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Notebook[{
Cell[TextData["BlackScholes.nb"], "Title",
  Evaluatable->False,
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Cell["Wilfrid S. Kendall", "Subtitle",
  Evaluatable->False,
  CellHorizontalScrolling->False,
  TextAlignment->Center],

Cell[TextData[
"Statistics, University of Warwick, \nCoventry CV4 7AL, UK."], "Subsubtitle",
  Evaluatable->False,
  CellHorizontalScrolling->False,
  TextAlignment->Center],

Cell[CellGroupData[{

Cell[TextData["Contact information:"], "SmallText",
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Cell[TextData["Email: w.s.kendall@warwick.ac.uk"], "Name",
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Cell[TextData["URL: http://www.warwick.ac.uk/statsdept/Staff/WSK"], "Name",
  Editable->False,
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Cell[CellGroupData[{

Cell[TextData["Acknowledgements:"], "SmallText",
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Cell[TextData[
"The research reported here was supported by EPSRC grants GR/71677 \
(Stochastic calculus in AXIOM using modules of stochastic differentials) and \
GR/L56831 (Perfect simulation in stochastic geometry), and a joint \
EPSRC/BBSRC research grant (Multi-strain species modelling and control via \
differential algebra reductions). This Mathematica notebook was constructed \
on a visit to MSRI Berkeley CA during its 1997-1998 program Stochastic \
Analysis. Finally, it is a pleasure to expres my gratitude to my friends \
Suzanne Scotchmer and Joseph Farrell for the generous hospitality they showed \
to me during my visit to MSRI."], "SmallText",
  Editable->False,
  Evaluatable->False,
  FontFamily->"Times New Roman",
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  FontColor->GrayLevel[0],
  Background->GrayLevel[1],
  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}]
}, Closed]],

Cell[TextData[
"Purpose: this is a Mathematica package (working under Mathematica3.0) which \
demonstrates a derivation of the Black-Scholes formula using Itovsn3."], 
  "SmallText",
  Evaluatable->False,
  FontFamily->"Times New Roman",
  FontSize->10,
  FontSlant->"Italic",
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Cell[TextData["References:"], "SmallText",
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  Background->GrayLevel[1],
  FontVariations->{"Underline"->False,
  "Outline"->False,
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Cell[TextData[
"Discussion of applications of symbolic Ito calculus and related software to \
problems in Mathematical Finance:"], "SmallText",
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Cell[CellGroupData[{

Cell[TextData[
"Derivation of Black-Scholes formula using an implementation of diffusion \
theory in Mathematica:"], "SmallText",
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Cell[TextData[
"J.M. Steele and R.A. Stine: \"Mathematica and Diffusions\", in Economic and \
Financial Modeling with Mathematica, edited by H. Varian, Springer-Verlag, \
New York, pp 192-213 (1993)."], "SmallText",
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Cell[TextData[
"Derivation of Duffie-Richardson quadratic hedging formula using Itovsn3:"], 
  "SmallText",
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Cell[TextData[
"WSK: \"Doing stochastic calculus with Mathematica\", in Economic and \
Financial Modeling with Mathematica, edited by H. Varian, Springer-Verlag, \
New York, pp 214-238 (1993)."], "SmallText",
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Cell[TextData[
"This notebook assumes no previous definitions in the Mathematica workspace. \
If this is not the case, then quit Mathematica and restart.\n\nFirst we \
ensure that the Itovsn3 package is loaded. "], "Text",
  Evaluatable->False],

Cell[BoxData[
    \(\(Needs["\<Itovsn3`\>"]; \)\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[
"Initialize Itovsn3 and set up basic semimartingales"], "Section",
  Evaluatable->False],

Cell[TextData[
"Initialize Itovsn3 and  set up a real Brownian motion  W  to serve as the \
innovation process for the share price:"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(ItoReset[t, dt]\), 
    \(\(BrownSingle[W, 0]; \)\)}], "Input",
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  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
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    InterpretationBox[GridBox[{
          {"\<\"Itovsn3  resetting ...\"\>"},
          {"\<\"Itovsn3  initialized\"\>"},
          {
            
