Mathematical Finance

Black-Scholes and Schrodinger

Eric28 Comments

In this post, I will perform some computations to demonstrate a relationship between the Black-Scholes PDE and the Schrodinger equation of quantum mechanics. This relationship is not new. You can, for example, find something similar discussed here. The purpose for writing the gory details here is that I hope we (that’s right.. you too!) can “categorify” the Black-Scholes framework.

The Black-Scholes PDE is derived in a million places. One of those places is a paper I wrote in May 2002 and published on the quantitative finance web site Wilmott.com.

Noncommutative Geometry and Stochastic Calculus
Applications in Mathematical Physics

I derived it in a slightly unusual way via noncommutative geometry. Regardless of how it is derived, the Black-Scholes PDE is given by

\partial_t V + \frac{\sigma^2 S^2}{2} \partial_S^2 V + r S \partial_S V - r V = 0

One day, for some reason that I forget, I was curious whether this could be written in a form that looked like Schrodinger’s equation

i\hbar\partial_t \psi = H \psi = \frac{p^2}{2m} \psi + U \psi = -\frac{\hbar^2}{2m} \partial_x\psi + U\psi

so I started chugging away with some algebraic gymnastics. The first thing is to rewrite the equation with the time derivative on one side and everything else on the other.

\partial_t V = -\frac{\sigma^2 S^2}{2} \partial_S^2 V - r S \partial_S V + r V

Comparing this to the Schrodinger equation

i\hbar\partial_t\psi = -\frac{\hbar^2}{2m} \partial_x^2\psi + U\psi

tempted me to complete the square in the BS PDE resulting in

\partial_t V = -\frac{\sigma^2}{2} \left[S\partial_S -\frac{1}{2}\left(1-\frac{2r}{\sigma^2}\right)\right]^2 V + \frac{\sigma^2}{8} \left(1 + \frac{2r}{\sigma^2}\right)^2 V.

Now letting

p = -i\sigma\left[\partial_x -\frac{1}{2}\left(1-\frac{2r}{\sigma^2}\right)\right],

where x = \log{S} and

U = \frac{\sigma^2}{8} \left(1 + \frac{2r}{\sigma^2}\right)^2

we find that the Black-Scholes PDE can be written as

\partial_t V = \frac{p^2}{2} V + U V.

Observation 1

The Black-Scholes PDE is a “Wick rotated” Schrodinger equation for a charged particle in an electromagnetic field, where the risk-free rate plays the role of a gauge connection.

Observation 2

The volatility \sigma plays the role of Planck’s constant \hbar while p satisfies the commutative relation

{}[x,p] = i\sigma.

Now let the games begin! 🙂

Note: Thanks to naumovic who pointed out in the comments below an algebraic mistake in an earlier version of this article.

28 thoughts on “Black-Scholes and Schrodinger”

  3. Glad to hear it. I keep meaning to write more: the review article that’s more than ten years out of date is still over a hundred pages long, so there’s not exactly a shortage of topics. I know I could get at least a couple posts out of the Fokker-Planck material, and then a couple more on shape invariance’s relation to the Laplace-Runge-Lenz vector, if only I could find the time to write about the LRL vector first. . . .

    Are you sure p shouldn’t be -i\sigma(\partial_x + \frac{r}{\sigma^2})? I don’t think the units work out in what you wrote, since \frac{r}{\sigma^2} is dimensionless (going by the first equation), but i\sigma \partial_x has units of time to the minus one-half.

    But it’s very late here and I should be in bed. . . .

    Reply

  4. Hi Blake,

    Thanks for pointing out the typo. I’ll correct it.

    The fact that the Black-Scholes equation can be mapped to the heat equation is well known in math finance and consequently, relates to Schrodinger via a Wick rotation. So nothing I did here was particularly new, but hope it will maybe generate some discussion.

    Fokker-Planck and path integrals also have an established role in mathematical finance as well, but I’ve never spent the time trying to see how everything is connected. Hopefully that will change! You know you will have at least one interested reader if you ever decide to write up some stuff. I liked the level of your other SUSY QM notes. It’s been ages since I thought about this stuff and its fun to dust off some of the cobwebs.

