What is Heston's equation?

Question

This paper mentions the elliptic Heston operator:

$Av:= -\frac y2(v_{xx}+2\rho\sigma v_{xy} + \sigma^2v_{yy}) - (c_0 - q - \frac y2)v_x + \kappa(\theta -y)v_y + c_0v$.

Then boundary value problem are discussed:

$Au=f \text{ on } \Omega \\ u = g \text{ on } \partial\Omega$

I would like to know how people use such Dirichlet conditions in mathematical finance.

Answer 1

From this abstract:

The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.

A simple Google search shows that there are only a handful of academics who even use this term. Your best bet may be to contact one of them directly for support. (They are unlikely to entertain a broad "what do I use this for" question, however.)

Answer 2

Expanding a bit on chrisaycock's answer, and noting in particular from the abstract

In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset.

we can see that this would be used to price those few rare cases of perpetual options.

The only traded examples I know of are perpetual convertible preferred securities, for example from Wells Fargo's offerings. Such securities are lightly traded by the market players and therefore not always analyzed using the full machinery of a stochastic vol model, even if they should be in principle.

In practice, these "perps" are so bond-like that it is often more useful to think of them as fixed-income instruments. The main concern with them is that the issuer will stop paying the dividends or change capital structure, so it is a bit ridiculous to spend one's time on a fancy stochastic vol model when all the interesting stochastic events have to do with unrelated variables such as alterations in capital structure.