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.

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