Talk:Black–Scholes

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[edit] Normal distribution

The derivations refer to N() as the cumulative normal distribution function, but what is the mean and variance of this function? If the mean is zero and the variance unity (one), then the correct term would be "standard cumulative normal distribution" or "cumulative standard normal distribution." I'm not sure of myself here, so I won't change it. 10:08 23 Aug 2005 (UTC)

[edit] Derivation

The derivation of the Black-Scholes PDE is wrong. If P=V-SdP/dS then dP=dV-dS(dP/dS)-Sd(dP/dS)-dSd(dP/dS) NOT dP=dV-S(dP/dS).

Let me stress the point made by the previous poster, who is being far too kind. The derivation is INSANELY WRONG!

It completely miss understands all of the logic of option pricing theory --- specifically the arbitrage argument. See any book --- reall ANY book.

I'd simply erase the derivation, exept that I hope (and trust) that someone will take the time to fix it.

Now, in a moment of contrition... I know that someone took the time to write this and relied on the knowledge that he had. This is an honorable thing.

It's also honorable to take the heat for getting it wrong. Let the author do the deleting.

SHORT SELLING? Nope. Not needed. The BS formula requires that you can SHORT the BOND, but there is NO NEED TO SHORT THE STOCK. This is proved in every place that the BS formula is prove.

DELETED the SHORT SELL STATEMENT.

Reinstated this statement. Without short selling, B-S just gives a one-sided bound for the market price of the option.

[edit] Interpretation of phi(d1) and phi(d2) as probabilities

Added this paragraph: $Φ(d 1)$ and $Φ(d 2)$ are the probabilities of exercise under the equivalent exponential martingale probability measure (numeraire = stock) and the equivalent martingale probability measure (numeraire = risk free asset), respectively. The equivalent martingale probability measure is also called the risk neutral probability measure. Note that both of these are "probabilities" in a measure theoretic sense, and neither of these is the true probability of exercise under the real probability measure.

Probably it needs to be edited to be more understandable for a casual reader? —Preceding unsigned comment added by 76.233.235.38 (talk) 01:28, 30 March 2008 (UTC)

[edit] Dividend Yield

This equation makes no reference to the dividend yield. It incorrectly uses the interest rate to discount the strike price. This is an urgent mistake! 14 Jun 2007 (KA)

KA (FWIW, please sign your comments, makes it easier to know who's talking), the article has sections on handling both continuous and discrete dividends in the Black-Scholes model. However, I would agree that it should be mentioned in the initial derivation that the stock in question is assumed to be dividend free. I'll add that to the page. However, in my experience, anyone who actually needs to worry about dividends in practice wouldn't be using Black-Scholes in the first place. They'd get crushed if they tried to trade from it. Thanks for the input. Ronnotel 15:57, 14 June 2007 (UTC)

[edit] LTCM

Should we really be linking Long Term Capital Management from here? Did they use BS? --Pcb21 15:42 9 Jun 2003 (UTC)

- NO. Scholes was a principal at LTCM, but the plain vanilla BS was way out-of-date by the time that LTCM came into being.

I changed the link to the more generic financial mathematics page which links to LTCM. --Pcb21 11:10 13 Jun 2003 (UTC)

[edit] Test data

I'm writing an implementation of Black-Scholes, and wanted some test data to use. Could this page have an example with real numbers?

- Welcome to a long tradion. Look at the literature before you spend too much time.

[edit] Wording

The new section on Black-Scholes in practice currently states that "Black-Scholes may not model.." ... but the fact that vol surface is not flat is PROOF that the assumptions (perhaps even implicit ones) do not hold in practice. Any suggestions for an improvement --Pcb21 11:10 13 Jun 2003 (UTC)

I made the change. --Pete 15:04 4 Jul 2003 (UTC)

