Barrier options under negative rates: complex numbers to the rescue

I stumbled upon an unexpected problem: the one touch barrier formula can break down under negative rates. While negative rates can sound fancy, they are actually quite real on some markets. Combined with relatively low volatilities, this makes the standard Black-Scholes one touch barrier formula blow up because somewhere the square root of a negative number is taken.

At first, I had the idea to just floor the number to 0. But then I needed to see if this rough approximation would be acceptable or not. So I relied on a TR-BDF2 discretization of the Black-Scholes PDE, where negative rates are not a problem.

Later, I was convinced that we ought to be able to find a closed form formula for the case of negative rates. I went back to the derivation of the formula, the book from Kwok is quite good on that. The closed form formula just stems from being the solution of an integral of the first passage time density (which is a simpler way to compute the one touch price than the PDE approach). It turns out that, then, the closed form solution to this integral with negative rates is just the same formula with complex numbers (there are actually some simplifications then).

It is a bit uncommon to use the cumulative normal distribution on complex numbers, but the error function on complex numbers is more popular: it's actually even on the wikipedia page of the error function. And it can be computed very quickly with machine precision thanks to the Faddeeva library.

With this simple closed form formula, there is no need anymore for an approximation. I wrote a small paper around this here.

Later a collegue made the remark that it could be interesting to have the bivariate complex normal distribution for the case of partial start one touch options or partial barrier option rebates (not sure if those are common). Unfortunately I could not find any code or paper for this. And after asking Prof. Genz (who found a very elegant and fast algorithm for the bivariate normal distribution), it looks like an open problem.