How Ricci Flow and Black-Scholes Relate Through Diffusion

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


# 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.

## Answer by vonjd (score 3, accepted)

https://quant.stackexchange.com/a/28228

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.

[Source]

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.

## Answer by Dendi Suhubdy (score 1)

https://quant.stackexchange.com/a/25067

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

Here is a similar intuition behind the Ricci flow

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 $\operatorname{Ric} -\frac{R}{3}g\rightarrow0$.