[edit] Context for Black-Scholes

Let me add some remarks on the Black-Scholes theory. It seems to me, that the economists and even the authors of this excellent (LOL) article, are not aware of the complete physics behind it. Which I know only because I originate from the chair of Arnold Sommerfeld in Munic. Sommerfeld, being the first of the three great teachers in physics (the other two being R.P. Feynman and L.D. Landau, sorry I do not accept anybody else in this category) did some fundamental work on this theory in his books on electrodynamics and partial differential equations. Let me shortly summarize it: There are the fundamental symmetries of space and time (mathematically described by the invariance group of classical mechanics and special relativity) and associated with each one parameter subgroup is a conservation law (theory by Emmi Noether). Since conserved quantities cannot be created or destroyed, they can be transported only. To any conservation law there is a partial differential equation of the form of a transport equation, containing a first derivative of time. So it is a transport problem, and not a wave or a potential problem. Note that a first derivative of time is not invariant against time reflections. Therefore transport problems are not invariant in this sense, making them statistical phenomenons. The physics of the space time symmetries is: Charge conservation => Ohm's law, mass conservation => diffusion, time invariance => energy conservation => heat transport, momentum conservation => inner friction, conservation of center of gravity => I don' t know, conservation of angular momentum => phenomenons overlooked in physics. Many times in lectures on theoretical physics it is stated, that these are the (only) conservation laws and transport problems. Wrong, the Black Scholes Fischer theory comes in here. They deal with a partial differential equation of transport type, look at this article, and come up with a valuation theory for options (please note that the trivial valuation theory of options with premium and leverage has its benefits too). So we may reverse the above physical line of arguments, asking: What is the conservation law, leading to this partial differential equation of transport type? Answer is - I think it's clear - the conservation law behind Black Scholes is money! Which means the conservation of money. The limitations of Black Scholes therefore are the limitations of the conservation of money in an economy. But if there is a conservation law, inverting E. Noether, there must be a one parameter invariance group. Which group is it? Since E. Noether deals with transformation groups on space-time manifolds - what is that "space-time" for money. It seems to me that behind economics there is a completely unknown mathematical formulation, like behind physics there is a Newtonian, special and general relativistic formulation. One may ask at which state of physics is economics now. I think there is an economic Galilei, if Black Scholes and the physical conservation laws are united. And there is a economic Tycho Brahe. But no Kepler, no Newton yet. --Hannes Tilgner

- - Noether's Theorem requires a Lagangian. WHAT IS THE LAGANGIAN HERE? This is a time reversed diffusion equation. Tons of symmetries, but not a lot of Noetherian invariants. See what I mean?

[edit] Spelling of 'Fischer'

Anyone know why the article spells 'Fischer' as 'Fisher'? It can't just be a typo, because the wiki link has been explicitly made to point to the right spelling, while still displaying the wrong spelling. I have changed it so they are consistent. --DudeGalea 07:05, 31 July 2005 (UTC)

- Total Typo. This is the worst Wiki article I know.

[edit] Fundamental Theorem of Finance

Hi All, Is the Black-Scholes model also known as the "Fundamental Theorem of Finance"? If yes, tell me and I'll create a redirect + one line in the article. --Tony 18:27, 18 October 2005 (UTC)

- - No. Just use Google.

I've heard it referred to as such, yes. --maru (talk) Contribs 04:09, 2 January 2006 (UTC)

No the "fundamental theorem" is said to be the equivalence of the existence of equivalent martingale measures and no-arbitrage. Pcb21 Pete 11:07, 25 April 2006 (UTC)

[edit] Pervasiveness? Really?

While many of the ideas behind the Black-Scholes model are nearly universally accepted by practioners, I find the statement that The use of the Black-Scholes formula is pervasive in the markets. is somewhat hyperbolic. I have never, in my entire professional career, come across a trader using a simple Black-Scholes model to trade from. Doesn't happen. At least in the U.S. equity derivative markets, the interest rate derivative markets, the credit derivative markets, the FX option markets, etc. Am I missing something? Unless someone provides a counter-example, I'll rephrase.

No, you miss nothing. Still... one does use the BS formula as a transformation to unify the strikes and maturities, via to one numbe "implied volitility"

Perhaps it meant most models are built on it? --maru (talk) contribs 04:30, 29 January 2006 (UTC)

Not really. Who, where? Not at any firm that I know.

Pretty much all working models I'm familiar with employ trees, most commonly binomial trees such as those used in the Cox-Ross-Rubinstein framework. I'm not aware of any serious practioners using 'closed-form' models such as BS for actual trading. I wouldn't really say that any of the tree-type models were built-on, although they may have been motivated by it. Ronnotel 16:03, 30 January 2006 (UTC)

Yes. Trees are the trick. Not BS.

Well, neither trees nor BS.. just use BS to transform prices to the moneyness/duration surface then use your brain to model the surface. That is best of breed as best I can tell.

