# What is Heston's equation?

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.

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## 2 Answers

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.)

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Thank you! I read this article. Actually it interested me because the author used weighted Sobolev spaces which I study. – nikita2 Oct 23 '12 at 14:48

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@nikita2 I do worry that the elliptic Heston operator is not a widely accepted term given that it doesn't appear in any peer-reviewed journal, and the few papers online seem to share the same particular co-author. If enough academics discover this work and agree with it, then the term may become more widely adopted; until then, you'll just get a lot of blank stares for using it in a sentence. – chrisaycock♦ Oct 23 '12 at 14:58

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.

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If I understood you correctly such models have little use and mathematicians just play with them. To do it clear, is there any sense to do research of PDE and SPDE in connection with mathematical finance? – nikita2 Oct 28 '12 at 10:08
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