# The Relation Between the Ricci flow and the Black-Scholes-Merton Equation

Grisha Perelman once wrote that

The Ricci-flow equation, a type of heat equation, is a distant relative of the Black-Scholes equation that bond traders around the world use to price stock and bond options.

Wilmot has derived from the BS Equation to the heat equation, but wonder if there is any proof that you can get the BS Equation from the Ricci flow.

• Could you include some details about the Ricci flow? How does the equation look what could be related to what? – Ric Mar 24 '16 at 7:53
• hmm, where did he write this, can you provide the citation? I actually doubt there is any real connection (no pun intended) but would certainly want to make sure that I do not contradict Mr. Perelman ;-) Oh and btw bond traders would not be the first who come up to mind when you mentioned the BS formulas. – g g Mar 24 '16 at 16:38
• Oh and this Wikipedia article (en.wikipedia.org/wiki/Ricci_flow) has something about the connection to diffusion. But I doubt this is more than a superficial coincidence. – g g Mar 24 '16 at 16:51
• If anything, this is an informal remark about the nature of these PDEs. Hence, there is no precise statement here to prove. BS can be cast as a heat equation, and Ricci flow can be reduced to it in a special case referenced in the above mentioned wiki. People try to use general heuristics applicable to heat equation for Ricci flow in general, but it's nothing more than heuristics. – LazyCat Mar 24 '16 at 17:33
• I doubt it's more than heuristics if Sasha Perrelman said so @LazyCat – Dendi Suhubdy Mar 24 '16 at 17:39

Well, this seems to be a popular account of these concepts but on a very high level the connection is the following:

The Ricci flow "is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric." [Wikipedia] Now the Black-Scholes equation is mathematically based on Geometric Brownian motion which describes the diffusion of the probability distribution of an underlying's price paths.

The connection between Black-Scholes and diffusion becomes especially clear when you have a look at how Black-Scholes' differential equation is solved by transforming it to the diffusion equation, see also this question and answers therein:
Transformation from the Black-Scholes differential equation to the diffusion equation - and back

So both, Ricci flow and Black-Scholes, are (based on) mathematical descriptions of diffusion models. I don't think that there is really anything more to it than this.

So here is an abrupt try find connections between them. I know this is incomplete and I hope someone else adds more/edits more into this:

The Ricci flow equation

$$\frac{dg}{dt} = - 2 Ric(g(t))$$

Both sides are the same type of object : at each point $p \in M$, a bilinear form on $T_pM$.

In terms of local coordinates this becomes

$$\frac{\partial g_ij}{\partial t}= - 2 R_{ij}$$

(Hamilton, 1982).

The heat equation in 3-D is

$$\frac{\partial f}{\partial t} = \nabla^2 f$$

The basic differences are

• the heat flow evolves an initial function $f_0$ towards a constant function
• Ricci flow evolves a Riemannian metric.

More on the Ricci flow by Bennett-chow

heat-type equations. The full curvature tensor $\operatorname{Rm}$ satisfies an equation of the form $\frac{\partial }{\partial t}\operatorname{Rm}=\Delta\operatorname{Rm}+q(\operatorname{Rm})$, where $q$ is a quadratic polynomial. Since $\operatorname{Rm}$ is a symmetric bilinear form on the vector space $\wedge^{2}T_{x}^{\ast}M$ at each point $x$, we have the notion of nonnegativity of $\operatorname{Rm}$. Since $q(\operatorname{Rm})$ satisfies a property sufficient for the maximum principle for systems to be applied, $\operatorname{Rm}\geq0$ is preserved under the Ricci flow. Generally, we can analyze the behavior of $\operatorname{Rm}$ by the maximum principle under various hypotheses.

Geometric application. In particular, when $n=3$ and $\operatorname{Ric} _{g_{0}}>0$, we have $\pi_{1}(M)=0$ and hence the universal cover $\tilde{M}$ is a homotopy $3$-sphere. Encouraged by this, Hamilton proved that the solution to the normalized Ricci flow exists for all time and converges to a constant positive sectional curvature metric; thus $M$ is diffeomorphic to a spherical space form. The main gonzo estimate is $\frac{|\operatorname{Ric}% -\frac{R}{3}g|^{2}}{R^{2}}\leq CR^{-\delta}$ for some $C$ and $\delta>0$. Intuitively, we expect $R\rightarrow\infty$ and hence $\operatorname{Ric} -\frac{R}{3}g\rightarrow0$.

• Let's not make a Ricci flow wiki here. The question is: did Perelman really write about the two together, and did he mean anything beyond general similarities. So far we know that it was mentioned once without a reference in a book targeted towards non-mathematicians. I gave you my opinion in the comments. It coincides with the answers the same question got on math.stackexchange and wilmott forums. If you want to dig further, I suggest you try to find a paper by Perelman where he mentions Black-Scholes. They should be mostly on arxiv. Or else write to O'Shea to check if he knows what he meant. – LazyCat Mar 24 '16 at 19:20
• I did try to find but there were no direct quotes on him, else than the book on Pointcare conjecture that I referred to you before. I'm still looking at this. – Dendi Suhubdy Mar 24 '16 at 19:21