Black-Scholes equation Variational / Weak form

Question

I am having difficulty deriving the weak formulation of the Black-Scholes Equation.

I have multiplied it with a test function phi and integrated over Omega. But results on the internet suggest integration by parts are used on the second integral, and then some calculations are skipped, and I don't obtain the same result as rest of the world apparently.

Remark: I'm aware, it is a backward parabolic partial differential equation.

Edit: Here is what I got so far;

Black-Scholes Equation:

$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0$

Variational / weak form:

$\int_\Omega \left(\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV \right)\phi dx$

< = >

(**) $\int_\Omega \frac{\partial V}{\partial t} \phi dx + \int_\Omega \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \phi dx - \int_\Omega rS \frac{\partial V}{\partial S} \phi dx- \int_\Omega rV \phi dx$

Then I found sources saying that by integration by parts on the second integral in the above equation, we will obtain the following:

(*) $\int_\Omega \frac{1}{2} \sigma^2 S^2 \frac{\partial^2}{\partial S^2} \phi dx = -\int_\Omega \frac{1}{2} \sigma^2 S^2 \frac{\partial V}{\partial S} \frac{\partial \phi}{\partial S} dx - \int_\Omega \sigma^2 S \frac{\partial V}{ \partial S} \phi dx$

That I simply don't understand. If I use partial integration on the right hand side of the equation above I get the following:

$\int_\Omega \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} \phi dx = \frac{1}{2} \sigma^2 S^2 \left(\frac{\partial^2 V}{\partial S^2} \frac{\partial \phi}{\partial S} - \int_\Omega \frac{\partial V}{\partial S} \frac{\partial \phi}{\partial S} dx\right)$

Question 1: What else is happening in (*) besides integrations by parts?

Considering (*) again, we can substitute the left hand side into (**):

$\int_\Omega \frac{\partial V}{\partial t} \phi dx -\int_\Omega \frac{1}{2} \sigma^2 S^2 \frac{\partial V}{\partial S} \frac{\partial \phi}{\partial S} dx - \int_\Omega \sigma^2 S \frac{\partial V}{\partial S} \phi dx + \int_\Omega rS \frac{\partial V}{\partial S} - rV \phi dx$

Reducing:

(***) $\int_\Omega \frac{\partial V}{\partial t} \phi dx -\int_\Omega \frac{1}{2} \sigma^2 S^2 \frac{\partial V}{\partial S}\frac{\partial \phi}{\partial S} dx - \int_\Omega (\sigma^2-r) S \frac{\partial V}{\partial S} \phi dx - \int_\Omega rS \frac{\partial V}{\partial S} - rV \phi dx$

question 2:

Is the equation (***) the correct variational form of the Black-Scholes equation? I ask since I found various sources giving different variational forms after applying integrations by parts (in neither of them I'm abble to dublicate since I end in a situation as above, where I'm missing a step in the integration by parts).

Afterwards I would switch the sign by replacing t by tau=T-t, and obtain the variational form of the forward in time parabolic equation.

Thank you in advance for your suggestions, and spelling correction.

Answer 1

The way the question is posed shows a lack of perhaps understanding of how to formulate the equations ?

First, the equations are not readable, that is probably why no one has answered yet.

$\int_\Omega \left(\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV \right)\phi dx$

should be replaced by:

$\int (\partial_t V + \frac{1}{2} \sigma^2 S^2 \partial^2_{S^2} V + rS \partial_S V - rV ) \cdot \phi$

or even better:

$$ \left ( \partial_t V + \frac{1}{2} \sigma^2 S^2 \partial^2_{S^2} V + rS \partial_S V - rV, \phi \right )_{L^2}$$

There are terms missing in the equations, what are the variables ? what is $x$ for example ?

Then there is no mathematical logic, you go from mathematical values (like I have written above or your $(**)$ ) to equations (like $(*)$), then back to just values... One speaks about Variational / weak form for EQUATIONS. In other words in most equations you miss an $=0$.

For these reasons, I do not want to continue on the equations you wrote because you might have meant something else than what you wrote. On the other hand, here are some thoughts in order to solve some of the issues some people face:

• What does weak formulation means ? you never mentioned weak derivative though it is exactly what this is all about.
• Ask yourself where you take $\phi$, in what space. If you directly take it in a Sobolev space, then you do not need to use any density, but most probably you will take it in $C_0^1$ which is dense under some norms in your Sobolev space. So it is something to check.
• Using the density $C_0^1$ in your bigger space, you will probably need to prove some linearity and continuity properties about your operators.
• Perhaps before writing the weak form, transform your problem into a classical parabolic PDE problem as you wanted to do at the end (switch order of operations). The issue here that you are facing (though if you would understand what spaces $\phi$ is taken from, it is not an issue), is because your parameters are non-constant.

If anyone passing by wants some more details about one thing, please ask in comment :) But the whole derivation is tedious to write from scratch while there are very good books about it (I learnt about it through the excellent https://www.springer.com/de/book/9783642354007)