Serving the Quantitative Finance Community

 
User avatar
complyorexplain
Topic Author
Posts: 121
Joined: November 9th, 2015, 8:59 am

Black Scholes Merton PDE

June 7th, 2020, 12:43 pm

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?
 
User avatar
Cuchulainn
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: Black Scholes Merton PDE

June 7th, 2020, 2:52 pm

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.
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: Black Scholes Merton PDE

June 8th, 2020, 2:57 am

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.
 
User avatar
Cuchulainn
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: Black Scholes Merton PDE

June 8th, 2020, 8:45 am

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
 
User avatar
complyorexplain
Topic Author
Posts: 121
Joined: November 9th, 2015, 8:59 am

Re: Black Scholes Merton PDE

June 8th, 2020, 12:17 pm

Thank you.
 
User avatar
Cuchulainn
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: Black Scholes Merton PDE

June 8th, 2020, 7:58 pm

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.
 
User avatar
complyorexplain
Topic Author
Posts: 121
Joined: November 9th, 2015, 8:59 am

Re: Black Scholes Merton PDE

June 8th, 2020, 9:22 pm

Thanks again!
 
User avatar
Cuchulainn
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: Black Scholes Merton PDE

June 8th, 2020, 11:12 pm

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).
 
User avatar
Collector
Posts: 2572
Joined: August 21st, 2001, 12:37 pm
Contact:

Re: Black Scholes Merton PDE

July 10th, 2020, 9:58 am

Screen Shot 2020-07-10 at 12.14.24 PM.png
 
User avatar
fyvr
Posts: 26
Joined: November 15th, 2015, 3:10 pm

Re: Black Scholes Merton PDE

August 24th, 2020, 4:44 pm

The continued existence of a 45 year traded market in the solutions to the PDE is itself a guarantee of uniqueness.
 
User avatar
Cuchulainn
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: Black Scholes Merton PDE

August 24th, 2020, 5:56 pm

The continued existence of a 45 year traded market in the solutions to the PDE is itself a guarantee of uniqueness.
And existence!
 
User avatar
Collector
Posts: 2572
Joined: August 21st, 2001, 12:37 pm
Contact:

Re: Black Scholes Merton PDE

March 24th, 2022, 12:06 pm

 
User avatar
bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: Black Scholes Merton PDE

March 24th, 2022, 3:48 pm

I was thinking that perhaps my command of the English language was failing me but was somewhat reassured by a ranked list of 113 different interpretations of ML that failed to include anything like “Modified Log”. On that basis alone, I’m hesitant to proceed further.