How signatures affect expected return and volatility: a rough model under transaction cost

We develop a general mathematical framework of stock prices based on rough path theory, a recent important extension of the classical Ito calculus. Thus, we propose a stock price model driven by a Hoelder continuous noise, understood in the sense of a rough differential equation. The no-arbitrage principle is then satisfied under the assumption of transaction costs as long as the driving noise is a sticky process.

This model offers the possibility of additional noises hidden in the signatures of rough paths, hence supporting the idea of mixture of a standard Brownian noise and another source of long memory noise, for instance a fractional Brownian motion. It also implies a nonlinear relation between the expectation and the variance of the logarithmic return. We show numerical evidence from stock indices and discuss the potential risk of model uncertainty where the ambiguity comes from the signatures of rough paths.

This is joint work with Jürgen Jost.