Talk:Black–Scholes model

Fundamental insight

From article: "The fundamental insight of Black-Scholes is that the option is implicitly priced if the stock is traded."

I think the main insight is that an option can be perfectly replicated by (continuous) trading in the underlying and the bond. 77.58.193.99 (talk) 19:52, 4 March 2009 (UTC)

I've removed that quoted statement mainly because it was out of place. But also, in retrospect, 77.58.xxx.xx's comment highlights a certain difficulty. I think what the writer of the quote above was trying to say is that it is not at all obvious that the option price is determined by the traded stock and riskfree bond. In fact, it's not obvious a priori there should be one mathematically correct price.

The problem with Black & Scholes' hedging derivation is that they assume from the start that the option price is a well-defined function of the stock price (and time). Then they end up concluding that there is only one option price! This circularity makes the conclusion not quite as amazing as it initially seems. Of course, there are other derivations now that avoid this issue, such as the one pointed out by anonymous above: replicating the option directly and showing the replicating portfolio's price satisfies the PDE. However, this shows we have to be careful in how we present this "insight" in this article. --DudeOnTheStreet (talk) 21:48, 17 April 2011 (UTC)

Substantial re-write & reorganization

I'd like to wikify this article. I'd like to present, first, the model, the PDE, and the call price without a lot of digression. I'd like to present it pretty much - except for the notation - as the authors laid it out it 1973. Subsequent sections can outline derivation, credit related work by Bachelier, for instance, and make the reader aware that people in industry still debate the relative merits of Black-Scholes from various perspectives. Some piece of this article is highly relevant to economics, but that's not easy to see the way it's now written. Any objection? Ernie shoemaker (talk) 06:35, 20 January 2009 (UTC)

I don't think that presenting it's like in original article is a good idea. The way BS is used is very different from what the authors thought while writing it. 77.58.193.99 (talk) 19:52, 4 March 2009 (UTC)

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)

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.

I would like to understand why this derivation is wrong, let alone "insanely wrong". I first will give an illustration why I feel the above comment seems to misunderstand basic concepts, before starting to argue about the more complicated parts of stochastic calculus: Say you have a share, with price S(t) at time t. Say you wish to follow the strategy: you wish to hold S(t) apples at time t. You will rebalance at t+dt. How many apples will you be holding before rebalancing at t+dt? Remarkably, you will still be holding S(t) apples! Why? Since S(t) is a constant in the strategy. It's not seen to be a constant - it IS a constant. The problem arises when you wish to rebalance the portfolio. You now wish to hold S(t+dt) apples. This will probably require the buying or selling of some of the apples. That is where the concept of self-financing comes in. If the change in the portfolio mathematically [ie: using the fact that your 'constant' S(t) has now changed to S(t+dt)], equals the ACTUAL change that occurs in the portfolio (where you are still only holding S(t)), the portfolio is self-financing and no extra financing is required to rebalance the portfolio. The important idea here is that you wish you have a self-financing portfolio.

This concept is extremely important if you are trying to replicate a derivative, as you wish to know that a certain amount of cash today, with no extra adding-in or removing of cash later, will replicate the derivative. This is where the stochastic calculus would come into the equation.

Going to the derivation given on the page for the PDE: As with the apples, the 'delta(t)' held of the share is a constant over (t,t+dt). This is not an approximation. The strategy states this: you hold 'delta(t)' over (t,t+dt). At (t+dt) you will still only have 'delta(t)' shares. The only confusion is about the self-financing nature of this portfolio. However... The proof does not claim anywhere that the portfolio must be self-financing. Moreover, the proof does NOT need the portfolio to be self-financing. The reason is that the goal is NOT about creating a portfolio that replicates the option payout. The goal is SOLELY about creating a risk-free portfolio over (t,t+dt). That's it! In-fact, if you were to try rebalance this portfolio at time (t+dt), it is not self-financing and would require the insertion of more money or the extraction of money. The point, however, is that the portfolio is risk-less over (t,t+dt), and hence over that tiny increment it accumulates at the risk-free rate. Moreover, since this risk-less portfolio can be created at ANY time t, the PDE holds for all t - this is identical to the derivations of PDE in Applied Mathematics - you find a PDE that explains some system, which is valid at any point in time. Hence, you can then use this PDE to solve for the original function.