            InterpretationBox[
              \("with time semimartingale "\[InvisibleSpace]t\),
              SequenceForm[ "with time semimartingale ", t],
              Editable->False]},
          {
            
            InterpretationBox[
              \("and time differential "\[InvisibleSpace]dt\),
              SequenceForm[ "and time differential ", dt],
              Editable->False]}
          },
        RowSpacings->1,
        ColumnSpacings->3,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
      TableForm[ {"Itovsn3  resetting ...", "Itovsn3  initialized", 
        SequenceForm[ "with time semimartingale ", t], 
        SequenceForm[ "and time differential ", dt]}]]], "Output"]
}, Open  ]],

Cell[TextData[
"Now we can use this to set up a geometric Brownian motion  S  to model the \
share price, together with a bond  B  which evolves in an entirely \
deterministic manner. "], "Text",
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  FontFamily->"Times New Roman",
  FontSize->12,
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  FontVariations->{"Underline"->False,
  "Outline"->False,
  "Shadow"->False}],

Cell[CellGroupData[{

Cell[BoxData[{
    \(alfa\  = \ \[Alpha]; \nmu\  = \ \[Mu]; \nsigma\  = \ \[Sigma]; \n
    SetAttributes[\[Sigma], Constant]\), 
    \(SetAttributes[\[Mu], Constant]\), 
    \(SetAttributes[\[Alpha], Constant]\), 
    \(Itosde[S, dS == sigma\ S\ dW + mu\ S\ dt, 1]\), 
    \(\(AddFixed[0, S, S0]; \)\), 
    \(Itosde[B, dB == alfa\ B\ dt, 1]\), 
    \(\(AddFixed[0, B, B0]; \)\), 
    \(\(ItoStatus[]; \)\)}], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \("---------------"\)], "Print"],

Cell[BoxData[
    \("Summary of current structure of stochastic differentials"\)], "Print"],

Cell[BoxData[
    \("- - - - - - - -"\)], "Print"],

Cell[BoxData[
    \("Current second-order structure of semimartingale differentials:"\)], 
  "Print"],

Cell[BoxData[
    TagBox[GridBox[{
          {"\<\"\"\>", "dB", "dS", "dW", "dt"},
          {"dB", "0", "0", "0", "0"},
          {"dS", "0", \(dt\ S\^2\ \[Sigma]\^2\), \(dt\ S\ \[Sigma]\), "0"},
          {"dW", "0", \(dt\ S\ \[Sigma]\), "dt", "0"},
          {"dt", "0", "0", "0", "0"}
          },
        RowSpacings->1,
        ColumnSpacings->3,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
      (TableForm[ #, TableHeadings -> {{dB, dS, dW, dt}, {dB, dS, dW, 
        dt}}]&)]], "Print"],

Cell[BoxData[
    \("- - - - - - - -"\)], "Print"],

Cell[BoxData[
    \("Current first-order structure of semimartingale differentials:"\)], 
  "Print"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {"dB", "dS", "dW", "dt"},
          {\(B\ dt\ \[Alpha]\), \(dt\ S\ \[Mu]\), "0", "dt"}
          },
        RowSpacings->3,
        ColumnSpacings->1,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
      TableForm[ {
        Times[ B, dt, \[Alpha]], 
        Times[ dt, S, \[Mu]], 0, dt}, TableDirections -> Row, 
        TableHeadings -> {{dB, dS, dW, dt}, {"Drifts:"}}]]], "Print"],

Cell[BoxData[
    \("- - - - - - - -"\)], "Print"],

Cell[BoxData[
    \("Current initial values:"\)], "Print"],

Cell[BoxData[
    InterpretationBox[GridBox[{
          {"B", "S", "W", "t"},
          {"B0", "S0", "0", "0"}
          },
        RowSpacings->3,
        ColumnSpacings->1,
        RowAlignments->Baseline,
        ColumnAlignments->{Left}],
      TableForm[ {B0, S0, 0, 0}, TableDirections -> Row, TableHeadings -> {{
        B, S, W, t}, {"Initially:"}}]]], "Print"],

Cell[BoxData[
    \("---------------"\)], "Print"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Construct self-financing trading strategy"], "Section",
  Evaluatable->False],