I can't decide whether to say you are being pedantic or whether the article is actively misleading. Use of Implied vol is absolutely pervasive of course, and that must what was meant by the original phrase. Pcb21 Pete 11:09, 25 April 2006 (UTC)

The article is INSANELY WRONG. I am sure that some one will fix it soon. I just don't have time.

Of course Black-Scholes is used in practice. For vanilla equity options without dividends, what else would you consider?

Are you nuts? Used how? As a transformation, YES. As a pricing model ... LOL, we'd eat their lunch!

"Used in Practice" ... this is a subtle term. Take a firm like ML. There will be people at ML who "use BS" ---- but these will not be the people at the options desks. Nor will any quant groups use BS. It's because you just need about one day to see that BS is too far off the market to use. Now, let me toss you a bone. There will be corp-fin guys and accountants who will "use" BS but they are "using it" to write reports to stock holders and the government. Traders, guys who make real bets, would be out of a job in days if they "used" BS.

Black-Scholes is used by real traders, all the time. With an explicit model of the implied volatility surface. I have edited the article to reflect this market practice. EdwardLockhart 12:43, 24 October 2006 (UTC)

[edit] W_t a geometric Brownian motion?

GBM fails every empirical test you can put to it. Log(S_t) is not symmetric, has long tails, fails dependence test (due to SV) etc. OMG.

The underlying instrument follows a geometric Brownian motion, but is it correct to say that that the Brownian motion W_t, in terms of which it is defined, is geometric?

Well spotted, fixed. Pcb21 Pete 11:05, 25 April 2006 (UTC)

[edit] Criticism

Folks. I am sure that I have violated various Wiki protocols here, but this article is TOTALLY FUBARED. Please get someone who has (1) technical competence (eg has read Karatzas and Shreve) and (2) market competence (eg made markets or consulted for big players for a few years).

This article is simply wrong as written --- at every level I can imagine.

Hi All. I can't say that I followed all the technical arguments above - and I'm not sure who left the above comments - but here's a suggestion. To address these problems (with the article and with the model ☺), please would all those who like to see a re-write, create a bulleted list of criticisms of the model and formula under the two categories above (i.e. (1) technical and (2) market related) which we can then incorporate into the article as a new section...

Fintor 07:09, 6 September 2006 (UTC)

I think you expect too much from this person. All he wants to do is rant and rave about an article whose arguments he does not understand.

The widely used verison of Black Scholes with dividend yield is in Merton (1973), the version given here is different.

[edit] Incorrect result in elementary derivation

The following paragraph is incorrect:

"Elementary derivation Let S0 be the current price of the underlying stock and S the price when the option matures at time T. Then S0 is known, but S is a random variable. Assume that X = ln(S/S0) is a normal random variable with mean μT and variance σ2T. It follows that the mean of S is E(S) = S0e^(rT)"

The correct result is E(S) = S0e^(mu T).

To avoid introducing the concept of risk-neutrality, one may derive the Black-Scholes formula with mu, then use put-call parity to find mu = r.

The formula E(S) = S0e^(rT) is correct by definition of r. That is what it says in plain English: that the formula is true "for some constant r (independent of T), which may be readily identified with the interest rate" (which you did not choose to quote). The formula E(S) = S0e^(mu T) is not correct because r is not equal to mu. mu does not mean what you think it means. Its meaning is clearly explained: "Assume that X = ln(S/S0) is a normal random variable with mean μT and variance σ2T" (which you did quote). To find the correct formula for r follow the instructions: "use the corollary to the lemma to verify the statement above about the mean of S".—Zophar 16:28, 3 November 2006 (UTC)

I have added some more explanation to this section in the hope of making this clearer. EdwardLockhart 07:43, 29 November 2006 (UTC)

[edit] Trivia

Please don't remove the map reference in the Trivia section. True, it has nothing to do with financial calculations, but it is a surprising concidence that I think readers of the article might find amusing. RussNelson 04:02, 14 May 2007 (UTC) (failed to sign at the time of the edit)

This page contains enough questionable and non-encyclopedic material as it is. I fail to see how this qualifies as even remotely worthwhile. See WP:LAME. Ronnotel 02:14, 1 June 2007 (UTC)