I am very much interested in hearing over people's opinions on this. I would say that the proof is completely correct; maybe it just needs a little more explanation. AnExplanation (talk) 20:02, 22 September 2008 (UTC)

Ok, i'll give you an opinion - it is wrong. The previous writers attempt at a justification is nonsense. Basically, all stochastically integrable processes are the limit of piecewise constant functions, and taking that limit you wind up with the rules of Ito calculus. You can't make up the rules to suit yourself. —Preceding unsigned comment added by 81.154.11.118 (talk) 00:04, 26 February 2009 (UTC)

I know if sounds like a correct derivation, and it actually gets to the right answer (and I think Hull's book actually presents this argument) but the underlying math is actually wrong. (This is an instance where 2 wrongs make a right) The 'wrongness' comes from not using the rules of stochastic calculus. You actually need to apply the Ito Product Rule which adds a few extra terms (that actually cancel to 0 in the correct derivation). See Shreve's book Stochastic Calculus for Finance II for very detailed use of stochastic calculus in the derivation. --159.53.110.142 (talk) 21:47, 3 March 2010 (UTC)

Interpretation of phi(d1) and phi(d2) as probabilities

Added this paragraph: $\Phi (d_{1})$ and $\Phi (d_{2})$ 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 editted to be more understandable for a casual reader? —Preceding unsigned comment added by 76.233.235.38 (talk) 01:28, 30 March 2008 (UTC)

Editing for the casual reader - Does the casual reader know about "Probability Measure", "Martingale", "Exponential Martingale", "Numeraire" and "Equivalent Probability Measure"? And if the reader is familiar with these terms, the first sentence is as clear as it could be! —Preceding unsigned comment added by 76.230.233.146 (talk) 02:19, 15 September 2008 (UTC)

'the probabilities of exercise' is too vague. Why more than one probability? How does one interpret the d1 & d2 probabilities (or, how do you distinguish one from the other)? —Preceding unsigned comment added by 98.238.25.195 (talk) 03:11, 27 November 2009 (UTC)

Exercise refers to exercising the option. The probability of exercise depends on the numeraire. Probability is dependent on the underlying measure. 206.230.48.50 (talk) 23:17, 23 October 2010 (UTC)

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)

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)

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.

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)

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?

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.

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)

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 editted the article to reflect this market practice. EdwardLockhart 12:43, 24 October 2006 (UTC)

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)

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.

Perhaps the criticisms of this paper should be represented. Martin (talk) 04:05, 28 March 2010 (UTC)

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)

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)

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)

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.

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)

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)

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).

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:

$dS_{t}=rS_{t}\,dt+\sigma S_{t}\,dW_{t}\,$

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).

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)

Here, here! I agree. And this form gives more information. drusus null 04:54, 2 December 2010 (UTC) —Preceding unsigned comment added by Drusus 0 (talk • contribs)

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

$d_{1}={\frac {\ln(F/K)+(\sigma ^{2}T)/2}{\sigma {\sqrt {T}}}}$

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)

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)

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)

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 $\mu$ and volatility $\sigma$:

$dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}\,$

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 $\mu$ and volatility $\sigma$:

$dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}\,$ (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)

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)

"Sholes." 67.173.10.34 (talk) 02:35, 21 April 2010 (UTC)Larry Siegel (not previously part of this discussion)

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 $\Phi$ 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)

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)

"used in practice"

This has come up again with a "citation needed" tag. Certainly a citation could be given. But the section on practice is very explicit that this formula gives a first approximation of the price. Also the Greeks are a major part of the model's use, as is implied volatility (which the section explicitly states). Is there a practitioner out there who does not use the Greeks or implied volatility?? It would be news to me if any sane practitioner (excluding pure retail speculators) didn't use these. Please let up on theoretical theological purity and look at the real world in this section. Smallbones (talk) 23:44, 3 July 2008 (UTC)

It's not clear what sort of citation would be acceptable. Trading operations do not generally publish their operating procedures. EdwardLockhart (talk) 09:58, 4 July 2008 (UTC)

The section contains a sentence "Black–Scholes pricing is widely used in practice", Paul Wilmott's article is given as a source. I guess Wilmott only says that pure B-S can be in several situations better that more sophisticated models and that many traders use models derived from B-S. I am affraid citation needed tag would be more appropriate. JanSuchy (talk) 20:21, 19 November 2008 (UTC)

Article a little dry?

