These notes were originally written for the Stochastic Analysis Seminar in the Department of Operations Research and Financial Engineering at Princeton University, in February of 2011. The seminar was attended and supported by members of the Research Training Group, with the author being...

Volatility products have become popular in the past 15 years as a hedge against market uncertainty. In particular, there is growing interest in options on the VIX volatility index. A number of recent empirical studies have examine whether there is significantly greater risk premium in VIX...

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility's hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be...

We explore the inversion of derivatives prices to obtain an implied probability measure on volatility's hidden state. Stochastic volatility is a hidden Markov model (HMM), and HMMs ordinarily warrant filtering. However, derivative data is a set of conditional expectations that are already...

This article explores the relationship between option markets for the S&P500 (SPX) and CBOE's Volatility Index (VIX). Results are obtained by using the so-called time-spread portfolio to replicate a future contract on the squared VIX. The time-spread portfolio is interesting because it provides...

We analyze the Merton portfolio optimization problem when the growth rate is an unobserved Gaussian process whose level is estimated by filtering from observations of the stock price. We use the Kalman filter to track the hidden state(s) of expected returns given the history of asset prices, and...

This paper shows how to recover stochastic volatility models (SVMs) from market models for the VIX futures term structure. Market models have more flexibility for fitting of curves than do SVMs, and therefore they are better-suited for pricing VIX futures and derivatives. But the VIX itself is a...