Cell[TextData[
"We now consider a trading strategy, holding  s[t]  units of the share  S  \
and  b[t]  units of the bond  B,  where  s  and  b  are allowed to vary \
smoothly with time but we require the strategy to be exactly self-financing. \
We establish a substitution rule to replace  V[S,t]  by this self-financing \
trading strategy."], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Strategy = s[t]\ S + b[t]\ B; \)\), 
    \(ValueDefinition = First[Solve[V[S, t] - Strategy == 0, b[t]]]\), 
    \(SelfFinancing = 
      First[Solve[\[PartialD]\_t s[t]\ S + \[PartialD]\_t b[t]\ B == 0, 
          \[PartialD]\_t b[t]]]\)}], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \({b[t] \[Rule] \(\(-S\)\ s[t] + V[S, t]\)\/B}\)], "Output"],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{
        RowBox[{
          SuperscriptBox["b", "\[Prime]",
            MultilineFunction->None], "[", "t", "]"}], "\[Rule]", 
        RowBox[{"-", 
          FractionBox[
            RowBox[{"S", " ", 
              RowBox[{
                SuperscriptBox["s", "\[Prime]",
                  MultilineFunction->None], "[", "t", "]"}]}], "B"]}]}], 
      "}"}]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Stochastic analysis of trading strategy"], "Section",
  Evaluatable->False],

Cell[TextData[
"Consider then the evolution of the value  V[S,t]  of the trading strategy \
over time: we know the following stochastic differential should be zero! \
Notice the self-financing condition means we know how to derive  b[t]  once  \
a[t]  is given."], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(sd = ItoD[V[S, t] - Strategy]; \)\), 
    \(SecondOrder = 
      First[Solve[\((ItoExpand[sd\ dW] /. SelfFinancing)\) == 0, s[t]]]\), 
    \(pde = 
      Expand[\(\(\(Drift[sd] /. SelfFinancing\) /. ValueDefinition\) /. 
              SecondOrder\)\/dt] == 0\)}], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    RowBox[{"{", 
      RowBox[{\(s[t]\), "\[Rule]", 
        RowBox[{
          SuperscriptBox["V", 
            TagBox[\((1, 0)\),
              Derivative],
            MultilineFunction->None], "[", \(S, t\), "]"}]}], "}"}]], "Output"],

Cell[BoxData[
    RowBox[{
      RowBox[{\(\(-\[Alpha]\)\ V[S, t]\), "+", 
        RowBox[{
          SuperscriptBox["V", 
            TagBox[\((0, 1)\),
              Derivative],
            MultilineFunction->None], "[", \(S, t\), "]"}], "+", 
        RowBox[{"S", " ", "\[Alpha]", " ", 
          RowBox[{
            SuperscriptBox["V", 
              TagBox[\((1, 0)\),
                Derivative],
              MultilineFunction->None], "[", \(S, t\), "]"}]}], "+", 
        RowBox[{\(1\/2\), " ", \(S\^2\), " ", \(\[Sigma]\^2\), " ", 
          RowBox[{
            SuperscriptBox["V", 
              TagBox[\((2, 0)\),
                Derivative],
              MultilineFunction->None], "[", \(S, t\), "]"}]}]}], "==", 
      "0"}]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[
"Recasting problem in terms of innovation Brownian motion"], "Section",
  Evaluatable->False],

Cell[TextData[
"We now have a linear second-order partial differential equation in  V  which \
we must solve. In principle one could write a solver for a class of equations \
including this example. However it seems more illuminating to proceed by a \
trial-and-error method.\n\nFirst observe that the argument  S  is constrained \
to be positive. This suggests we should re-cast the argument using the known \
solution of  S  in terms of its innovation  W. First we check the solution \
(again, we could have built a solver!): "], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(SS = Exp[sigma\ W + mu\ t - \(sigma\^2\ t\)\/2]; \)\), 
    \(\(sd = ItoD[S - SS] /. S \[Rule] SS; \)\), 
    \(Together[Drift[sd] /. S \[Rule] SS] == 0\), 
    \(\((ItoExpand[sd\^2] /. S \[Rule] SS)\) == 0\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    \(True\)], "Output"],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]],

