When Variance is Negative

Michael has been writing on correlation matrices in his recent posts. Is there also a relevance in option pricing?

Obviously, there are multi-asset options, e.g., equity basket options, on the market. If we assume that each of the underlyings follows an individual geometric Brownian motion (Black-Scholes world) with correlated Wiener processes, then the value of the option satisfies the multi-dimensional Black-Scholes equation

When the variance covariance matrix is positive definite, then this equation is a backwards heat equation. With appropriate end conditions (the payoff of the multi-asset option), it obtains a uniques solution which depends continuously on the data.

If the space dimension is not too large (not larger than 4 oder 5), it can be solved numerically by finite elements or by convolution of the payoff with the Green kernel of the PDE (a multi-dimensional lognormal density).

Alternatively, Monte Carlo techniques could be applied.

What happens if, for some reason, the variance-covariance matrix turns out to be not positive definite meaning that at least one of the eigenvalues is not positive. For the sake of simplicity of arguing, let this eigenvalue be negative.

Finite elements
In this case, the finite element method tries to solve an equation which is a backwards equation with end condition for the directions corresponding to the positive eigenvalues and a forwards equation with end condition for the directions corresponding to the negative eigenvalues. This is severely ill-posed as we pointed out in our Traunsee example.

Green's kernel
Naively applying the multidimensional normal distribution when one of the eigenvalues is negative leads to a density which is not a probability density as it does not integrate to one. (But I like the 3d plot.)
This is not exactly the bell shape one would expect.

Monte Carlo
Monte Carlo simulation typically takes uncorrelated random numbers and transforms them to the correlated ones by applying the (roughly speaking) Cholesky square root ot he variance covariance matrix. This is the good news: If an eigenvalue is negative, then the Cholesky decomposition will terminate with an error.

Practical relevance
But is this of practical relevance? We have learnt that a variance covariance matrix is at least positve semidefinite.

Coming next Wednesday: Negative eigenvalues in practical finance.