this formula by the banks? So the Nobel Prize was justly awarded? Very reassuringDelahays (talk) 07:56, 5 November 2017 (UTC)

Adding one's software
User: LAA78 SPA is trying to put/market his own software on the Black Scholes Page. He emailed me after revert to say so. An encyclopedia is not a bulletin board. Limit-theorem (talk) 23:51, 8 January 2018 (UTC)


 * User: Limit-theorem An encyclopedia is a place for new ideas and contributions. I do not see any Black-Scholes calculators available that use numerical methods.  All the calculators available that I have seen use the close-form solution.  How can I "put/market" my own software when I am making it available for free and placing it in the public domain?  LAA78  —Preceding undated comment added 00:10, 9 January 2018 (UTC)


 * @LAA78: If you're drawing traffic to your own website, that's promotion. If you're trying to get your name out or increase awareness of your product, that's promotion. There does not have to be a commercial motive for the spam policies to apply. —C.Fred (talk) 00:24, 9 January 2018 (UTC)

Good point, C.Fred! Then why do you have a "Computer Implementations" section? Who posted the different links? There are multiple duplicates of the same type of software, which isn't very productive. Lol. Sad. This place is sad. No more Wikipedia donations for me. Can I get my money back? LAA78 —Preceding undated comment added 00:32, 9 January 2018 (UTC)
 * I cannot revert today but, unless someone does it, post on ANI. Note that SPAs are systematically banned.Limit-theorem (talk) 01:12, 9 January 2018 (UTC)

Please speak English, Limit-theorem. I'm not a millennial. And please address my contention on the implementation issue regarding numerical methods. Do you know anything about numerical methods? LAA78 — Preceding unsigned comment added by 108.48.189.58 (talk) 02:16, 9 January 2018 (UTC)

Needless specification of units of years on rates and times.
I would just like to point out that the model works correctly for time intervals expressed in any unit u as long as interest rates are expressed in 1/u. One needs simply to convert whenever appropriate, just as in any dimensional problem. Specifying that these must be in years is incorrect and confusing. There must be some source which can be cited for this. 50.121.51.32 (talk) 14:13, 27 February 2019 (UTC)


 * That is the norm is practically all papers, including the original one. Limit-theorem (talk) 05:50, 28 February 2019 (UTC)

riskless profit
When the article says (under "Assumptions on the market") "there is no way to make a riskless profit", shouldn't that be "riskless profit greater than the risk-free interest rate"? Jmichael ll (talk) 16:33, 10 June 2019 (UTC)

Tone template addded
This article has the tone of a math textbook, with instructional language and extensive use of "we". Wikipedia is an encyclopedia, not a textbook. Please see What_Wikipedia_is_not.

Ira Ira Leviton (talk) 14:42, 25 July 2020 (UTC)

d_1 and d_2 vs d_+ and d_-
The bit with the Black/Scholes formula uses both $$d_1$$ and $$d_2$$ as well as $$d_{+} $$ and $$d_{-} $$ notation interchangeably. Both are possible, but it should be consistent within the article. Smeyen (talk) 16:59, 31 March 2023 (UTC)