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Module[{$CellContext`bPath$, $CellContext`maxPath$, \ $CellContext`ddPath$, $CellContext`maxDD$}, SeedRandom[$CellContext`seed$$]; $CellContext`bPath$ = Accumulate[ RandomReal[ NormalDistribution[$CellContext`mu$$/250, $CellContext`sig$$/Sqrt[ 250]], $CellContext`days$$]]; $CellContext`maxPath$ = Drop[ FoldList[Max, 0, $CellContext`bPath$], 1]; $CellContext`ddPath$ = $CellContext`maxPath$ - \ $CellContext`bPath$; $CellContext`maxDD$ = Drop[ FoldList[Max, 0, $CellContext`ddPath$], 1]; ListLinePlot[{$CellContext`bPath$, $CellContext`maxDD$}, AxesLabel -> {"Days", "Return (cumulative %)"}, Filling -> Axis, ImageSize -> {450, 300}, ImagePadding -> {{45, 45}, {25, 25}}, PlotRange -> All]], "Specifications" :> {{{$CellContext`mu$$, 6.5, "mean (annual %)"}, -20, 40, 0.1, Appearance -> "Labeled"}, {{$CellContext`sig$$, 7., "standard deviation (annual %)"}, 1, 25, 0.5, Appearance -> "Labeled"}, {{$CellContext`days$$, 765, "days"}, 25, 1000, 1, Appearance -> "Labeled"}, {{$CellContext`seed$$, 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The jagged blue line is the cumulative return of the \ daily return series. The red line is the maximum drawdown to date of the \ series. Adjusting the mean and standard deviation sliders demonstrates how \ the cumulative return and maximum drawdown change with respect to these \ parameters for a given underlying set of random shocks. Adjusting the \"new \ random case\" slider allows you to see different random cases to get a sense \ of how variable a return series can be for a single set of parameters. Note \ that in general the greater the ratio of mean to standard deviation, the \ smoother the return evolution is and the smaller the maximum drawdown is.\ \>", "Text"] }, Close]] }, Open ]], Cell[CellGroupData[{ Cell["DETAILS", "Section", CellFrame->{{0, 0}, {1, 0}}, CellFrameColor->RGBColor[0.87, 0.87, 0.87], FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.597406, 0, 0.0527047]], Cell[TextData[{ "A geometric Brownian motion is the ", StyleBox["de facto ", FontSlant->"Italic"], "standard model for stock price evolution. It is locally represented by the \ simple stochastic differential equation" }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ FractionBox["dS", "S"], TraditionalForm]], "InlineMath"], "=\[Mu]dt+", "\[Sigma]dz" }], "Text"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ StyleBox["S", FontSlant->"Plain"], TraditionalForm]], "InlineMath"], " is the stock price, \[Mu] is the \"drift\" parameter and \[Sigma] is the \ standard deviation. The equation says that over independent time increments \ of size \[CapitalDelta]t, the stock price's fractional change is normally \ distributed with mean \[Mu]\[CapitalDelta]t and standard deviation \[Sigma]", Cell[BoxData[ FormBox[ SqrtBox["\[CapitalDelta]t"], TraditionalForm]], "InlineMath"], ". 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The plot shows that \ the maximum drawdown is tightly linked to the information ratio of the series.