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

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)