While I am sure that the mathematical explanations on this page are useful for some, for a general encyclopaedia - which this is - I think the article needs to explain in words what the formula does, why it is important, what its limitations are, and how it is used in financial markets. I got more answers to these questions by reading this talk page than I did reading the main article, which doesn't directly address any of them. Unless anyone has a maths degree, they won't bother reading the rest and it won't explain anything to them. So definitely by all means keep all the detail, but I would be grateful if someone who knows about it could write some prose. Thanks. Art Markham (talk) 01:03, 26 January 2009 (UTC)

I second that! We really need a better introduction and something like a summary or anything at all that explains what exactly this is to non-experts. 148.142.66.110 (talk) 13:58, 22 June 2010 (UTC)

Links

I don't understand why the website pricing-option.com is always deleted on this page. This website is about option pricing models and the user can test the Black & Scholes models.

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]

I agree with the comment about the number of links above, and will try to move as many links as possible to the footnotes (or delete unnecessary links). Ulner (talk) 20:56, 10 April 2009 (UTC)

Needs a criticism section

There have been severe criticisms of the model and its application at this point (May, 2009), inasmuch as markets have been not quite log-normal in their behavior, and risk seems to have been misjudged to a significant extent. I don't see a criticism section, though these are widespread in Wikipedia, and the section titled "Black–Scholes in practice" doesn't begin to express widespread negative opinions about the results of that practice. Has one been removed? 69.228.171.40 (talk) 23:39, 20 May 2009 (UTC)

No mention of Dividend Yield?

In the Black-Scholes formula section there is absolutely no mention of the dividend yield. It just uses the spot price seeming to assume that δ = 0. It should either say explicitly that it assumes no dividends or it should use Se^(-δ(T-t)) (for cont.) or S - PV(Div) (for discrete) as the coefficient to N(d1). In my actuarial study manual it just uses F^p(S) to stand as the pre-paid forward price of the asset so as to make the formula work for both continuous and discrete dividends. We should considering making that change to the article as well.--Jersey Devil (talk) 20:07, 19 September 2009 (UTC)

Wait I see now that in the Black-Scholes model section is says explicitly that "The stock does not pay a dividend (see below for extensions to handle dividend payments)". I was presented in my book with the General Black-Scholes formula including pre-paid forward price of the stock to include the dividend yield. I had never heard that the Black-Scholes model itself assumed no dividends.--Jersey Devil (talk) 20:21, 19 September 2009 (UTC)

First articulation of the formula?

The intro is literally true, but misleading:

Fischer Black and Myron Scholes first articulated the Black–Scholes formula in their 1973 paper, "The Pricing of Options and Corporate Liabilities."

To most readers, this would imply that the formula first appeared in TPoOaCL; in fact, it had been published many times in the 60s. This article should really mention that B&S have their name on this formula not because they produced it but because of their robust derivation. See Haug's Models on Models for a good history of the many authors who published the exact "BS" formula (going back to '64).

--Wragge (talk) 16:26, 22 November 2009 (UTC)

Is it possible to read Haug's Models on Models online - or to find online some list of articles where the formula is published before 1973? Regards Ulner (talk) 21:47, 22 November 2009 (UTC)

Wragge, are you referring to the section "The Partly Ignored and Forgotten History" from Haug's Models on Models, page 34--44? Ulner (talk) 21:08, 19 January 2010 (UTC)

Introduction

Propose adding a more intuitive, non-mathematical explanation to the introduction, as below. Fintor (talk) 09:27, 19 January 2010 (UTC)

The term Black–Scholes refers to three closely related concepts:

• The Black–Scholes model describes how the price of a share changes with time: it is a mathematical model of the market for an equity, in which the equity's price is a stochastic process.
• The Black–Scholes PDE describes how the price of an option on the share changes as a function of time and the share price: it is a partial differential equation which (in the model) must be satisfied by the price of a derivative on the equity.
• The Black–Scholes formula returns the fair price of the option on the share: it is the result obtained by solving the Black–Scholes PDE for a European call option.

