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Parabolic PDEs (e.g. heat equation) are closely linked to finance via the Feynman Kac Theorem.

Do other types of PDEs appear in quant finance? Elliptic PDEs don't contain a time dimension (so perhaps for perpetual options?) I can't think of an example for the wave equation (two time derivatives) or other hyperbolic PDEs?

Are there special numerical treatments for these other types? Crank Nicolson FD schemes seem to be popular but restricted to parabolic PDEs.

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PDE Classification (Background)

Linear second-order PDEs can be classified as either elliptic, parabolic or hyperbolic. A general PDE in two dimensions for $u=u(x,y)$ would look like $$Au_{xx}+2Bu_{xy}+Cu_{yy}+Du_x+Eu_y+Fu+G=0.$$ The PDE is called

  • hyperbolic if $B^2-AC>0$ (prime example: Wave equation with $a^2\Delta u=u_{tt}$ ),
  • parabolic if $B^2-AC=0$ (prime example: Heat equation with $a\Delta u=u_t$),
  • elliptic if $B^2-AC<0$ (prime example: Laplace equation with $\Delta u=0$).

where $\Delta$ denotes the Laplacian.

This definition resembles quadratic forms. Of course, the coefficients $A,B,C$ can depend on $x$ and $y$ and a PDE can be of different types over its domain. In more than two dimensions, we can use the eigenvalues of the coefficient matrix to define the PDE's type.

PDEs and Option Values

Most PDEs in quantitative finance govern the value of an option, $V=V(t,X_{1,t},...,X_{n,t})$ (in its continuation region), subject to (free) boundary conditions. In an arbitrage-free market, these PDEs stem from the pricing rule $$\mathbb{E}^\mathbb{Q}[\text{d}V]=rV\text{d}t.$$ Assuming that the underlying follows a (multi-dimensional) Itô process and thus abstracting from jumps (which give rise to PIDEs), Itô's Lemma states $$\text{d}V = \left(\frac{\partial V}{\partial t}+\sum_{i=1}^n\mu_{i,t}\frac{\partial V}{\partial X_{i,t}}+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n \sigma_{i,t}\sigma_{j,t}\rho_{i,j,t}\frac{\partial^2 V}{\partial X_{i,t}\partial X_{j,t}}\right)\text{d}t+\sum_{i=1}^n \sigma_{i,t}\frac{\partial V}{\partial X_{i,t}}\text{d}W_{i,t}.$$ Thus, option values satisfy in general the PDE \begin{align*} \frac{\partial V}{\partial t}+\sum_{i=1}^n\mu_{i,t}\frac{\partial V}{\partial X_{i,t}}+\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n \sigma_{i,t}\sigma_{j,t}\rho_{i,j,t}\frac{\partial^2 V}{\partial X_{i,t}\partial X_{j,t}}-rV=0. \tag{$\star$} \end{align*}

In a complete market, this PDE can be derived by dynamic hedging. The Feynman-Kac Theorem provides a lot of economic intuition by linking the PDE to an expectation operator.

Parabolic PDEs

In the Black and Scholes model, we have one risk source, $n=1$ with $\mu_t=rS_t$ and $\sigma_t=\sigma S_t$. Thus, the PDE ($\star$) collapses to the standard \begin{align*} \frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2} \sigma^2S^2\frac{\partial^2 V}{\partial S^2}-rV=0. \end{align*}

This PDE is parabolic because there is no $V_{tS}$ term and no $V_{tt}$ term. So the classification scheme would be $$B^2-A\cdot C=0^2-\frac{1}{2}\sigma^2S^2\cdot 0=0.$$ Indeed, it's well-known how to transform this PDE into the heat equation. This is, in fact, how Black and Scholes solved the PDE in their original JPE paper.

In the stochastic volatility model from Heston, we have two risk sources ($n=2$) and thus, the PDE ($\star$) collapses to \begin{align*} \frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\kappa(\theta-v)\frac{\partial V}{\partial v}+\frac{1}{2} vS^2\frac{\partial^2 V}{\partial S^2}+ \rho\xi v S\frac{\partial^2 V}{\partial S\partial v}+\frac{1}{2} \xi^2v\frac{\partial^2 V}{\partial v^2}-rV=0. \end{align*} Clearly, choosing $\kappa=\xi=0$ recovers the model from Black and Scholes. This PDE is again parabolic. However, the PDE cannot be reduced to $\Delta u=u_t$ because the cross-correlation term cannot be eliminated by substitutions (``degenerate PDE'').

