Stochastic Collocation - Old And New
Thomas Roos recently put a preprint on SSRN called Simple, Flexible, Analytic, Arbitrage Free Volatility Interpolation. Being interested in the subject, I had a detailed look at it. It turns out that Thomas stumbled upon spline stochastic collocation without realizing it.
There are a few differences in his approach:
- The optimization is on the x’s instead of the y’s, meaning the strike axis is fixed.
- An original approach to avoid spurious modes, although I would have liked more details on it, with concrete examples of the penalty. Penalties are often challenging to get right.
- A nice analysis of the asymptotic behavior in the wings, showing that an exponentially quadratic form for extrapolation corresponds to linear slopes in implied variance.
I had also explored a fixed strike axis initially (even back in 2014 - I uploaded some old notes on SSRN around this). I found it to be somewhat unstable back then, but it may well be that I did not put enough efforts into it, especially since optimizing the y’s instead of the x’s allowed for a straightforward use of B-splines, which is very attractive. It is very direct to impose monotonicity with B-splines.
The main advantage of optimizing on the x’s instead of the y’s is that the knots are defined in the more usual strike space.
Now I find interesting that someone else stumbled upon essentially the same idea starting from another point of view. There may be more merits to this parameterization than I thought.
The challenges I found with stochastic collocations were:
- spurious spike when the collocation function derivative is close to zero which happens on occasion. This is not really like an extra mode - it is not smooth. Penalty or minimal slope are possible workarounds, but challenging to make robust.
- dependency on the choice of knots, especially if nearly exact interpolation is required.
- more minor, non-predictable runtime: the non-linear optimization is sometimes very fast, sometimes much slower. This is the case of most arbitrage-free interpolation techniques.