External links to computer implementations

Someone seems to add this link http://www.pidolphin.com/optionsEUStockCalculator.jsp to several wiki pages repetedly, after people (including me) delete the link. One reason against inclusion is that no details is given about how the implementation has been done. Ulner (talk) 21:21, 22 July 2010 (UTC)

May i ask the reason why you blocking web sites (www.pidolphin.com) which provides free advanced financial calculator in many asset classes ? There are many external links in many wiki pages which is not blocked or verified. PiDolhin is purely an educational web site based on financial engineering text books. It is unfair to be blocked. I am sure many financial engineering students will use www.pidoplhin.com when they study financial (derivatives) theorems. regards —Preceding unsigned comment added by 80.254.146.36 (talk) 12:57, 26 August 2010 (UTC)

Criticism section

I think a criticism section is interesting for many readers, and for this reason I put the critism section back. Ulner (talk) 19:58, 11 October 2010 (UTC)

I invite all other contributors to give comments whether you think the current criticism section is good. Is the Taleb-Haug arguments relevant etc? Ulner (talk) 20:01, 11 October 2010 (UTC)

Agreed that a section addressing criticism is important, but the criticism must be specific to Black-Scholes, or it doesn't belong here. Statements like "All models assume a normal distribution, therefore all models are wrong" are not a criticism of Black-Scholes, they are a criticism of quantitative finance. —Preceding unsigned comment added by Betterfinance (talk • contribs) 11:49, 12 October 2010 (UTC)

Black-Scholes is the most prominent model, and for this reason I think it is good to keep a Criticism section both in this article and in quantitiative finance. If you criticize all models assuming normal dist, you also criticize B-S (as well as lots of other models). Ulner (talk) 23:54, 23 October 2010 (UTC)

New formulation

I am happy to see that Drusus have important ideas for how to improve this article, but I think we should discuss this a little here before continuing this revision.

I reverted to version from 16:51 by 2 November 2010 139.149.1.231. Drusus introduced non-standard notation e.g. d and d' etc, as well as made the article more general treating for example put and call options at the same time. This generality makes the article harder to read in my opinion.

The article can certainly be improved, but we should use standard terminology (d1 and d2).

I encourage Drusus to give some background about his changes here so we can discuss the best possible formulations of the article. I also encourage Drusus to write small edit comments which will make it easier to understand the reasons for his edits. Ulner (talk) 16:31, 6 November 2010 (UTC)

I think it is okay to use 'ln' because this expression has a unique meaning, where on the other hand 'log' can be interpreted both as the natural logarithm as well as 10-logarithm. Ulner (talk) 16:33, 6 November 2010 (UTC)

First: Notation Section: In the Notation section Drusus introduces the letters O and D for underlying asset and for the derivative (treating call and put at the same time). This is less clear because it is unnecessarily abstract. Furthermore, the notation section is much harder to read and more confusing with all special cases discussed inside paranthesis. The different indentation also makes this section less clear than the earlier example.

Second: Concerning the "Mathemtical model section" I think the flow in the earlier formulation is better. "Some minor calculations" is unnecessary. The section (1) and (2) makes things harder to read.

Third: It is hard to carefully study all other changes because the reason behind the changes have not been explained in edit comments (which would be good).

It would be nice if Drusus (and others) are interested to discuss how to improve the article here and different possible formulations. Best regards Ulner (talk) 21:31, 18 November 2010 (UTC)

I reverted to the old version. It is easier to understand and uses less advanced notation. As written earlier by anonymous IP "rv changes by Drusus0 which introduced several errors and use non-standard notation". I don't think Drusus has introduced any errors, but I think the notation he uses is non-standard and makes things harder to understand. Anyhow, I am very interested in discussing the suggestions by Drusus part by part; it would be very interesting to know the reasons behind his edits (no reasons has been given in his edit comments). Ulner (talk) 20:52, 19 November 2010 (UTC)

Hi (Ulner?). I had made changes, because the ontology was not clear/transparent, and the math was a bit(!) messy. Judge for yourselves by viewing the notation and mathematical model sections in my edit [[2]], and compare with what they are now; which is more logically presented. For example, compare my intro to the mathematical model section: I actually state what it is we are setting out to do.

The thing to be determined is the price of the derivative, ${\mathcal {D}}$, viz. how $V$ evolves over time. To do this, we rely on the assumptions on the underlying asset, as well as market equilibrium.

In the current version, a fresh learner would be lost as to what is going on:

Per the model assumptions above, we assume that the price of the underlying asset (typically a stock) follows a geometric Brownian motion. That is,

$dS=\mu S\,dt+\sigma S\,dW\,$

where $W$ is Brownian—the $dW$ term here stands in for any and all sources of uncertainty in the price history of the stock.