Other PDEs such as the Fokker-Planck PDE are also parabolic. The PDE associated to the HJB framework also tends to be parabolic.

Elliptic PDEs

The ``problem'' with the PDEs above is that there is a first-order time derivative, but no cross time-space derivative and no higher time derivatives. Thus, the PDEs always resemble parabolic PDEs.

It's quite difficult to artificially fiddle in higher order time derivatives. It's however quite easy to just drop the time derivative altogether. If the option value $V=V(X_{1,t},...,X_{n,t})$ is independent of time, then the two previous PDEs collapse to \begin{align*} r S\frac{\partial V}{\partial S}+\frac{1}{2} \sigma^2S^2\frac{\partial^2 V}{\partial S^2}-rV=0, \end{align*} \begin{align*} rS\frac{\partial V}{\partial S}+\kappa(\theta-v)\frac{\partial V}{\partial v}+\frac{1}{2} vS^2\frac{\partial^2 V}{\partial S^2}+ \rho\xi v S\frac{\partial^2 V}{\partial S\partial v}+\frac{1}{2} \xi^2v\frac{\partial^2 V}{\partial v^2}-rV=0. \end{align*} The first one is simply an ODE (to be precise, a Cauchy-Euler ODE) and the second PDE is an elliptic PDE (it still cannot be reduced to $\Delta u=0$).

For options to be independent of time, they need to be perpetual (i.e. never expire). These options are obviously not traded but they always appear in real options applications and capital budgeting decisions. The early exercise (free) boundary represent a difficulty in solving these elliptic PDEs. Another real options example would be a perpetual exchange option where both assets follow a geometric Brownian motion (recall Margrabe's formula). The corresponding PDE is \begin{align} rS\frac{\partial V}{\partial S} + rK \frac{\partial V}{\partial K} + \frac{1}{2}\sigma_S^2S^2 \frac{\partial^2 V}{\partial S^2} + \rho\sigma_S\sigma_K SK \frac{\partial^2 V}{\partial S\partial K} + \frac{1}{2}\sigma_K^2K^2 \frac{\partial^2 V}{\partial K^2} - rV=0. \end{align} Due to the lack of a time derivative, elliptic PDEs tend to describe a state in ``equilibrium'' (not the economic meaning of the word).

Hyperbolic PDEs

For the aforementioned reasons, it's difficult to think of many applications of the wave equation in finance. Higher order time derivatives normally do not appear. The only exception I am aware of includes interest rate models, where current time and time to maturity are state variables. Santa Clara and Sornette (1998, RFS) study second-order stochastic PDEs, where there is a propagation term. Physically, it resembles a vibrating elastic string (the paper's title includes ``stochastic string shocks'').

Summary

PDEs are linked to conditional expectations and thus deeply related to quantitative finance. Because option values depend on the time to maturity, they typically include a time derivative alongside spatial derivatives (depending on risk sources). Thus, most PDEs are parabolic. Perpetual options occur in real options examples and tend to be elliptic. Hyperbolic PDEs are rare but occur in interest rate applications.

Regarding numerical methods and finite differences, you can use them for any of these types. You tend to have different boundary conditions for different types and therefore a different implementation. For example, for parabolic PDEs you can go back in time step-by-step (highlighting the relationship between finite differences and multinomial trees) whereas you find all grid points for elliptic PDEs in one go by solving one linear equation system (e.g. LUP decomposition). Because of optimal exercise, iterative scheme may be necessary though.

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    $\begingroup$ Very nice summary, especially the paragraph PDEs and Option Values, that's a very neat and concise linkage of risk-free returns and pricing PDEs! $\endgroup$ – Daneel Olivaw Feb 9 at 15:58
  • $\begingroup$ @DaneelOlivaw Thank you very much. :) It's a really nice three-line-argument to write down the PDE for a particular model and it shows, yet again, how all the pieces in finance fit nicely together. However, I guess the standard derivation via dynamic hedging has the benefit that it directly tells you how your replicating portfolio looks like. $\endgroup$ – Kevin Feb 9 at 17:14

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