Exponentiation of Conditional Expectations Under Stochastic Volatility

28 Pages Posted: 8 Jun 2017 Last revised: 20 Oct 2018

See all articles by Elisa Alos

Elisa Alos

University of Pompeu Fabra - Department of Economics

Jim Gatheral

CUNY Baruch College

Rados Radoicic

CUNY Baruch College

Date Written: October 16, 2018

Abstract

We use the Itô Decomposition Formula (see Alòs (2012)) to express certain conditional expectations as exponentials of iterated integrals. As one application, we compute an exact formal expression for the leverage swap for any stochastic volatility model expressed in forward variance form. As another, we show how to extend the Bergomi Guyon expansion to all orders in volatility of volatility. Finally, we compute exact expressions under rough volatility, obtaining in particular the fractional Riccati equation for the rough Heston characteristic function. As a corollary, we compute a closed-form expression for the leverage swap in the rough Heston model which can be used for fast calibration.

Keywords: Stochastic volatility, Conditional expectations, Exponentiation, Rough volatility

JEL Classification: C2, C3, C4, C6

Suggested Citation

Alos, Elisa and Gatheral, Jim and Radoicic, Rados, Exponentiation of Conditional Expectations Under Stochastic Volatility (October 16, 2018). Available at SSRN: https://ssrn.com/abstract=2983180 or http://dx.doi.org/10.2139/ssrn.2983180

Elisa Alos

University of Pompeu Fabra - Department of Economics ( email )

c/o Ramon Trias Fargas 25-27
08005 Barcelona
Spain
34 93 542 19 25 (Phone)
34 93 542 17 46 (Fax)

Jim Gatheral (Contact Author)

CUNY Baruch College ( email )

Department of Mathematics
One Bernard Baruch Way
New York, NY 10010
United States

Rados Radoicic

CUNY Baruch College ( email )

One Bernard Baruch Way
New York, NY 10010
United States

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