The payoff of an option $V(S,T)$ at maturity is known. To find its value at an earlier time we need to know how $V$ evolves as a function of $S$ and $t$. By Itō's lemma for two variables we have

Hmmm. You see that the present version says (after minor distracting calculations which themselves could be placed elsewhere instead of coming in front of the main point) "to determine V", without it even being said that our goal is to determine V. Compare with my version. I think Grice would be more pleased with mine.

Anyway, I should give my apologies for something: I had only recently become engaged in Wiki by 2010, so I was not aware that we had to explain our edits: I see the value of this now.

But continuing on from the above. The driving force in my edits was to organise the information in a way that someone who is pretty unfamiliar with the whole ordeal of market-pricing, could see the data laid out thoroughly (hence my reason for being very explicit with the ontology), arranged logically (hence my reason for indenting things the way I had) and communicated clearly (e.g. highlighting the real essence of how Black-Scholes works, and subduing as minor calculations those parts which really are not). I am coming at this from the view point of a critical thinker and not as a financial student: when I sought to understand Black-Scholes, it took me some time to wade through the clutter and uncover for myself the objects at play, what it was exactly that we ultimately wanted to calculate, and what the crucial assumptions were (prevention of delta-hedging, from which everything follows almost trivially). After going through that effort, I though it would be a good idea to better communicate the explanation in this way (and hopefully satisfy Grice's Maxims!).

I have come to realise, however, that my changes might not be accepted by a financial audience; and the way it is now must be good for many people (though I still hold that my version is more logically presented/better communicated — if I didn't, I wouldn't have made those changes in November 2010). Hence a good compromise might be that I would like to see an alternative article for those who are like-minded, whilst the present article remains as the main Black-Scholes page.

— Sincerely, drusus null 13:57, 31 January 2011 (UTC)

Have a second page covering the same topic is generally discouraged per WP's policy prohibiting content forks and it likely be nominated for deletion. Every page on WP is meant to be written for a general audience. Ronnotel (talk) 14:07, 9 December 2010 (UTC)

Make this into a FEATURED article?

I think we should try to improve this article to make it a featured article; this is the single most important concept in financial mathematics and of great general interest. Ulner (talk) 22:54, 7 November 2010 (UTC)

First suggestion: Try to move external links into footnotes if possible. Ulner (talk) 22:58, 7 November 2010 (UTC)

It would take a lot of work to turn it into a Featured Article. Not that I'm against work, but the kind of work that would mainly be required is not the kind most people are able and willing to contribute....namely, there needs to be more history, overview of current practices (well-sourced with reliable sources that would take a great effort to collect). There also needs to be more work on synthesizing the content in a good expository style. Right now the article is just a hodge-podge of stuff drive-by authors have slapped onto the page.

Having seen some Featured Articles get demoted and observed the FA process (from afar), it seems to be quite an intensive process; without someone to spend a lot of time (basically someone that doesn't have to work a 9-5 job, it seems) shepherding the article through, the article would not make it far through the process. --DudeOnTheStreet (talk) 20:19, 4 May 2011 (UTC)

I think you are right, but we should try to improve that article a little bit anyhow if possible! :) What is the biggest problem with the current article? Perhaps the proof is strange in a Wikipedia article? Ulner (talk) 20:51, 5 May 2011 (UTC)

I think proofs can appear in mathematical articles, even Featured Articles (see for example Euclidean algorithm). But notice how that article is very extensive and complete. The current Black-Scholes article doesn't even convey to the lay reader that often pricing with the BS model is done using some numerical method and there is no closed-form formula. There is no explanation of how to price American options. There is no mention of common exotics. I think it's also necessary to fill out the material on martingale pricing. --DudeOnTheStreet (talk) 18:04, 6 May 2011 (UTC)

I agree with you! The problem is that the proof (or discussion of proof) is taking up a too big part of the article. Ulner (talk) 21:18, 7 May 2011 (UTC)

There used to be separate articles on the Black-Scholes formula, equation, and model. I think that might be the best thing. "Black-Scholes" should redirect to "Black-Scholes model" which would explain the basic set-up: basic history, terms, assumptions, and implications/contradictions. "Black-Scholes formula" would give the formula, explain several ways of deriving it, and link to related formulae. "Black-Scholes equation" would explain its derivation(s) and meanings, and can go into solutions with various payoff conditions. --DudeOnTheStreet (talk) 01:37, 3 June 2011 (UTC)