Abstract:
Universal classes of dynamic processes based on neural networks and signature methods have recently entered the area of stochastic modeling and Mathematical Finance. This has opened the door to robust and more data-driven model selection mechanisms, while first principles like no arbitrage still apply.
In the first part of the talk we focus on signature SDEs whose characteristics are linear functions of a primary underlying process, which can range from a (market-inferred) Brownian motion to a general multidimensional tractable stochastic process. The framework is universal in the sense that any classical model can be approximated arbitrarily well and that the model characteristics can be learned from all sources of available data by simple methods. Indeed, we derive formulas for the expected signature in terms of the expected signature of the primary underlying process.
In the second part we focus on a stochastic volatility model where the dynamics of the volatility are described by linear functions of the (time extended) signature of a primary underlying process. Under the additional assumption that this primary process is of polynomial type, we obtain closed form expressions for the squared VIX by exploiting the fact that the truncated signature of a polynomial process is again a polynomial process. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log-price and the squared VIX as linear functions of the signature of the corresponding augmented process. For both SPX and VIX options we obtain highly accurate calibration results.
The talk is based on joint works with Christa Cuchiero, Guido Gazzani, and Janka Möller.