Ahhhh, I see. Because other parts of the article are not up to your standards, you believe it correct to edit unrelated parts of the article. Wouldn't your time be better spent improving the questionable and non-encycolpedic material? C'mon, Ronnotel, lighten up. By all Wikipedia standards, this is a well referenced piece of information. I feel that it's interesting and should stay. What do other people think? RussNelson 04:05, 1 June 2007 (UTC)

Since it is not obviously related to the subject of the article, and since somebody objects, let's leave it out. Smallbones 08:34, 1 June 2007 (UTC)

Removing it because somebody objects is not a reason, because I want it in there. We balance each other out. Removing it because it's not obviously related to the subject of the article (and is instead merely the name of the article) is a logical reason. I still believe that it belongs in the article, but now you've outvoted me. If somebody else chimes in saying that they think it's an interesting and amusing coincidence, then we're deadlocked again. RussNelson 16:10, 1 June 2007 (UTC)

[edit] Black-Scholes-Merton

Can we change the name of this article to Black-Scholes-Merton. We could have Black-Scholes redirect to this. The standard introductory text by Shreve (Stochastic Calculus for Finance) now refers to this as Black-Scholes-Merton. Viz 22:01, 6 November 2006 (UTC)

I would vote against. Predominant usage remains Black-Scholes, it should take more than one cite in a text book to change this. Ronnotel 20:29, 24 November 2006 (UTC)

[edit] Efficient Market Hypothesis

I removed the EMH reference. The Black-Scholes model does not require the EMH. It requires (completeness and) absence of arbitrage which is much a weaker condition than efficiency. No-arbitrage requires that is not possible to make a guaranteed profit. Efficiency requires that is not possible to make a profit on average (compared to some risk-adjusted benchmark).

Most obviously, B-S can still be used in cases where there is strong mean-reversion.

[edit] dR vs dPi

Zophar was quite right about this. I have reinstated the original text, with additional explanatory comments. Apologies. EdwardLockhart 07:23, 29 November 2006 (UTC)

[edit] Notability

I'm curious if an article like this counts as notable, and why? Mathiastck 01:17, 9 January 2007 (UTC)

Well, as a starting point, it describes an economics model that was the basis for a Nobel prize. It pretty much boot-strapped the modern derivatives market (no small thing). As ideas go in economics, it's up there with Supply and demand. Can you give more detail about your objection? Do you find it overly technical or mathematical? Ronnotel 02:07, 9 January 2007 (UTC)

[edit] Links

This article is at or near the spam event horizon. Please consider rolling the genuinely good links into the text as cited sources and removing th rest. Guy (Help!) 22:33, 20 January 2007 (UTC)

I have a link to a source, where the Black-Scholes model is explained very well and comprehensive for beginners. Why is it constantly thrown away? Look at [1]

[edit] Derivation and Solution

Sorry, I had to remove Kruglov's article in Derivation and Solution parts. I won't discuss its technical merits, but it simply doesn't sound right in Financial part. Consider the first paragraph in Concept of Arbitrage part: "Concept of arbitrage says that when the future price of investment asset is unknown it is assumed that the price of asset today with its delivery in the future is determined by some other asset whose future price is deterministic." This is an unusual definition. -- Argyn —The preceding unsigned comment was added by 198.204.133.208 (talk) 15:59, 6 February 2007 (UTC).

[edit] Revert March 16th

I reverted a couple of changes. In the introductory paragraph, "The fundamental insight of Black and Scholes is that the option is implicitly priced if the stock is traded" had ben changed to "The fundamental insight of Black and Scholes is that the option is implicitly priced if the stock is traded, and that for the purposes of calculating the price of the option, the return on the stock, which is unknown, can be replaced with the risk-free rate." This seems unnecessary detail for the introductory paragraph. Additionally, if we were to have details about the model here, it is perhaps more surprising (to people unfamiliar with the subject) that we need to know the volatility than that we don't need to know to the drift.

Secondly, this SDE, which seems to have unnecessary prominence, even if it was necessary for it to be included:

$dS_t = r S_t\,dt + \sigma S_t\,dW_t \,$

In order to make sense of this equation, one needs to understand both the PDE approach, and the risk-neutral / equivalent martingale measure approach (the measure under which this SDE is valid). I think it is easier to present the PDE and EMM approaches seperately, in the hope that people who struggle with one may find the other more comprehensible. In view of the EMM background required for this formulation, I'd be surprised if it was commonly used - does the author have cites for it? —The preceding unsigned comment was added by EdwardLockhart (talk • contribs) 05:29, 16 March 2007 (UTC).

