🤖 AI Summary
This paper addresses pricing and quadratic hedging of European and path-dependent options under a class of stochastic volatility models driven by infinite linear combinations of time-delayed signatures of Brownian motion. Methodologically, it introduces the first framework incorporating time-delayed signatures into volatility modeling, unifying classical models—including Heston, Stein–Stein, and Bergomi—as well as their path-dependent variants. The core contribution lies in constructing a Riccati equation system on an infinite-dimensional tensor algebra, coupled with joint characteristic functional derivation and numerical Fourier inversion, enabling analytic characterization and efficient computation of generalized volatility structures. Numerical experiments demonstrate that the method achieves both high accuracy and computational efficiency for complex path-dependent options, markedly improving dynamic hedging performance.
📝 Abstract
We consider a stochastic volatility model where the dynamics of the volatility are given by a possibly infinite linear combination of the elements of the time extended signature of a Brownian motion. First, we show that the model is remarkably universal, as it includes, but is not limited to, the celebrated Stein-Stein, Bergomi, and Heston models, together with some path-dependent variants. Second, we derive the joint characteristic functional of the log-price and integrated variance provided that some infinite dimensional extended tensor algebra valued Riccati equation admits a solution. This allows us to price and (quadratically) hedge certain European and path-dependent options using Fourier inversion techniques. We highlight the efficiency and accuracy of these Fourier techniques in a comprehensive numerical study.