Monday, November 09, 2015

Broken Internet?

There is something funny going on with upcoming generic top level domains (gTLDs), they seem to be looked up in a strange manner (at least on latest Linux). For example:

ping chrome 


ping nexus 


While existing official gTLDs don't (ping dental returns "unknown host" as expected). I first thought it was a network misconfiguration, but as I am not the only one to notice this, it's likely a genuine internet issue.

Strange times.

Sunday, November 01, 2015

Holiday's read - DFW - Everything and more

I am ambivalent towards David Foster Wallace. He can write the most creative sentences and make innocuous subjects very interesting. At the same time, i never finished his book Infinite Jest, partly because the characters names are too awkward for me so that i never exactly remember who is who, but also because the story itself is a bit too crazy.
I knew however that a non fiction book on the subject of infinity written by him would make for a very interesting read. And I have not been disappointed. It's in between maths and philosophy going back to the Greeks up to Gödel through a lot of Cantor following more or less the historical chronology.
Most of it is easy to read and follow, except the last part around sets and transfinite numbers. This last part is actually quite significant as it tries to explain why we still have no satisfying theory around the problems raised by infinity especially in the context of a Sets theory. I did not expect to learn much around the subject, I was disappointed. The book showed me how naive I was and how tricky the concept of infinity can be.
While I found the different explanations around Zeno's paradox of the arrow very clever, there is one other view possible: the arrow really does not move at each instant (you could think of those as a snapshot) but an interval of time is just not a simple succession of instants. This is not so far of Aristotle attack, but the key here is around what is an interval really. DFW suggests slightly this interpretation as well p144 but it's not very explicit.
I had not heard about Kronecker's conception that only integers were mathematically real (against decimals, irrationals, infinite sets). I find it very appropriate in the frame of computer science. Everything ends up as finite integers (a binary representation) and we are always confronted to the process of transforming the continuous, that despite all its conceptual issues is often simpler to reason in to solve concrete problems, to the finite discrete.

Wednesday, September 30, 2015

Crank-Nicolson and Rannacher Issues with Touch options

I just stumbled upon this particularly illustrative case where the Crank-Nicolson finite difference scheme behaves badly, and the Rannacher smoothing (2-steps backward Euler) is less than ideal: double one touch and double no touch options.

It is particularly evident when the option is sure to be hit, for example when the barriers are narrow, that is our delta should be around zero as well as our gamma. Let's consider a double one touch option with spot=100, upBarrier=101, downBarrier=99.9, vol=20%, T=1 month and a payout of 50K.
Crank-Nicolson shows big spikes in the delta near the boundary
Rannacher shows spikes in the delta as well
Crank-Nicolson spikes are so high that the price is actually a off itself.

The Rannacher smoothing reduces the spikes by 100x but it's still quite high, and would be higher had we placed the spot closer to the boundary. The gamma is worse. Note that we applied the smoothing only at maturity. In reality as the barrier is continuous, the smoothing should really be applied at each step, but then the scheme would be not so different from a simple Backward Euler.

In contrast, with a proper second order finite difference scheme, there is no spike.
Delta with the TR-BDF2 finite difference method - the scale goes from -0.00008 to 0.00008.
Delta with the Lawson-Morris finite difference scheme - the scale goes from -0.00005 to 0.00005
Both TR-BDF2 and Lawson-Morris (based on a local Richardson extrapolation of backward Euler) have a very low delta error, similarly, their gamma is very clean. This is reminiscent of the behavior on American options, but the effect is magnified here.