Talk:Black–Scholes equation

The mathematics in this article is horrible, even if standard for economics and/or finance. It is essentially unreadable by someone outside of the field even with expertise in mathematical PDE.

Specifically, example 1 is the "boundary condition"

$C(S,\,T)\to S$ as $S\to \infty$

is not a boundary condition. It is at best an asymptotic condition and should be written as such using standard mathematical notation. Otherwise, it is a nonsensical mathematical symbolism that states as S approaches infinity the function of S approaches S, which is of course infinity. This clearly makes no sense.

Example 2 is the function (let $K=1$),

$u_{0}(y)=e^{{\rm {max}}(y,\,0)}-1$.

There are only 2 elements in this maximum function. So, the result of applying the max function is either 0 or y. If the result is 0 then the whole function $u_{0}(y)=0$. In this case, the result of the convolution integral is always zero. On the other hand, if the result of applying the max function is y, then the function,

$u_{0}(y)=e^{y}-1$,

convoluted as indicated, results in

$e^{x+{\frac {1}{2}}\sigma ^{2}\tau }-1$,

and there is no cumulative distribution function $N(x)$ involved.

Example 3 is more minor but nonetheless is of concern. The boundary condition

$C(0,\,t)=0\quad {\text{for all}}\quad t$,

should in fact read

$C(0,\,t)=0\quad {\text{for all}}\quad t\in [0,\,T)$.

This is because it is well known, or ought to be well known that as $t\uparrow T$ the fundamental solution to the heat equation becomes a Dirac delta distribution. Thus for the function, C(S, t),

$\lim _{t\uparrow T}C(S,t)=C(S,\,T)=S-K$,

which is consistent with,

$C(S,\,T)={\rm {max}}\{S-K,\,0\}\quad {\text{iff}}\quad S\geq K$.

Notice that,

$\lim _{S\downarrow 0}\lim _{t\uparrow T}C(S,\,t)=-K\neq \lim _{t\uparrow T}\lim _{S\downarrow 0}C(S,\,t)=0\quad {\text{whenever}}\quad K\neq 0$.

Can someone please edit this disaster of mathematics so that it is readable by someone not already in the field of finance/economics? M. A. Maroun 20:42, 28 March 2016 (UTC)

For example 3 that you give, your limit statement is incorrect. $C(S,T)=\max\{S-K,0\}\neq S-K$ in general. As a call option assumes that the strike price $K$ is positive (otherwise you would always exercise), you find $C(0,T)=\max\{-K,0\}=0\neq -K$. Zfeinst (talk) 20:09, 29 March 2016 (UTC)

Ok that is what is missing from the original source, I see now, i.e. S, K > 0. --M. A. Maroun 21:00, 29 March 2016 (UTC) — Preceding unsigned comment added by MMmpds (talk • contribs)

Also, example 3 is not incorrect per se, as I calculated it from the solution function itself. You make my point exactly, namely that

${\rm {max}}\{S-K,\,0\}\neq S-K$, i.e.

$\delta (x)\neq 0$,

unless of course x is known to never be zero. That is, the boundary data force the solution to be not even continuous from the left toward its right most value in t.--M. A. Maroun 21:11, 29 March 2016 (UTC)

Resolved despite terrible notation

Using the Fourier convolution theorem, I found the error in the article. The initial function is missing a Heaviside unit-step function. See Heaviside step function for more background. This is a consequence of enforcing the boundary data C(S, T) =max{S - K, 0}. Explicitly, one should note (in x, τ coordinates) for τ=0,

$u(x,\,0)=:u_{0}(x):=K\left(e^{{\rm {max}}\{x,\,0\}}-1\right)=K\left(e^{x}-1\right)\,H(x)\quad {\text{given}}\quad x\in \mathbb {R}$,

where H(x) is the Heaviside step function, i. e.,

$u(x,0)=u_{0}(x)=0,\quad \forall \;\;x<0.$.

I am editing the article to reflect this fact.

