Phorgy Phynance

Phynance 101

with 4 comments

If you have any questions related to anything I discuss here or anything related to mathematical phynance in general, please feel free to ask in a comment here.

Written by Eric

July 24, 2008 at 11:18 am

4 Responses

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  1. Hello Mr. Forgy,

    I think your site is full of interesting topics! As I have a (albeit only a master’s) degree in physics and I somehow ended up working in the derivatives business, I have been wondering about the following for quite a while; perhaps you can shed some light on it?

    Is it possible to arrive directly at the Schrodinger equation in the continuum limit by constructing a complex binomial lattice with a (constant) diffusion sigma = sqrt(i*hbar/m), where i is the complex number, hbar is planck’s constant, and m the particle’s mass?

    Cheers, Frido

    Frido Rolloos

    December 28, 2008 at 3:56 am

  2. Hey! That’s “Dr Forgy”! *just kidding* πŸ™‚

    Thanks for your question Frido. Yes, just as you can arrive at Black Scholes via a continuum limit of the binary tree, you can also arrive at Schrodinger via a continuum limit of a complex binary tree. My favorite way to see this is via my paper:

    Financial Modeling Using Discrete Stochastic Calculus
    https://phorgyphynance.files.wordpress.com/2008/06/discretesc.pdf

    In this way, you see it via the fact that the “calculus” itself on a tree converges to the continuum calculus, so anything you build on a tree will converge in the continuum limit. In particular, on a general tree, the discrete calculus results in four commutative relations:

    {}[dx,x] = \frac{(\Delta x)^2}{\Delta t} dt

    {}[dx,t] = [dt,x] = \Delta t dx

    {}[dt,t] = \Delta t dt

    I showed two different continuum limits.

    1.) \Delta x = c \Delta t, \Delta t \to 0 (leads to exterior calculus)
    2.) (\Delta x)^2 = \Delta t, \Delta t\to 0 (leads to stochastic calculus)

    There is a third continuum limit you could take, i.e.

    3.) (\Delta x)^2 = i \Delta t, \Delta t\to 0 (leads to Schrodinger)

    This is just the “Wick rotation” of 2.)

    There are other ways to see quantum stuff from continuum limits. You might enjoy the “Feynman checkerboard” if you haven’t seen it. There, the Dirac equation pops out of a complex random walk. Very neat.

    By the way, if you decide to have a look at some of my papers, I’m happy to try to answer any questions you might have.

    Cheers and have a happy new year!

    phorgyphynance

    December 28, 2008 at 8:05 am

  3. Mr. Dr. Forgy, thanks for your speedy reply! πŸ™‚

    It’s been a while since I’ve done some noncommutative calculus (wrote my thesis on it), but definitely will read your papers on noncommutative geometry & finance. I could be wrong, but as far as I know many theorists in the physics community are not aware that there is a discrete structure approximating, or perhaps underlying, the Schrodinger eqt. and that discrete calculus is an elegant way to derive/show it.

    Happy new year to you too, and will most probably ask you more questions in the new year!

    Frido Rolloos

    December 28, 2008 at 9:37 am

  4. Hey Eric,

    You might wanna take a quick look at this paper: Graphical models for correlated defaults. It’s not everyday that you see an algebraic geometer writing a paper on CDOs!

    Cheers

    -R

    Rod Carvalho

    July 9, 2009 at 1:57 am


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