Forecasting trends with asset prices

Performance Analysis of the Optimal Strategy Under Partial Information

“…The figure 2 represents the asymptotic Sharpe ratio of the optimal strategy with partial information as a function of the trend mean reversion speed λ and of the signal-to-noise ratio. Since λ ∈ [1,252] and SNR< 1, Equation (24) is satisfied and this Sharpe ratio is an increasing function of SNR and a decreasing function of λ. Moreover, the maximal value is inferior to 0.2.…”

“…If the signalto-noise ratio is inferior to 1, which corresponds to a trend standard deviation inferior to the volatility of the risky asset, the asymptotic Sharpe ratio of the optimal strategy with complete information is inferior to 0.5. Now, suppose that λ ∈ [1,252] and that the trend is an unobservable process. The figure 2 represents the asymptotic Sharpe ratio of the optimal strategy with partial information as a function of the trend mean reversion speed λ and of the signal-to-noise ratio.…”

“…This corresponds to the steady-state Kalman filter. The following proposition (see [1] for a proof) gives a first continuous representation of the steadystate Kalman filter:…”

“…Here, we find that the steady-state trend estimatorμ is an Ornstein Uhlenbeck process. In practice, the parameters (λ, σ µ , σ S ) are unknown and must be estimated (see [1] where the authors assess the feasibility of forecasting trends modeled by an unobserved meanreverting diffusion). In this paper, we assume that the parameters are known.…”