    Reply

  5. Let’s see, from what I remember, the most “elementary” way to go from a point-particle theory to a string theory is via the Lagrangian formulation. Write yourself an action in terms of a proper time and then up the dimension for an action in terms of a world-sheet area. All this Black-Scholes stuff is being done in a Hamiltonian formalism; like the heat Fokker-Planck Equation, Black-Scholes transforms to Schrödinger. This book looks relevant.

    Reply

  7. Yeah, that book does look interesting. I wouldn’t be surprised if he(?)’s already worked out all the details of what I’m trying to do. It’s fun to reinvent things anyway 🙂

    The author of the book also has a bunch of papers on the arxiv that should give some hints.

    Reply

  9. Hi Eric,

    These are pretty interesting posts. The idea that r/\sigma^2 is your connection term actually makes sense if you think of the connection as pulling your pricing kernel in the time direction. I suspect the factor r there comes from assuming absence of arbitrage. Otherwise it will be the growth rate of the underlier…

    On a similar note, you might find the following paper by Smolin interesting: a gauge theory for almost equilibrium markets.

    Click to access 0902.4274v1.pdf

    Also, at the expense of expanding the size of the comment, the non-commutative explanation for ito calculus was refreshing. I am sure you know there is some connection of the ito rule with how you discretize the stochastic integral: the ito rule follows when you define the stoch integral as a limit of
    \sum f(x_i) (x_{i+1} – x_i )

    where x_i is the random variable denoting the value of the random walk process at t_i.

    If instead you use the mid-point rule, you will get a calculus where leibnitz rule is satisfied.

    Any idea how to connect these two ideas together?

    Also, your forward arrow and backward arrow partials feel a lot like kolmogorov’s forward and backward equation.

    Very interesting collection of posts, thank you.

    Reply

  10. Hi Corporate Serf,

    Thanks for the link to Smolin’s paper! You know the cat is out of the bag when Lee Smolin starts writing finance papers. I never thought the day would come 🙂

    Regarding the discrete rule, that is actually what first made me realize the relation between noncommutative geometry and stochastic calculus. To be honest, I was a little disappointed to later find that Dimakis and Mueller-Hoissen beat me to it 🙂

    In “My Papers” (link here on the blog), I have a another paper that presents the discrete version of the original paper that you might find interesting.

    The idea of connections and parallel transport involving the “time value of money” is a fun thing to think about.

    Reply

  11. Another thing…

    Regarding the relation between the Ito integral and Stratonovich intergal, have a look at that discrete paper. On page 6, I say:

    “If you are familiar with stochastic calculus, this might make you think of the differences in the definitions of the Stratonovich and Ito integrals. It is no coincidence!”

    Reply

  13. Eric,
    I did glance thru some of your postings at the wizard’s den :).

    Here is another connection which might be relevant to your goal of doing non-commutative yield curves. Look at the derivation of the market price of risk under several fixed income models. Essentially you look at the martingale measures for different numeraires, namely zero coupon bonds maturing at different times in the future.

    If you go thru the derivation, eg in http://www.sam.sdu.dk/~cmu/noter/IntroFIA.pdf around page 32 / 33 and equation 3.6 (just picking something from my bookmarks; the derivation is fairly standard), it seems to me that the reasoning itself is very similar to gauge invariance arguments. The argument itself depends crucially on the simultaneous equilibrium in all maturities, so it would be interesting to do the same assuming some sort almost equilibrium assumption, say, as in Smolin’s paper. I would imagine that your formalism would definitely make some these calculations simpler.

    The use of path integral type calculations is actually not very new. Dash has a fat book which is very readable for physicists (under the not so imaginative name of “Quantitative Finance and Risk something for Physicists”, or something like that). That has a chapter explaining the path integral calculation and the connection to feynman kac. The convergence properties are apparently much easier than for quantum mechanics because of the heat kernel structure.

