The elusive reference: the Lamperti transform

Without knowing that it was a well known general concept, I first noticed the use of the Lamperti transform in the Andersen-Piterbarg "Interest rate modeling" book p292 "finite difference solutions for general phi".

 

Pat Hagan used that transformation for a better discretization  of the arbitrage free SABR PDE model.

I then started to notice the use of this transformation in many more papers. The first one I saw naming it "Lamperti transform" was the paper from Ait-Sahalia "Maximum likelyhood estimation of discretely sampled diffusions: a closed-form approximation approach". Recently those closed form formulae have been applied to the quadrature method (where one integrates the transition density by a quadrature rule) in "Advancing the universality of quadrature methods to any underlying process for option pricing". There is also a recent interesting application to Monte-Carlo simulation in "Unbiased Estimation with Square Root Convergence for SDE Models".

So the range of practical applications is quite large. But there was still no reference. A google search pointed me to a well written paper that describes the application of the Lamperti transform to various stochastic differential equations, showing its limits "From State Dependent Diffusion to Constant Diffusion in Stochastic Differential Equations by the Lamperti Transform". 

Gary then blogged about the Lamperti transform and various papers from Lamperti, but does not say which one is the source.

After going through some, I noticed that Lamperti's 1964 "A simple construction of certain diffusion processes" seemed to be the closest, even though it seems to go beyond stochastic differential equations.

Today, I found a paper referencing this paper explicitly when presenting the transformation of a stochastic process to a unit diffusion in "Density estimates for solutions to one dimensional Backward SDE's". In addition it also references one exercise of the Karatzas-Schreve book "Brownian motion and Stochastic calculus", which presents again the same idea, without calling it Lamperti transform.

The elusive reference: the Lamperti transform

Without knowing that it was a well known general concept, I first noticed the use of the Lamperti transform in the Andersen-Piterbarg "Interest rate modeling" book p292 "finite difference solutions for general phi".

 

Pat Hagan used that transformation for a better discretization  of the arbitrage free SABR PDE model.

I then started to notice the use of this transformation in many more papers. The first one I saw naming it "Lamperti transform" was the paper from Ait-Sahalia "Maximum likelyhood estimation of discretely sampled diffusions: a closed-form approximation approach". Recently those closed form formulae have been applied to the quadrature method (where one integrates the transition density by a quadrature rule) in "Advancing the universality of quadrature methods to any underlying process for option pricing". There is also a recent interesting application to Monte-Carlo simulation in "Unbiased Estimation with Square Root Convergence for SDE Models".

So the range of practical applications is quite large. But there was still no reference. A google search pointed me to a well written paper that describes the application of the Lamperti transform to various stochastic differential equations, showing its limits "From State Dependent Diffusion to Constant Diffusion in Stochastic Differential Equations by the Lamperti Transform". 

Gary then blogged about the Lamperti transform and various papers from Lamperti, but does not say which one is the source.

After going through some, I noticed that Lamperti's 1964 "A simple construction of certain diffusion processes" seemed to be the closest, even though it seems to go beyond stochastic differential equations.

Today, I found a paper referencing this paper explicitly when presenting the transformation of a stochastic process to a unit diffusion in "Density estimates for solutions to one dimensional Backward SDE's". In addition it also references one exercise of the Karatzas-Schreve book "Brownian motion and Stochastic calculus", which presents again the same idea, without calling it Lamperti transform.

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.