Jim Gatheral is a researcher in the field of mathematical finance , who has contributed to the study of volatility as applied to the pricing and risk management of derivatives. A recurrent subject in his books and papers is the volatility smile , and he published in a book The Volatility Surface based on a course he taught for six years at New York University , along with Nassim Taleb. More recently his work has moved in the direction of market microstructure , especially as applied to algorithmic trading. In March ,  Jim Gatheral left his position at Merrill Lynch to assume a tenured full professor position at the Financial Engineering Masters Program  at Baruch College ,  where he is teaching volatility surface modeling and market microstructure. Prior to this, he worked at Bank of America and Bankers Trust  before heading the Equity Quantitative Analytics group at Merrill Lynch in , where he was a managing director for 17 years.
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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. 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. 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. SVI is thus shown to provide a parsimonious but realistic description of the volatility surface, facilitating analysis of its dynamics. 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. The Volatility Surface March Abstract These lectures will survey recent work on the parameterization of volatility surfaces and the modeling of their dynamics.
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The Department of Mathematics
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Prof. Jim GATHERAL (Baruch College, City University of New York)