Cell[TextData[
"Now we work through the derivation of the pde, but this time using  SS  not  \
S,  and viewing the value as  V1[W,t]  instead of  V[S,t],  in other words \
viewed as a function of  W  not of  S: "], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\(Strategy = s[t]\ SS + b[t]\ B; \)\), 
    \(\(ValueDefinition = First[Solve[V1[W, t] - Strategy == 0, b[t]]]; 
    \)\), 
    \(\(SelfFinancing = 
      First[Solve[\[PartialD]\_t s[t]\ SS + \[PartialD]\_t b[t]\ B == 0, 
          \[PartialD]\_t b[t]]]; \)\), 
    \(\(sd = ItoD[V1[W, t] - Strategy]; \)\), 
    \(\(SecondOrder = 
      First[Solve[\((ItoExpand[sd\ dW] /. SelfFinancing)\) == 0, s[t]]]; 
    \)\), 
    \(pde = 
      Expand[\(\(\(Drift[sd] /. SelfFinancing\) /. ValueDefinition\) /. 
              SecondOrder\)\/dt] == 0\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    RowBox[{
      RowBox[{\(\(-\[Alpha]\)\ V1[W, t]\), "+", 
        RowBox[{
          SuperscriptBox["V1", 
            TagBox[\((0, 1)\),
              Derivative],
            MultilineFunction->None], "[", \(W, t\), "]"}], "+", 
        FractionBox[
          RowBox[{"\[Alpha]", " ", 
            RowBox[{
              SuperscriptBox["V1", 
                TagBox[\((1, 0)\),
                  Derivative],
                MultilineFunction->None], "[", \(W, t\), "]"}]}], 
          "\[Sigma]"], "-", 
        FractionBox[
          RowBox[{"\[Mu]", " ", 
            RowBox[{
              SuperscriptBox["V1", 
                TagBox[\((1, 0)\),
                  Derivative],
                MultilineFunction->None], "[", \(W, t\), "]"}]}], 
          "\[Sigma]"], "+", 
        RowBox[{\(1\/2\), " ", 
          RowBox[{
            SuperscriptBox["V1", 
              TagBox[\((2, 0)\),
                Derivative],
              MultilineFunction->None], "[", \(W, t\), "]"}]}]}], "==", 
      "0"}]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Further simplification using drift term"], "Section",
  Evaluatable->False],

Cell[TextData[
"This is a considerable improvement, as we can now recognize an orthodox heat \
equation hiding in there. It looks like we can improve matters still further \
by adding in a drift term! We work with  V2[WW,t]  where  WW = W - \
((alfa-mu)/sigma)*t,  instead of  V1[W,t]:"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(WW = W - \(\((alfa - mu)\)\ t\)\/sigma\), 
    \(\(Strategy = s[t]\ SS + b[t]\ B; \)\), 
    \(\(ValueDefinition = First[Solve[V2[WW, t] - Strategy == 0, b[t]]]; 
    \)\), 
    \(\(SelfFinancing = 
      First[Solve[\[PartialD]\_t s[t]\ SS + \[PartialD]\_t b[t]\ B == 0, 
          \[PartialD]\_t b[t]]]; \)\), 
    \(\(sd = ItoD[V2[WW, t] - Strategy]; \)\), 
    \(\(SecondOrder = 
      First[Solve[\((ItoExpand[sd\ dW] /. SelfFinancing)\) == 0, s[t]]]; 
    \)\), 
    \(pde = 
      Expand[\(\(\(Drift[sd] /. SelfFinancing\) /. ValueDefinition\) /. 
              SecondOrder\)\/dt] == 0\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    \(W - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma]\)], "Output"],

Cell[BoxData[
    RowBox[{
      RowBox[{
      \(\(-\[Alpha]\)\ V2[W - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma], t]\), 
        "+", 
        RowBox[{
          SuperscriptBox["V2", 
            TagBox[\((0, 1)\),
              Derivative],
            MultilineFunction->None], "[", 
          \(W - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma], t\), "]"}], "+", 
        RowBox[{\(1\/2\), " ", 
          RowBox[{
            SuperscriptBox["V2", 
              TagBox[\((2, 0)\),
                Derivative],
              MultilineFunction->None], "[", 
            \(W - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma], t\), "]"}]}]}], 
      "==", "0"}]], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Further simplification using Feynman-Kac formalism"], "Section",
  Evaluatable->False],

