Exponentiation of Conditional Expectations Under Stochastic Volatility

We use the Alòs Decomposition Formula 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... [ view full abstract ]

We use the Alòs Decomposition Formula 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.