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Estimate Process Parameters from Data

Estimate Process Parameters from Data

The following data represents daily returns for a hypothetical stock simulated from a FractionalBrownianMotionProcess from January 1995 to January 2005.

Wolfram Language code: stock = TemporalData[Automatic, {{{-0.3899417643995551, -0.38462203153532065, -0.6033605470747778, -0.668664481720854, -1.1671704895998636, -1.2002033110763917, -1.4639760171102851, -1.337626834769846, -1.2029070005703169, -0.8678385639814625, -0 ... 45.72073826799363, 45.41777128863214, 45.80619096502528, 45.987339287154356}}, {{2997907200, 3313526400, 86400}}, 1, {"Continuous", 1}, {"Continuous", 1}, 1, {ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}}}, False, 9.];
Wolfram Language code: Labeled[DateListPlot[stock["Path"], PlotStyle -> Thick, ImageSize -> 400, Joined -> True, Filling -> -30], Column[{Style["Simulated Daily Returns", Bold, FontSize -> 18, FontFamily -> "Helvetica"], Style["1995 to 2005", Bold, FontSize -> 16, FontColor -> Gray, FontFamily -> "Helvetica"]}], {{Top, Left}}]
The estimated Hurst exponent suggests a long memory process:
Wolfram Language code: pars = FindProcessParameters[stock, FractionalBrownianMotionProcess[h]]
This is exemplified by the CorrelationFunction of the estimated process:
Wolfram Language code: ListPlot[Table[CorrelationFunction[FractionalBrownianMotionProcess[h] /. pars, 1, j], {j, 100}], Filling -> Axis, PlotRange -> All]