\ \>", "Text"] }, Close]], Cell[CellGroupData[{ Cell["THIS NOTEBOOK IS THE SOURCE CODE FROM", "Text", CellFrame->{{0, 0}, {0, 0}}, CellMargins->{{48, 10}, {4, 28}}, CellGroupingRules->{"SectionGrouping", 25}, CellFrameMargins->{{48, 48}, {6, 5}}, CellFrameColor->RGBColor[0.87, 0.87, 0.87], FontFamily->"Helvetica", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.597406, 0, 0.0527047]], Cell[TextData[{ "\"", ButtonBox["Brownian Motion Path and Maximum Drawdown", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/\ BrownianMotionPathAndMaximumDrawdown/"], None}, ButtonNote-> "http://demonstrations.wolfram.com/BrownianMotionPathAndMaximumDrawdown/"],\ "\"", " from ", ButtonBox["the Wolfram Demonstrations Project", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/"], None}, ButtonNote->"http://demonstrations.wolfram.com/"], "\[ParagraphSeparator]\[NonBreakingSpace]", ButtonBox["http://demonstrations.wolfram.com/\ BrownianMotionPathAndMaximumDrawdown/", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/\ BrownianMotionPathAndMaximumDrawdown/"], None}, ButtonNote-> "http://demonstrations.wolfram.com/BrownianMotionPathAndMaximumDrawdown/"] }], "Text", CellMargins->{{48, Inherited}, {0, Inherited}}, FontFamily->"Verdana", FontSize->10, FontColor->GrayLevel[0.5]], Cell[TextData[{ "Contributed by: ", ButtonBox["Neil Chriss", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/author.html?author=Neil+Chriss"], None}, ButtonNote-> "http://demonstrations.wolfram.com/author.html?author=Neil+Chriss"] }], "Text", CellDingbat->"\[FilledSmallSquare]", CellMargins->{{66, 48}, {2, 4}}, FontFamily->"Verdana", FontSize->10, FontColor->GrayLevel[0.6], CellID->29868623], Cell[CellGroupData[{ Cell[TextData[{ "A full-function Wolfram ", StyleBox["Mathematica", FontSlant->"Italic"], " system (Version 6 or higher) is required to edit this notebook.\n", StyleBox[ButtonBox["GET WOLFRAM MATHEMATICA \[RightGuillemet]", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.wolfram.com/products/mathematica/"], None}, ButtonNote->"http://www.wolfram.com/products/mathematica/"], FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[1, 0.42, 0]] }], "Text", CellFrame->True, CellMargins->{{48, 68}, {8, 28}}, CellFrameMargins->12, CellFrameColor->RGBColor[0.87, 0.87, 0.87], CellChangeTimes->{3.3750111182355957`*^9}, ParagraphSpacing->{1., 1.}, FontFamily->"Verdana", FontSize->10, FontColor->GrayLevel[0.411765], Background->RGBColor[1, 1, 1]], Cell[TextData[{ "\[Copyright] ", StyleBox[ButtonBox["Wolfram Demonstrations Project & Contributors", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/"], None}, ButtonNote->"http://demonstrations.wolfram.com/"], FontColor->GrayLevel[0.6]], "\[ThickSpace]\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", StyleBox[ButtonBox["Terms of Use", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/termsofuse.html"], None}, ButtonNote->"http://demonstrations.wolfram.com/termsofuse.html"], FontColor->GrayLevel[0.6]], "\[ThickSpace]\[ThickSpace]\[ThickSpace]|\[ThickSpace]\[ThickSpace]\ \[ThickSpace]", StyleBox[ButtonBox["Make a new version of this Demonstration \ \[RightGuillemet]", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/participate/upload.jsp?id=\ BrownianMotionPathAndMaximumDrawdown"], None}, ButtonNote->None], FontColor->GrayLevel[0.6]] }], "Text", CellFrame->{{0, 0}, {0, 0.5}}, CellMargins->{{48, 10}, {20, 50}}, CellFrameMargins->{{6, 0}, {6, 6}}, CellFrameColor->GrayLevel[0.6], FontFamily->"Verdana", FontSize->9, FontColor->GrayLevel[0.6]] }, Open ]] }, Open ]] }, Editable->True, Saveable->False, ScreenStyleEnvironment->"Working", CellInsertionPointCell->None, WindowSize->{710, 650}, WindowMargins->{{Inherited, Inherited}, {Inherited, 0}}, WindowElements->{ "StatusArea", "MemoryMonitor", "MagnificationPopUp", "VerticalScrollBar", "MenuBar"}, WindowTitle->"Brownian Motion Path and Maximum Drawdown - 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