MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

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

Hoang Duc Luu and Jürgen Jost


We develop a general mathematical framework, based on rough path theory, that can incorporate the empirically observed nonlinear mean-variance relation of the logarithmic return in a systematic manner. This model offers the possibility of an additional noise hidden in the rough path lift, hence supporting the idea of mixture of a Gaussian noise that is close to a standard Brownian motion and another source of long memory noise (a fractional Brownian motion for instance), that can account for the multi-scaling phenomenon in financial data. The no-arbitrage 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 rough path lifts, as well as the problem of cooperation. Our models are supported by empirical evidence from financial data and in particular, can explain some stylized fact (a parabolic lower bound of a mean-variance relation) that has not been explained before.

Nov 19, 2019
Dec 3, 2019
stock price, expected return, volatility, noise, rough path theory, rough differential equations, no-arbitrage, risk

Related publications

2023 Repository Open Access
Hoang Duc Luu and Jürgen Jost

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

In: SIAM journal on financial mathematics, 14 (2023) 3, pp. 879-909