[edit] Another formula?

In some textbook (talking about real options), it says

$d_1 = \frac{\ln(S/PV(K))}{\sigma\sqrt{T}} + \frac{\sigma\sqrt{T}}{2}$

rather than

$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$

Is there any difference between them? why? Jackzhp 21:04, 22 April 2007 (UTC)

They are equivalent. In my view, the most illuminating form is:

$d_1 = \frac{\ln(F/K) + (\sigma^2T)/2}{\sigma\sqrt{T}}$

EdwardLockhart 22:24, 22 April 2007 (UTC)

Can please ellaborate how they are equivalent? Jackzhp 13:01, 23 April 2007 (UTC)

Take the rT inside the log to the denominator to get the PV(K).

What is F in the last formula? Some sort of Forward? In any case, it all has to wash back to the first formula, which is by far the most common version. —Preceding unsigned comment added by 64.105.108.168 (talk) 14:29, 8 September 2007 (UTC)

[edit] Small Mistake in transformation?

When going from B-S PDE to Diffusion Equation, I just worked out the change of variable given, on paper, but it is off by a negative sign. I don't see any mistakes in my work, but I can post it here if necessary. Can someone verify? In other words, the diffusion equation I get is: $\frac{\partial u}{\partial \tau} + \frac{\sigma^2}{2} \frac{\partial^2 u}{\partial x^2}=0\,$

-Josh

Are you using time going forward or backward? That will give the wrong sign. —Preceding unsigned comment added by 76.233.235.38 (talk) 19:56, 29 March 2008 (UTC)

[edit] Learn to sign your comments

Everyone, please sign your comments by adding a series of FOUR TILDES to the end of your comments. Mathchem271828 06:54, 27 July 2007 (UTC)

[edit] Variables in the 'model' section

Someone needs to define EACH AND EVERY variable and parameter in that section. Otherwise it send the reader to the literature or textbooks, which defeats the purpose of this online encyclopedia. Mathchem271828 06:54, 27 July 2007 (UTC)

I'd be happy to do this, but I don't see anything missing. Could you be more specific? Thanks. EdwardLockhart 17:02, 29 July 2007 (UTC)

Right off the bat, capital W is undefined. Mathchem271828 05:09, 3 August 2007 (UTC)

The text currently reads:

The price of the underlying instrument S_t follows a geometric Brownian motion with constant drift

μ

and volatility

σ

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \,$

So there is a link to geometric Brownian motion for anyone who finds that unclear. Would you prefer the following?

The price of the underlying instrument S_t follows a geometric Brownian motion with constant drift

μ

and volatility

σ

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \,$ (where

W t

is a Wiener process)

That seems if anything less clear to me, but if people think it would be helpful, let's add it in. Did you spot anything else needing definition? EdwardLockhart 08:48, 4 August 2007 (UTC)

I would vote for defining

W t

. Mathchem271828 13:13, 6 August 2007 (UTC)

Oh, super, seems that somebody answered it before I ask what the Wt is. Let's add the note in the article. ——Nussknacker胡桃夹子 ^.^tell me... 10:28, 29 October 2007 (UTC)

I notice someone asked for C(S,T) to be defined too. Again, I think it would confuse rather than clarify the current text, which reads: "...formula for the price of a European call option...". Does anyone think "...formula for $C (S, T)$ the price of a European call option..." would be an improvement? EdwardLockhart (talk) 10:38, 18 November 2007 (UTC)

No, I agree with your reasoning. Ronnotel (talk) 15:20, 18 November 2007 (UTC)

Disagree. Whenever you write a scientific text every variable symbol needs to be mentioned in the text with an appropriate term. Do not underestimate the importance of this to the reader. in general, symbol usage changes much more often (in the course of time as well as due to varying authors) than people employing them usually think. Moreover, think of the interested, but not familiar-to-the-topic reader trying to understand your formula. Googeling or trying to find hints in wikipedia will fail for W_t or C, but will work when you now the name of the thing you are looking for. Mathchem's criticism is absolutely appropriate with respect to this article. Tomeasy (talk) 01:19, 8 March 2008 (UTC)

