(* Content-type: application/mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 7.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
NotebookFileLineBreakTest
NotebookDataPosition[       145,          7]
NotebookDataLength[     14913,        492]
NotebookOptionsPosition[     14356,        469]
NotebookOutlinePosition[     14699,        484]
CellTagsIndexPosition[     14656,        481]
WindowFrame->Normal*)

(* Beginning of Notebook Content *)
Notebook[{
Cell[BoxData[
 RowBox[{
  RowBox[{"V", "=", 
   RowBox[{
    RowBox[{"g", "[", 
     RowBox[{"r", ",", "t"}], "]"}], 
    RowBox[{"H", "[", 
     RowBox[{
      FractionBox["S", 
       RowBox[{"g", "[", 
        RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}]}], 
  ";"}]], "Input",
 CellChangeTimes->{{3.5015223162844534`*^9, 3.5015223597535677`*^9}, {
  3.5015224398011475`*^9, 3.501522440582407*^9}}],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
  "construct", " ", "the", " ", "convertable", " ", "bond", " ", "pricing", 
   " ", "equation"}], " ", "*)"}]], "Input",
 CellChangeTimes->{{3.5015223841444407`*^9, 3.501522396644537*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"CBPE", "=", 
  RowBox[{
   RowBox[{
    RowBox[{"D", "[", 
     RowBox[{"V", ",", 
      RowBox[{"{", 
       RowBox[{"t", ",", "1"}], "}"}]}], "]"}], "+", 
    RowBox[{
     FractionBox["1", "2"], 
     SuperscriptBox["\[Sigma]", "2"], 
     SuperscriptBox["S", "2"], 
     RowBox[{"D", "[", 
      RowBox[{"V", ",", 
       RowBox[{"{", 
        RowBox[{"S", ",", "2"}], "}"}]}], "]"}]}], "+", 
    RowBox[{"\[Rho]", " ", "\[Sigma]", " ", "S", " ", "w", " ", 
     RowBox[{"D", "[", 
      RowBox[{"V", ",", 
       RowBox[{"{", "S", "}"}], ",", 
       RowBox[{"{", "r", "}"}]}], "]"}]}], "+", 
    RowBox[{
     FractionBox["1", "2"], 
     SuperscriptBox["w", "2"], 
     RowBox[{"D", "[", 
      RowBox[{"V", ",", 
       RowBox[{"{", 
        RowBox[{"r", ",", "2"}], "}"}]}], "]"}]}], "+", 
    RowBox[{"r", " ", "S", " ", 
     RowBox[{"D", "[", 
      RowBox[{"V", ",", "S"}], "]"}]}], "+", " ", 
    RowBox[{
     RowBox[{"(", 
      RowBox[{"u", " ", "-", " ", 
       RowBox[{"\[Lambda]", " ", "w"}]}], ")"}], " ", 
     RowBox[{"D", "[", 
      RowBox[{"V", ",", "r"}], "]"}]}], "-", 
    RowBox[{"r", " ", "V"}]}], "//", "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.5015223750661993`*^9, 3.501522432519804*^9}, {
  3.5015224790202074`*^9, 3.50152249944228*^9}, {3.5015225389582186`*^9, 
  3.501522575552309*^9}, {3.5019325866969395`*^9, 3.5019325883063354`*^9}}],

