How signatures affect expected return and volatility: a rough model under transaction cost
Hoang Duc Luu and Jürgen Jost
Contact the author: Please use for correspondence this email.
Submission date: 19. Nov. 2019 (revised version: December 2019)
Keywords and phrases: stock price, expected return, volatility, noise, rough path theory, rough differential equations, no-arbitrage, risk
Download full preprint: PDF (2032 kB)
We develop a general mathematical framework, based on rough path theory, a recent important extension of the classical Itô calculus, that can incorporate the empirically observed nonlinear relation between the expected logarithmic return and its variance in a systematic manner. Thus, we propose a stock price model driven by a Hölder continuous noise, understood in the sense of a rough differential equation. This model offers the possibility of additional noises hidden in the signatures of rough paths, hence supporting the idea of mixture of a stan-dard Brownian noise and another source of long memory noise (a fractional Brownian motion for instance), and enabling to account for the multi-scaling phenomenon in financial data. The noarbitrage principle is then satisfied under the assumption of transaction costs as long as the driving noise is a sticky process. We also discuss the potential risk of model uncertainty where the ambiguity comes from the signatures of rough paths. Our models are supported by the numerical data from stock indices.