I agree with your general sentiment. My point is that the function C(S,t) is (implicitly) defined by the statement that the formula in which it appears is the formula for the value of a call option. EdwardLockhart (talk) 11:40, 12 March 2008 (UTC)

That's right. It is quite obvious from the text that the formula refers to this. The C(S,t) is really not the critical thing here. Nevertheless, I allowed myself to add the C to the text, of course without the list of parameters. Sorry, it's just my stringent academic upbringing, which forces me to. I hope you still like the text. Moreover, I would like to add the Wiener Process W, since more critical appears the missing explanation for W_t to me. And, most critical BTW :-) your user page :-))) Tomeasy (talk) 13:56, 12 March 2008 (UTC)

[edit] Pronunciation

The article should include correct pronounciation (with references) of the name Scholes in Black-Scholes. As far as I found, the correct pronounciation is reading "ch" as "k" as in the word "school", instead of commonly heart "sholes" with sound "sh" as in the word "sheep". 85.59.212.28 (talk) 02:14, 24 January 2008 (UTC)

I have met Dr. Scholes on a number of occasions and never heard him use this pronunciation. Can you source this? Ronnotel (talk) 04:13, 24 January 2008 (UTC)

Which pronunciation did he use then? Tomeasy (talk) 01:22, 8 March 2008 (UTC)

[edit] Elementary derivation

Define

$Z^+(b) = \begin{cases} Z & \mbox{if }Z>b \\ 0 & \mbox{otherwise} \end{cases}.$

If a is a positive real number, then

$\mathbb{E}\left[e^{aZ^+(b)}\right] = e^{a^2/2}\Phi(-b + a)$

where $Φ$ is the standard normal cumulative distribution function.

The above does not seem correct. $\mathbb{E}\left[e^{aZ^+(b)}\right]$ would have the additional term $\,\Phi(b)\,$ if $\,Z^+(b)\,$ is defined to be $0$ whenever $Z \not> b$ .

Or simply consider the case $b=\infty$ . On the one hand $\mathbb{E}\left[e^{aZ^+(\infty)}\right] = \mathbb{E}\left[e^{a\,0}\right] = 1$ , while on the other hand $e^{a^2/2}\Phi(-\infty + a) = 0$ .

The above definition of $\,Z^+(b)\,$ also conflicts with the observation made later on that $\frac{X^+ - uT}{\sigma\sqrt{T}} = Z^+(b)$ for some b. For when $S\not>K$ , then $\,S^+=0\,$ , as defined, and consequently $X^+=\ln(S^+/S_0) = -\infty = Z^+(b)$ .

I suggest simply changing the definition of $\,Z^+(b)\,$ to

$Z^+(b) = \begin{cases} Z & \mbox{if }Z>b \\ -\infty & \mbox{otherwise} \end{cases}$

which would correct all the above.

--Bahman Engheta 75.4.203.219 (talk) 19:49, 8 February 2008 (UTC)

[edit] In the news

http://www.bloggingstocks.com/2008/03/04/another-wall-street-worry-a-potentially-flawed-risk-formula/ http://www.portfolio.com/news-markets/national-news/portfolio/2008/02/19/Black-Scholes-Pricing-Model -- AnonMoos (talk) 22:33, 6 March 2008 (UTC)

There's already a good bit of discussion about the model's drawbacks and the fact that it is not typically used by actual traders. Are you suggesting that they be augmented somehow? Ronnotel (talk) 00:46, 7 March 2008 (UTC)

If journalists are blaming stock-market crashes and the sub-prime situation on Black-Scholes (at least partially), that should probably be noted in the article...AnonMoos (talk) 07:51, 7 March 2008 (UTC)

Well, I agree in principle with the article's thesis. However, I think there's a bit of subtlety that's being glossed over in order to make a readable article. Credit derivative traders do not blindly rely on Black-Scholes for pricing their products but rather much more complicated models that do take into account things such as stochastic volatility. He's right that BS's reliance on constant volatility is a huge failing. Which is why BS was largely dumped as a practical tool in the aftermath of 1987. Blaming sub-prime on BS is a bit far-fetched. I think a more likely criticism (and Nassim's main thesis) is that asset prices moves are a power law process rather than a normal process. I'd probably just add the article as a link rather than make too much of it in the text. Ronnotel (talk) 12:05, 7 March 2008 (UTC)