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complyorexplain
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Joined: November 9th, 2015, 8:59 am

Black Scholes Merton PDE

BS write in their original 1973 paper that there is only one formula w(x,t) that satisfies the PDE (their equation 7) subject to the boundary condition of their equation (8).

They don't actually prove the uniqueness of their result. Is it necessary to prove it? Every uniqueness proof of other PDEs I have looked at is very complicated. Is there a simple one of the BSM PDE? Does it actually matter?

Cuchulainn
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Re: Black Scholes Merton PDE

See Friedman's book  "Partial Differential Equations of Parabolic Type".
Some PDEs may have multiple solutions (for example, nonlinear BS PDE) and some PDEs may have no solutions, at least not the ones that satisfy your requirements.

A popular plan B is to use the high-falutin' word ansatz which is Morse code for "let's assume this PDE has a solution and take it from there". A bit like freshman separation of variables technique.

Alan
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Re: Black Scholes Merton PDE

I second looking at Friedman, also his SDE book. Once you've got rigorous conditions for the validity of the Feynman-Kac type probabilistic solution,

$u(x,t) = E_{x,t}[g(X_T)]$, where $g(\cdot)$ is a payoff function, then uniqueness becomes trivial. After all, if $u_1$ and $u_2$ are two solutions with the same payoff, then

$(u_1 - u_2)(x,t) = E_{x,t}[ 0 ] = 0$.

The general rule for uniqueness of PDE solutions associated to SDE's of the form $dS = a(S,t) S dW$, is that you need $a(\cdot,t)$ bounded on the positive S-axis. That's certainly true for the BS GBM process where $a = \sigma$, a constant. You can violate that rule and still be ok if the payoff is bounded, but combining violations with call option payoffs will lead to (non-uniqueness) trouble. (Also, there will be big problems if S=0 is not either unreachable or absorbing).

Does it matter? This kind of thing is quite useful for knowing when generalized models are problematic. For example, the CEV process $dS = S^{\gamma} dW$ with $\gamma > 1$ has non-unique call option prices. Google `strict local martingale' for literature.

Cuchulainn
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Re: Black Scholes Merton PDE

And Alan's book, Appendix 1.2 Vol I discusses it as well, for PDEs in finance.

For nonlinear PDE, this influential paper.
https://www.researchgate.net/publicatio ... _Equations

complyorexplain
Topic Author
Posts: 70
Joined: November 9th, 2015, 8:59 am

Re: Black Scholes Merton PDE

Thank you.

Cuchulainn
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Re: Black Scholes Merton PDE

Another way to prove uniqueness is by energy inequalities (standard in PDE) for initial boundary value problems. A potted version

0
(1) du/dt = Lu (L elliptic)

1
multiply (1) by u on both sides and use Green's function (fancy name for integration by parts) and get  the boundary conditions

2
Integrate now in time, use Gronwall and you get

Norm of solution <= input data.

I am writing a paper an initial ideas started here.
https://forum.wilmott.com/viewtopic.php?f=4&t=102228

Sorry i don't have more time at the moment. The approach is weak form of Finite Element Method. bit of research
Last edited by Cuchulainn on June 8th, 2020, 11:04 pm, edited 1 time in total.

complyorexplain
Topic Author
Posts: 70
Joined: November 9th, 2015, 8:59 am

Re: Black Scholes Merton PDE

Thanks again!

Cuchulainn
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Re: Black Scholes Merton PDE

You're welcome.
A good example of what I mean is section 4.1.2. Most of my points are worked out. Once you get the hang of it it becomes automatic.
http://www.csc.kth.se/utbildning/kth/ku ... pdated.pdf

(I did this for my MSc thesis way back for Friedrichs' systems and the BCs just rolled out  like a dream).

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