Abstract: We introduce an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier-Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati-Volterra equations, which can be efficiently approximated using a multi-factor approximation technique. Prices of options on the underlying and its volatility index can then be obtained using Fourier-inversion techniques. We show that a parsimonious setup, characterized by a power kernel and an exponential law for the jumps, is able to simultaneously capture the behavior of the implied volatility smile for both S&P 500 and VIX options. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self-exciting jumps in order to jointly calibrate S&P 500 and VIX smiles. This is joint work with Alessandro Bondi (Scuola Normale Superiore di Pisa) and Simone Scotti (Università di Pisa).

Zoom information:

https://us02web.zoom.us/j/5241873990?pwd=NW1abU84OVR3Sitxa3puZWlTUnNPdz09

Meeting ID: 524 187 3990

Passcode: mfap

Abstract:

Predictable Forward Performance Processes (PFPPs) are stochastic optimal control frameworks for an agent who controls a dynamically evolving system but can only prescribe the system dynamics for a short period ahead. This is a common scenario in which the controlling agent must re-calibrate her model for the underlying system periodically through time. PFPPs allow the agent to form time-consistent optimal policies over time horizons that span multiple estimation periods. In this talk, I prove the existence of PFPPs in complete markets and show that the main step in their construction is solving a one-period problem involving an integral equation. I will discuss a new solution method for this integral equation using the Fourier transform. For PFPPs with completely monotonic inverse marginal functions, the integral equation has a unique solution that is obtained in closed form.