--M. A. Maroun 20:25, 29 March 2016 (UTC) --M. A. Maroun 19:39, 29 March 2016 (UTC)

Adding the Heaviside step function is unnecessary. If $x<0$ then $e^{\max\{x,0\}}-1=0$, so you would multiply $0$ by $0$.Zfeinst (talk) 20:04, 29 March 2016 (UTC)

Yes you are correct that having both is redundant. I am editing the article to reflect this. I see the issue. I regard the Heaviside function as a bona fide distribution, for which the calculus makes sense. I never thought of regarding the max function as a distribution. I read it algebraically. But as you point out given that x is a real number ranging over all real values, one has that it indeed can be regarded as equivalent to the Heaviside function.

--M. A. Maroun 20:47, 29 March 2016 (UTC)

Only One Source?!?

Is this article a purely plagiarized article from Hull? Granted, Hull is well thought of and I own a copy. It is at another home so I can't check but if that's the case, we really need to add additional sources and citations to this page.Geoff918 (talk) 15:50, 26 March 2018 (UTC)

            Reference 4 was added. With one of the original articles in hand, one can independently verify the claims, and content in Hull are indeed correct and consistent.     - M. A. Maroun 23:50, 3 August 2019 (UTC)

No Definitions

What is u()? What is C()? How do these relate back to S and V? 'mu' appears but is never defined.

107.184.51.14 (talk) 21:27, 3 January 2020 (UTC)

Reply: Since

$V:\;\mathbb {R} ^{2}\to \mathbb {R} :\;(S,\,t)\mapsto V(S,\,t)$

$u:\;\mathbb {R} ^{2}\to \mathbb {R} :\;(x,\,\tau )\mapsto u(x,\,\tau ),$

then one has that $u(x,\,\tau )=V(S(x,\,\tau ),\,t(x,\,\tau ))$.

C is the non-exponential part of u. See Solving the Black-Scholes PDE. M. A. Maroun 23:37, 6 January 2020 (UTC)

Incorrect formula?

I have seen several papers on Black-Scholes and this article uniquely has the first term with the opposite sign. It seems to be missing the preceding "-".

For examples see:

https://warwick.ac.uk/fac/cross_fac/complexity/study/emmcs/outcomes/studentprojects/akinyemi.pdf

https://www.researchgate.net/publication/227624203_A_simple_approach_for_pricing_Black-Scholes_barrier_options_with_time-dependent_parameters — Preceding unsigned comment added by Kallax (talk • contribs) 11:36, 24 February 2020 (UTC)

Connections to diffusion processes

In stochastic process theory, diffusion equations typically have two forms -- a forward equation which tells us how a probability distribution evolves forward from some initial condition under the random walk, and a backward equation which tells us about the initial conditions given the current distribution. These equations are usually adjoints of each other. The constant-coefficient heat equation is special because it is self self-adjoint, and the same equation forward and backward, but the Black--Scholes equation is not constant coefficient. Backward equations appear in Markov decision theory where we are trying to plan for an uncertain future.

When I look closely, this seems like a backward equation, but in this description, there is no mention of this distinction between forward and backward equations, or any connection to it. Can somebody help clarify? — Preceding unsigned comment added by Uscitizenjason (talk • contribs) 20:20, 10 November 2022 (UTC)

Redirect issue

Linking from Parabolic partial differential equation - Wikipedia

Why is there a redirect?

I guess the redirect is hyphen versus emdash, but surely fixable by a Bot?

Use of the word derivative

The article uses the word derivative to mean both a financial instrument (like an option) and the mathematical operation. This is potentially confusing. In some places we have both uses in the same sentence. I suggest adding, after the first para: "In the remainder of this article we talk about options, although some of what follows can also apply to other derivatives." and then changing all subsequent "derivative"s (financial instrument) to "option"s. — Preceding unsigned comment added by Blitzer99 (talk • contribs) 18:34, 17 November 2023 (UTC)

Darcourse (talk) 11:59, 18 March 2023 (UTC)