Cell[TextData[
"It looks as if we could get an even neater answer by adding in a multiplier \
to take out the  -alfa V2  term (this is equivalent to using the Feynman-Kac \
trick). Thus we now work with  VV = Exp[alfa*(t-t1)]*V3[WW,t]  instead of  \
V1[W,t]  (so we are discounting relative to the value of the bond at time  \
t1):"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(SetAttributes[t1, Constant]\), 
    \(VV = Exp[alfa\ \((t - t1)\)]\ V3[WW, t]\), 
    \(\(Strategy = s[t]\ SS + b[t]\ B; \)\), 
    \(\(ValueDefinition = First[Solve[VV - Strategy == 0, b[t]]]; \)\), 
    \(\(SelfFinancing = 
      First[Solve[\[PartialD]\_t s[t]\ SS + \[PartialD]\_t b[t]\ B == 0, 
          \[PartialD]\_t b[t]]]; \)\), 
    \(\(sd = ItoD[VV - Strategy]; \)\), 
    \(\(SecondOrder = 
      First[Solve[\((ItoExpand[sd\ dW] /. SelfFinancing)\) == 0, s[t]]]; 
    \)\), 
    \(pde = 
      Expand[\(\(\(Drift[sd] /. SelfFinancing\) /. ValueDefinition\) /. 
              SecondOrder\)\/dt] == 0\)}], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    \(E\^\(\((t - t1)\)\ \[Alpha]\)\ 
      V3[W - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma], t]\)], "Output"],

Cell[BoxData[
    RowBox[{
      RowBox[{
        RowBox[{\(E\^\(\((t - t1)\)\ \[Alpha]\)\), " ", 
          RowBox[{
            SuperscriptBox["V3", 
              TagBox[\((0, 1)\),
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              MultilineFunction->None], "[", 
            \(W - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma], t\), "]"}]}], 
        "+", 
        RowBox[{\(1\/2\), " ", \(E\^\(\((t - t1)\)\ \[Alpha]\)\), " ", 
          RowBox[{
            SuperscriptBox["V3", 
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                Derivative],
              MultilineFunction->None], "[", 
            \(W - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma], t\), "]"}]}]}], 
      "==", "0"}]], "Output"]
}, Open  ]]
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Cell[CellGroupData[{

Cell[TextData[
"Recognition of Brownian kernel and explicit solution"], "Section",
  Evaluatable->False],

Cell[TextData[
"We can now recognize  V3  as solving the Kolmogorov backwards differential \
equation solved by the well-known Brownian heat kernel. Consequently we know \
the general form for  V3:  integrals with respect to  S1  and  t1  of the \
following (and notice the option we establish so that this can be eventually \
rephrased in terms of the terminal share price  S1  rather than the terminal \
innovation  W1!):"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[{
    \(TerminalShare = 
      W1 \[Rule] \(Log[S1] - \((mu\ t - \(sigma\^2\ t\)\/2)\)\)\/sigma\), 
    \(V3 = 
      Exp[\(-\(\((#1 - \((W1 - \(\((alfa - mu)\)\ t1\)\/sigma)\))\)\^2\/\(2\ 
                    \((t1 - #2)\)\)\)\)]\/\@\(2\ \[Pi]\ \((t1 - #2)\)\)&\)}], 
  "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(W1 \[Rule] 
      \(\(-t\)\ \[Mu] + \(t\ \[Sigma]\^2\)\/2 + Log[S1]\)\/\[Sigma]\)], 
  "Output"],

Cell[BoxData[
    \(Exp[
          \(-\(\((#1 - \((W1 - \(\((alfa - mu)\)\ t1\)\/sigma)\))\)\^2\/\(2\ 
                  \((t1 - #2)\)\)\)\)]\/\@\(2\ \[Pi]\ \((t1 - #2)\)\)&\)], 
  "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Verification of answer"], "Section",
  Evaluatable->False],

Cell[TextData["Substituting back shows we have got the answer:"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(Together[First[pde]] == 0\)], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(True\)], "Output"]
}, Open  ]]
}, Open  ]],

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Cell[TextData["Rewriting of solution in terms of share price  S"], "Section",
  Evaluatable->False],

Cell["\<\
So now we know the kernel which should be integrated (as a function of  W1)  \
to yield solutions to Black-Scholes-type problems. Notice that subject to \
regularity conditions such an integration yields solutions whatever function \
of  W1  is integrated against the kernel. This allows us to derive \
self-financing strategies delivering arbitrary functions of  W1  at time  t1.