    My interests currently involve trying to understand slightly off equilibrium situations. Say some calculation involving relative relaxation times of different securities. Might make the analysis of statistical arbitrage algorithms somewhat more principled than “dude, it happened in the past”.

    Reply

  14. I’m sorry but I don’t think that the operator (x\partial_x)^2 is equal to the operator x^2 (\partial_x)^2, so that you’re very first schroedinger like calculation is false.

    Reply

    • Yes, you’re right. The idea can be saved though. Completing the square still goes through (as we know it must) so the conclusion of the article is still in tact. I will fix the algebraic error with reference to the fact you pointed it out.

      We simply need to go back and note that x^2(\partial_x)^2 = (x\partial_x)^2 - (x\partial_x) and complete the square again.

      Reply

  15. Hi naumovic,

    Thanks for checking that. I’ll have a look. It is possible that I made an error. I’ll follow up when I have a moment.

    Cheers

    Reply

  16. Did you ever read that book called “Quantum Finance”? Is it related to this stuff? I just bumped into a mention of it today. There’s also an arXiv paper by that title. I have no idea if this stuff makes sense, or if it’s related to your ideas. In general I get a bit skeptical when someone sticks the adjective “quantum” in front of some seemingly unrelated word, but who knows?

    Reply

  17. On a somewhat separate note, I’ve finally gotten up the courage to read Frieden’s book Physics from Fisher Information. It has some sentences you might find interesting, e.g. “As will become evident below, Schroedinger’s mysterious Lagrangian term was simply Fisher’s data information.” I’ll have to see if I can understand this.

    Reply

    • Sounds very interesting.

      By the way, I no longer have the freedom for extra curricular activities, e.g. personal email, forums, blogs, etc, during the day, so it took a while for me to see this comment.

      Reply

      • Okay. I hope you’re busy with interesting things!

        I will do a bunch more posts on information geometry in a while, and maybe even tackle the information geometry of Brownian motion and/or Schroedinger’s equation.

      • So far so good. I’m now working in insurance, which is a bit of a change.

        I’m obviously looking forward to the information geometry stuff. I still hope to relate it to commutation relations if possible. For instance, beginning with Leibniz and nilpotency, i.e. a differential algebra, together with commutation relations, and nothing more, you can derive both the heat equation and the Schrodinger equation.

        The heat equation follows from

        [dx,x] = dt

        and the Schrodinger equation follows from

        [dx,x] = i dt.

  18. Eric,

    You might like this book by Henry-Labordere (if I have got the spelling right): http://www.amazon.com/gp/product/1420086995/ref=ord_cart_shr?ie=UTF8&m=ATVPDKIKX0DER

    He maps all of traditional finance into geometry, like you do. The connection between ito and stratonovich differentials show up exactly as a connection (I think, it is still a riemannian geometry but of course you can have different connections, I am still thumbing through this). Measure changes are somehow gauge transformations. Couple of tricks I learnt from here are heat kernel expansions (you get nifty asymptotics very quickly) and a few very geometric derivations of thigs you already know, like dupire’s local volatility, expression of implied volatility as path integrals (averages) of local volatility.

    There is also a bit at the end (that seems like I need to work through a bit) where he uses things like functional integrals and so on for quick derivations of convexity correction for CMS’s; which sounds like about right.

    I am pretty excited about how some of the complicated asymptotics (implied vols at wings and so on) become much more intuitive and connected through the physics derived techniques.

    archi

    Reply

    • Also forgot to mention, he describes a way of factoring local volatility model equation using supersymmetry to classify all solvable models.

      Another commentator made a similar comment up thread.

      Reply

    • Thanks for the reference CS. The book looks cool. I’ll try to get my hands on it. Maybe a Christmas gift to myself 🙂

      By the way, I’ve moved to Asia and have been in HK a little over a year now. I’m having a bit of fun recently pricing interest rate options in China.

      Reply

      • Cool. You are not the only one to move to asia.

        BTW, I have moved away from rates world some time ago, but have people started to use different funding curves and the embedded convexity at all?

      • I don’t know about other people, but I have. It is very interesting looking at rates in Asia.

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