Exponentiation of conditional expectations under stochastic volatility

We use the Itô Decomposition Formula (see Alòs [A decomposition formula for option prices in the Heston model and applications to option pricing approximation. Finance Stoch., 2012, 16(3), 403–422]) 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 derive a smile expansion analogous to that of Bergomi and Guyon to all orders. 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.