Cell[BoxData[
 RowBox[{
  FractionBox["1", 
   RowBox[{"2", " ", 
    SuperscriptBox[
     RowBox[{"g", "[", 
      RowBox[{"r", ",", "t"}], "]"}], "3"]}]], 
  RowBox[{"(", 
   RowBox[{
    RowBox[{
     SuperscriptBox[
      RowBox[{"g", "[", 
       RowBox[{"r", ",", "t"}], "]"}], "4"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{
        RowBox[{"-", "2"}], " ", "r", " ", 
        RowBox[{"H", "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "+", 
       RowBox[{"2", " ", 
        RowBox[{
         SuperscriptBox["H", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}]}], ")"}]}], 
    "+", 
    RowBox[{
     SuperscriptBox[
      RowBox[{"g", "[", 
       RowBox[{"r", ",", "t"}], "]"}], "3"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{"2", " ", "r", " ", "S", " ", 
        RowBox[{
         SuperscriptBox["H", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "+", 
       RowBox[{
        RowBox[{"H", "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}], " ", 
        RowBox[{"(", 
         RowBox[{
          RowBox[{"2", " ", 
           RowBox[{
            SuperscriptBox["g", 
             TagBox[
              RowBox[{"(", 
               RowBox[{"0", ",", "1"}], ")"}],
              Derivative],
             MultilineFunction->None], "[", 
            RowBox[{"r", ",", "t"}], "]"}]}], "+", 
          RowBox[{"2", " ", 
           RowBox[{"(", 
            RowBox[{"u", "-", 
             RowBox[{"w", " ", "\[Lambda]"}]}], ")"}], " ", 
           RowBox[{
            SuperscriptBox["g", 
             TagBox[
              RowBox[{"(", 
               RowBox[{"1", ",", "0"}], ")"}],
              Derivative],
             MultilineFunction->None], "[", 
            RowBox[{"r", ",", "t"}], "]"}]}], "+", 
          RowBox[{
           SuperscriptBox["w", "2"], " ", 
           RowBox[{
            SuperscriptBox["g", 
             TagBox[
              RowBox[{"(", 
               RowBox[{"2", ",", "0"}], ")"}],
              Derivative],
             MultilineFunction->None], "[", 
            RowBox[{"r", ",", "t"}], "]"}]}]}], ")"}]}]}], ")"}]}], "-", 
    RowBox[{"2", " ", 
     SuperscriptBox["S", "2"], " ", "w", " ", "\[Rho]", " ", "\[Sigma]", " ", 
     RowBox[{"g", "[", 
      RowBox[{"r", ",", "t"}], "]"}], " ", 
     RowBox[{
      SuperscriptBox["g", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"1", ",", "0"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{"r", ",", "t"}], "]"}], " ", 
     RowBox[{
      SuperscriptBox["H", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"2", ",", "0"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{
       FractionBox["S", 
        RowBox[{"g", "[", 
         RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "+", 
    RowBox[{
     SuperscriptBox["S", "2"], " ", 
     SuperscriptBox["w", "2"], " ", 
     SuperscriptBox[
      RowBox[{
       SuperscriptBox["g", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"1", ",", "0"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"r", ",", "t"}], "]"}], "2"], " ", 
     RowBox[{
      SuperscriptBox["H", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"2", ",", "0"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{
       FractionBox["S", 
        RowBox[{"g", "[", 
         RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "+", 
    RowBox[{"S", " ", 
     SuperscriptBox[
      RowBox[{"g", "[", 
       RowBox[{"r", ",", "t"}], "]"}], "2"], " ", 
     RowBox[{"(", 
      RowBox[{
       RowBox[{
        RowBox[{"-", "2"}], " ", 
        RowBox[{
         SuperscriptBox["g", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"0", ",", "1"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"r", ",", "t"}], "]"}], " ", 
        RowBox[{
         SuperscriptBox["H", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "-", 
       RowBox[{"2", " ", 
        RowBox[{"(", 
         RowBox[{"u", "-", 
          RowBox[{"w", " ", "\[Lambda]"}]}], ")"}], " ", 
        RowBox[{
         SuperscriptBox["g", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"r", ",", "t"}], "]"}], " ", 
        RowBox[{
         SuperscriptBox["H", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "-", 
       RowBox[{
        SuperscriptBox["w", "2"], " ", 
        RowBox[{
         SuperscriptBox["H", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}], " ", 
        RowBox[{
         SuperscriptBox["g", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"2", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"r", ",", "t"}], "]"}]}], "+", 
       RowBox[{"S", " ", 
        SuperscriptBox["\[Sigma]", "2"], " ", 
        RowBox[{
         SuperscriptBox["H", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"2", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{
          FractionBox["S", 
           RowBox[{"g", "[", 
            RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}]}], ")"}]}]}],
    ")"}]}]], "Output",
 CellChangeTimes->{
  3.501522380035013*^9, {3.5015224252697754`*^9, 3.5015224330510607`*^9}, 
   3.5015224998172846`*^9, 3.501522543286399*^9, 3.5015225764116945`*^9, {
   3.5019326016814423`*^9, 3.5019326048221073`*^9}, 3.5019331118420467`*^9}]
}, Open  ]],

Cell[BoxData[
 RowBox[{"(*", " ", 
  RowBox[{
  "lets", " ", "put", " ", "in", " ", "the", " ", "fact", " ", "that", " ", 
   RowBox[{"g", "[", 
    RowBox[{"r", ",", "t"}], "]"}], " ", "satisfied", " ", "the", " ", "bond",
    " ", "pricing", " ", "equation", " ", "by", " ", "replacing", " ", "the", 
   " ", "time", " ", "derivative", " ", "of", " ", "g"}], " ", 
  "*)"}]], "Input",
 CellChangeTimes->{{3.5019325337902684`*^9, 3.501932579306284*^9}, {
  3.501932734932572*^9, 3.501932745073327*^9}}],

