Advanced Smile Asymptotics: Geometry, Geodesics, and All That

This is the title of Chapter 12 of  “Option Valuation under Stochastic Volatility II”. Because of some interest in this chapter, I decided to offer it as a stand-alone Kindle ebook at the various Amazons around the world. The image below links to

This was an interesting exercise for me as it was my first attempt at a Kindle book and I learned some things.

First, I had been somewhat reluctant to embrace that format as I thought that fairly complex book material (equations, graphics, etc.) just didn’t work. However, using the “Print replica” format, which I did, you just get the original pdf displayed. Viewing the results on the Kindle reader on my iPad, I think it’s quite readable. (You can ask Amazon to “Send you a free sample” to view it on your own Kindle app prior to any actual purchase). But, if anyone does experience readability problems on any device, I’d appreciate hearing about it.

Second, the process was quite fast. Within about three hours of starting, I had a book.

What’s this chapter about? It discusses the general principles for the construction of the limiting time-0 option smile in stochastic volatility models. It constructs in detail the limiting smile for the case where the volatility process is of CEV-type with an arbitrary volatility power p. Solutions are found using ideas from Riemannian geometry and related theory. It’s fascinating to see how basic Riemannian geometry of curved spaces — in particular, the geodesic distance from a point to a line — leads very naturally to the small-time behavior of option smiles.

From the cover background above, you can see that surfaces of positive, negative, and zero Gaussian curvature occur in the smile solution.

The ebook length is 76 pages and it’s also available in a traditional paperback version.

The Equity Risk Premium (forthcoming book) — slides posted

Yesterday I gave a Zoom talk for a Business school seminar at the Stevens Institute of Technology in New Jersey. My thanks go out to Dan Pirjol, Zack Feinstein, and the other folks from the Business school at Stevens, both for the invitation and attending.

I summarized results from my forthcoming book: “The Equity Risk Premium: Unified Modeling + Python Automation”.
If you’re interested in a preview, here are the slides:

Download “The Equity Risk Premium: slides: Stevens”

THE-EQUITY-RISK-PREMIUM.Stevens.pdf – Downloaded 1077 times – 1.92 MB

Proof of non-convergence of the short maturity expansion for the SABR model

The SABR model has been around for a relatively long time and it’s often studied via a small-maturity power series. In fact, all stochastic volatility models have a double power series expansion, for the Black-Scholes implied volatility, in the time-to-maturity T and the log-moneyness. With my coauthor Dan Pirjol, we prove that this series (taking the lognormal SABR model at the at-the-money point) is strictly asymptotic: non-convergent for any T>0. This new research preprint is posted to the arXiv here

Update: The final published version (in the journal Quantitative Finance) has limited (1st 50 clicks) free availability: here  

Update 2: Here is a zipped file containing the Mathematica code for most of the figures in the published article: FiguresPaper

For some background, the non-analyticity of the diffusion kernel factor, e^{-d^2/T} at T=0 leads many to believe strict asymptotics in T is the only possibility for option values and related solutions for diffusion models. As we explain in the paper, that’s not correct. For example, a simple case is the at-the-money Black-Scholes option value

V_{BS}(T) = {\text{erf}}(2^{-3/2} \sqrt{T}),

using the error function. After division by a \sqrt{T} , this is an entire function in the complex T-plane.

In principle, similar “nice” behavior could occur in the SABR model. That is, the value function or some simple normalized version of it, could have had a power series that converged for
|T| \lt R in the complex T-plane. Here \{ R: 0 \lt R \le \infty\} denotes a convergence radius. In fact, we prove this doesn’t happen.