How rough path lifts affect expected return and volatility: a rough model under transaction cost
Hoang Duc Luu and Jürgen Jost
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Submission date: 19. Nov. 2019 (revised version: May 2023)
Keywords and phrases: stock price, expected return, volatility, noise, rough path theory, rough differential equations, no-arbitrage, risk
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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.