ABSTRACT
In this chapter we introduce non-parametric methods for modeling the volatility surface and discuss their strengths and weakness. We begin with a review of the Breeden-Litzenberger technique and then proceed to develop an optimization-based approach where the probabilities are solved for. Weighted Monte Carlo, one such optimization framework, is then introduced. Emphasis is placed on the relationship between certain key options structures and the risk-neutral CDF and PDF respectively. Additionally, the relationship between an observed volatility smile, or skew, and the corresponding risk-neutral density are analyzed in detail. Importantly, we not only discuss the techniques for estimating a risk-neutral density from a set of options prices, but also discuss how one might juxtapose this with an econometrically driven estimate of the physical density. When doing this, significant emphasis is placed on the potential benefits and challenges of making such a comparison.
