# Talk:Black–Scholes model

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## 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 (talkcontribs) 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
${\displaystyle \Pi =-V+{\frac {\partial V}{\partial S}}S.}$
Over the time period ${\displaystyle [t,t+\Delta t]}$, the total profit or loss from changes in the values of the holdings is:
${\displaystyle \Delta \Pi =-\Delta V+{\frac {\partial V}{\partial S}}\,\Delta S.}$

What happened to the term ${\displaystyle S\Delta {\frac {\partial V}{\partial S}}}$? 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 ${\displaystyle d\Pi =-dV+{\frac {\partial V}{\partial S}}\ dS.}$ 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)

## Non-smooth solution

Why isn't C(S, t) = max{SD 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–ScholesBlack–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, "Black–Scholes model" has been a redirect to "Black–Scholes". (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

Question Note
There is muddle with terms. https://en.wikipedia.org/wiki/Black–Scholes_equation (here we can find C as price and as value)

https://en.wikipedia.org/wiki/Black–Scholes_model

https://en.wikipedia.org/wiki/Option_time_value

Why is ${\displaystyle V(S,T)}$ known? We don't know the S_T which is geometric Brownian motion, so we cannot calculate (S_T - K).

As I can see, there are 2 options in portfolio: 1st - the short call (sold, written), 2nd -- the long call (bought). So by V(S,T) author means the value of 1st short call. Right?

https://en.wikipedia.org/wiki/Black–Scholes_equation

quote: The payoff of an option ${\displaystyle V(S,T)}$ at maturity is known.

95.132.143.157 (talk) 19:13, 22 November 2015 (UTC)

The exercise price is the strike price is generally written as ${\displaystyle K}$ but only equals ${\displaystyle S_{0}}$ if it is at the money
The payoff or payout can be ${\displaystyle S_{T}-K}$, though that is really a futures contract. If it is a European call option then the payoff is ${\displaystyle (S_{T}-K)^{+}}$ or a European put option is ${\displaystyle (S_{T}-K)^{-}}$. If we truly want general then the payoff is ${\displaystyle V(S_{T},T)}$.
${\displaystyle V(S,T)}$ is known because for any actualized price value ${\displaystyle S\in \mathbb {R} }$ at the terminal time ${\displaystyle T}$ 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)
Quote
Does this help? Yes.
The exercise price is the strike price is generally written as ${\displaystyle K}$ but only equals ${\displaystyle S_{0}}$ if it is at the money Suppose someone bought out-of-the-money call option. Price of underlying asset goes up https://img-fotki.yandex.ru/get/3710/240791000.0/0_1a663a_c1a8cc94_orig.gif , ${\displaystyle S_{T}>S_{0}}$ but ${\displaystyle (S_{T}-K)<0}$ . What should option holder do? He have already paid intrinsic value as part of premium. So he can buy underlying asset at the price less than K. Is it correct?
the value of the option is equal to its payoff which is known Option value is the synonym of payoff. This statement can not be the proof. Ok, then why payoff is known? At moment t=0 of option buying only S_0 is known . Or we are reviewing the moment t=T ?
Let first examine european call option: https://img-fotki.yandex.ru/get/3102/240791000.0/0_1a6662_28c5c637_orig.gif . On this image what is option value, what is payoff, and what is option price? And what actually authors are trying to find by this formula: ${\displaystyle C(S,t)=N(d_{1})S-N(d_{2})Ke^{-r(T-t)}}$ (https://en.wikipedia.org/wiki/Black–Scholes_model) ? And why does not this solution match with solution ${\displaystyle u(x,\tau )=Ke^{x+{\frac {1}{2}}\sigma ^{2}\tau }N(d_{1})-KN(d_{2})}$ on this page https://en.wikipedia.org/wiki/Black–Scholes_equation ?

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

Can anybody explain me how is next formule obtained ${\displaystyle C(S,t)=N(d_{1})S-N(d_{2})Ke^{-r(T-t)}}$ ? 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 ${\displaystyle df=\left({\frac {\partial f}{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial B_{t}^{2}}}\right)dt+{\frac {\partial f}{\partial B_{t}}}\,dB_{t}}$ (http://www.math.tamu.edu/~stecher/425/Sp12/brownianMotion.pdf) . By substituting ${\displaystyle B_{t}}$ to ${\displaystyle S}$ and ${\displaystyle dS}$ to ${\displaystyle dS=\mu Sdt+\sigma SdW}$ authors get ${\displaystyle dV=\left(\mu S{\frac {\partial V}{\partial S}}+{\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}\right)dt+\sigma S{\frac {\partial V}{\partial S}}\,dW}$ (https://en.wikipedia.org/wiki/Black–Scholes_equation)

But what is next? How do they obtain Black–Scholes equation and ${\displaystyle C(S,t)=N(d_{1})S-N(d_{2})Ke^{-r(T-t)}}$ ??? 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 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.