Libor and the Negative Eigenvalue Trap

In my previous post Negative Eigenvalues in Practical Finance I gave a few examples of variance-covariance matrices that are not positive definite. One of them was using a wrong correlation ansatz in the Libor market model. What happens if you as a quant are requested to write a numerical solver for such an equation with possibly negative eigenvalues?

The model problem
We consider the following partial differential equation with initial and boundary conditions

When both eigenvalues (lambda_1, lambda_2) are positive, this is an anisotropic heat equation that can be solved numerically, e.g., by a finite difference scheme. If the Courant-Friedrichs-Levy condition (that restricts the length of the time step) is satisfied, even an explicit time-stepping scheme is stable and will converge.

Initial Condition for the model problem.

All eigenvalues positive
If we set lambda_1 = 1, lambda_2 = 0.1, and take 50 grid points in every space direction and a time step of 1/10000, then we obtain at t = 1

Note the scale: The Dirichlet condition V=0 at all boundaries draws the solution towards zero.

Second eigenvalue negative
For lambda_1 = 1, lambda_2 = - 0.1, we obtain for t=0.02, 0.04, 0.06:

The solution explodes. The scale on the third plot is 10^19.

Negative eigenvalue closer to zero
Maybe the second eigenvalue was "too negative"? What happens for lambda_2 = - 0.01?


No chance to stabilize it
The reason for these oscillations does not lie in the explicit scheme but in the ill-posedness of the equation. In the Traunsee example, the instability of the backwards heat equation was analysed in more detail.