Prof. Jim GATHERAL (Baruch College, City University of New York)

The Volatility Surface (March 16-17-19 2015)

Abstract

These lectures will survey recent work on the parameterization of volatility surfaces and the modeling of their dynamics.  After reviewing the basics of volatility modeling, I will motivate the SVI ("stochastic volatility inspired") parameterization of the volatility surface.  I will show how to fit SVI to option prices whilst ensuring no static arbitrage.  We will see that volatility surfaces have a characteristic shape that is not well described by conventional Markovian stochastic volatility models, with or without jumps.  I will then review our recent econometric analysis of the time series of realized variance, working out its implications for options pricing.  I will demonstrate the remarkable consistency of the resulting non-Markovian stochastic volatility model with both the historical time series of realized variance and the volatility surface.  Last but not least, example R code will be provided to illustrate the main points.

Lecture 1. The Volatility Surface: Static and Dynamics (PDF)

(March 16, 17:30 - 19:30, Lecture Theatre 340)

In the first lecture I will start with a brief introduction to R and iPython notebook.  After defining the volatility surface, I will plot examples of typical volatility surfaces.  Using the Bergomi-Guyon expansion, I will show how various features of the volatility surface relate to the joint dynamics of the volatility surface and the underlying. We will see that conventional Markovian stochastic volatility models are consistent neither with observed characteristics of the volatility time series nor with the shape of the volatility surface.  iPython notebooks for all three lectures will be made available afterwards for easy reproduction of much of the analysis.

• Download related documents - lecture 1

Lecture 2. The SVI Arbitrage-Free Volatility Surface Parameterization (PDF)

(March 17, 17:30 - 19:30, Clore Lecture Theatre)

In the second lecture I will show how to calibrate the widely-used SVI parameterization of the implied volatility surface in such a way as to guarantee the absence of static arbitrage. In particular, I will exhibit a large class of arbitrage-free SVI volatility surfaces with simple closed-form representations.  I will demonstrate the high quality of typical SVI fits with numerical examples using recent SPX options data.  SVI is thus shown to provide a parsimonious but realistic description of the volatility surface, facilitating analysis of its dynamics. 

• Download related documents - lecture 2

Lecture 3. Rough Volatility (PDF)
Rough Volatility AND THE CONNECTION BETWEEN HISTORICAL AND IMPLIED VOLATILITY

(March 19, 17:30 - 19:30, Lecture Theatre 308)

In the final lecture I will present our recent work on rough volatility.  We will explore further the time series of historical volatility, studying its scaling properties which we will find lead to a natural model for the underlying, the RFSV model.  We then show how the RFSV model can be used to price claims on both the underlying and integrated volatility.  We will analyze in detail a simple case of this model, the rBergomi model. In particular, we will find that the rBergomi model fits the SPX volatility surface markedly better than conventional Markovian stochastic volatility models, and with fewer parameters. Finally using SVI fits, we show that actual SPX variance swap curves seem to be consistent with model forecasts, with particular dramatic examples from the weekend of the collapse of Lehman Brothers and the Flash Crash.  

• Download related documents - lecture 3

References:

1. Arbitrage-Free SVI Volatility Surfaces, with Antoine Jacquier, Quantitative Finance, 14(1) 59-71, (2014).
2. Volatility is Rough, with Thibault Jaisson and Mathieu Rosenbaum,  SSRN (2014)
3. Pricing under Rough Volatility, with Christian Bayer and Peter Friz, SSRN (2015)