Cell[CellGroupData[{

Cell[BoxData[
 RowBox[{"t1", "=", 
  RowBox[{
   RowBox[{"CBPE", "/.", 
    RowBox[{"{", 
     RowBox[{
      RowBox[{
       SuperscriptBox["g", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"0", ",", "1"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"r", ",", "t"}], "]"}], "\[Rule]", 
      RowBox[{
       RowBox[{
        FractionBox[
         RowBox[{"-", "1"}], "2"], 
        SuperscriptBox["w", "2"], 
        RowBox[{
         SuperscriptBox["g", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"2", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"r", ",", "t"}], "]"}]}], "-", 
       RowBox[{
        RowBox[{"(", 
         RowBox[{"u", "-", 
          RowBox[{"\[Lambda]", " ", "w"}]}], ")"}], 
        RowBox[{
         SuperscriptBox["g", 
          TagBox[
           RowBox[{"(", 
            RowBox[{"1", ",", "0"}], ")"}],
           Derivative],
          MultilineFunction->None], "[", 
         RowBox[{"r", ",", "t"}], "]"}]}], "+", 
       RowBox[{"r", " ", 
        RowBox[{"g", "[", 
         RowBox[{"r", ",", "t"}], "]"}]}]}]}], "}"}]}], "//", 
   "Simplify"}]}]], "Input",
 CellChangeTimes->{{3.501932607369015*^9, 3.5019326263691416`*^9}, {
  3.5019326617288303`*^9, 3.501932724010621*^9}, {3.50193307215423*^9, 
  3.5019330727479877`*^9}, {3.501933652737291*^9, 3.501933653299798*^9}}],

Cell[BoxData[
 RowBox[{
  FractionBox["1", 
   RowBox[{"2", " ", 
    SuperscriptBox[
     RowBox[{"g", "[", 
      RowBox[{"r", ",", "t"}], "]"}], "3"]}]], 
  RowBox[{"(", 
   RowBox[{
    RowBox[{"2", " ", 
     SuperscriptBox[
      RowBox[{"g", "[", 
       RowBox[{"r", ",", "t"}], "]"}], "4"], " ", 
     RowBox[{
      SuperscriptBox["H", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"0", ",", "1"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{
       FractionBox["S", 
        RowBox[{"g", "[", 
         RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "+", 
    RowBox[{
     SuperscriptBox["S", "2"], " ", 
     SuperscriptBox["\[Sigma]", "2"], " ", 
     SuperscriptBox[
      RowBox[{"g", "[", 
       RowBox[{"r", ",", "t"}], "]"}], "2"], " ", 
     RowBox[{
      SuperscriptBox["H", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"2", ",", "0"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{
       FractionBox["S", 
        RowBox[{"g", "[", 
         RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "-", 
    RowBox[{"2", " ", 
     SuperscriptBox["S", "2"], " ", "w", " ", "\[Rho]", " ", "\[Sigma]", " ", 
     RowBox[{"g", "[", 
      RowBox[{"r", ",", "t"}], "]"}], " ", 
     RowBox[{
      SuperscriptBox["g", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"1", ",", "0"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{"r", ",", "t"}], "]"}], " ", 
     RowBox[{
      SuperscriptBox["H", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"2", ",", "0"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{
       FractionBox["S", 
        RowBox[{"g", "[", 
         RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}], "+", 
    RowBox[{
     SuperscriptBox["S", "2"], " ", 
     SuperscriptBox["w", "2"], " ", 
     SuperscriptBox[
      RowBox[{
       SuperscriptBox["g", 
        TagBox[
         RowBox[{"(", 
          RowBox[{"1", ",", "0"}], ")"}],
         Derivative],
        MultilineFunction->None], "[", 
       RowBox[{"r", ",", "t"}], "]"}], "2"], " ", 
     RowBox[{
      SuperscriptBox["H", 
       TagBox[
        RowBox[{"(", 
         RowBox[{"2", ",", "0"}], ")"}],
        Derivative],
       MultilineFunction->None], "[", 
      RowBox[{
       FractionBox["S", 
        RowBox[{"g", "[", 
         RowBox[{"r", ",", "t"}], "]"}]], ",", "t"}], "]"}]}]}], 
   ")"}]}]], "Output",
 CellChangeTimes->{
  3.5019326205878706`*^9, {3.5019327198386927`*^9, 3.5019327248075066`*^9}, 
   3.5019330731229925`*^9, 3.5019331153577166`*^9, 3.501933654081058*^9}]
}, Open  ]]
},
WindowSize->{868, 727},
WindowMargins->{{0, Automatic}, {Automatic, 0}},
FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)",
StyleDefinitions->"Default.nb"
]
(* End of Notebook Content *)

(* Internal cache information *)
(*CellTagsOutline
CellTagsIndex->{}
*)
(*CellTagsIndex
CellTagsIndex->{}
*)
(*NotebookFileOutline
Notebook[{
Cell[545, 20, 409, 13, 49, "Input"],
Cell[957, 35, 235, 5, 31, "Input"],
Cell[CellGroupData[{
Cell[1217, 44, 1404, 40, 68, "Input"],
Cell[2624, 86, 7045, 226, 202, "Output"]
}, Open  ]],
Cell[9684, 315, 503, 10, 52, "Input"],
Cell[CellGroupData[{
Cell[10212, 329, 1440, 45, 47, "Input"],
Cell[11655, 376, 2685, 90, 88, "Output"]
}, Open  ]]
}
]
*)

(* End of internal cache information *)