Fractional order stochastic differential equation driven by a fractional Brownian motion (FSDE) is defined as follows:
For comparison, the hedge effectiveness index [e.sup.*] with different lag hedge ratios [h.sup.*.sub.t-[tau]], [tau] = 0,1,2 ([tau] = 0 represents contemporaneous time, [tau] = 1 is 1-day-later lag time, and [tau] = 2 is 2-day-later lag time), and then evaluate the variance reduction over the conventional MV method, the hedge ratio of MV models in (7), the hedge ratio of SDE model in (13), and the hedge ratio of FSDE model in (32).
Both SDE model and FSDE model outperform the conventional MV model, with the improvement over the MV model by 0.94% and 0.52%, respectively.
To test our models and address the effectiveness of our SDE model and FSDE model, it may be much interesting by consideration and application when the futures are less correlated with the underlying asset.
The hedging effectiveness of the SDE-MV model outperforms the MV model and the FSDE model; thus the SDE-MV model performs better both in sample and out of sample when the Hurst parameter of the spot is larger than the futures.
The frame termination method aims to enhance the efficiency and the stability of the FSDE. The reader terminates the current frame depending on whether the efficiency of the remaining slots is less than a threshold value or not.
If [[eta].sub.ir]<min(li,ri), the reader terminates the current frame and allocates a new one whose length is determined by the FSDE based on [n.sub.ir].
In order to verify the performance of the FSDE with AFE, the efficiency is compared among three methods, i.e., the FSDE with AFE, the Q algorithm, and FSDE.
Step3: Making use of [r.sub.ip], the reader determines the length [v.sub.i+1] of the frame [F.sub.i+1] by the FSDE and runs the AFE in the end of each slot of frame [F.sub.i+1.]