Exponentiation of Conditional Expectations Under Stochastic Volatility
Quantitative Finance, Vol. 20, No. 1, 13-27, 2020.
28 Pages Posted: 8 Jun 2017 Last revised: 9 Jan 2020
Date Written: October 16, 2018
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: Suggested Citation