Rewritten in terms of the current share price  S the kernel is as follows \
(but note here that in  Solve  Mathematica 3.0 warns about multiple \
solutions!):\
\>", "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(VV = 
      \(VV /. First[Solve[SS == S, W]]\) /. Log[s_\^2] \[Rule] 2\ Log[s]\)], 
  "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(Solve::"ifun" \( : \ \) 
      "Inverse functions are being used by \!\(Solve\), so some solutions may \
not be found."\)], "Message"],

Cell[BoxData[
    \(E\^\(\((t - t1)\)\ \[Alpha] - 
          \((\(-W1\) - \(t\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma] + 
                  \(t1\ \((\[Alpha] - \[Mu])\)\)\/\[Sigma] - 
                  \(2\ t\ \[Mu] - t\ \[Sigma]\^2 - 2\ Log[S]\)\/\(2\ 
                      \[Sigma]\))\)\^2\/\(2\ \((\(-t\) + t1)\)\)\)\/\(\@\(2\ 
            \[Pi]\)\ \@\(\(-t\) + t1\)\)\)], "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Analysis of value of option at time zero"], "Section",
  Evaluatable->False],

Cell[TextData[
"We are principally interested in the value of this at time zero:"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(InitialKernel = Simplify[InitialValue[0, VV]]\)], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(E\^\(\(-t1\)\ \[Alpha] - 
          \((t1\ \((\[Alpha] - \[Mu])\) - W1\ \[Sigma] + Log[S0])\)\^2\/\(2\ 
              t1\ \[Sigma]\^2\)\)\/\(\@\(2\ \[Pi]\)\ \@t1\)\)], "Output"]
}, Open  ]],

Cell[TextData[
"Using integration against  W1,  this tells us the value of the strategy we \
must start with at time 0 in order to achieve value  f[S1]  at time  t1,  \
since integration of the kernel multiplied by a function of  S1  \
(equivalently,  W1)  against  W1  still yields a solution of the partial \
differential equation, and therefore still yields a value process achievable \
by a self-financing strategy. We first compute the value of  W1  required to \
obtain  S1:"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(S1 = \(SS /. W \[Rule] W1\) /. t \[Rule] t1\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
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    \(E\^\(t1\ \[Mu] + W1\ \[Sigma] - \(t1\ \[Sigma]\^2\)\/2\)\)], "Output"]
}, Open  ]],

Cell[TextData[
"Thus the value required at time  0  to achieve  f[S1]  at time  t1  is given \
by"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\( f[S1]\ InitialKernel\) 
        \[DifferentialD]W1\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    FractionBox[
      RowBox[{
        SubsuperscriptBox["\[Integral]", 
          InterpretationBox[\(-\[Infinity]\),
            DirectedInfinity[ -1]], 
          InterpretationBox["\[Infinity]",
            DirectedInfinity[ 1]]], 
        \(\(E\^\(\(-t1\)\ \[Alpha] - 
                \((t1\ \((\[Alpha] - \[Mu])\) - W1\ \[Sigma] + Log[S0])\)\^2
                    \/\(2\ t1\ \[Sigma]\^2\)\)\ 
            f[E\^\(t1\ \[Mu] + W1\ \[Sigma] - \(t1\ \[Sigma]\^2\)\/2\)]\) 
          \[DifferentialD]W1\)}], \(\@\(2\ \[Pi]\)\ \@t1\)]], "Output"]
}, Open  ]],

Cell[TextData[
"This values the option  f[S1]  at time  t1.  Here is how it works if instead \
we value an option on buying the share at price  Soption  at time  t1.  \
Firstly we calculate the value  Woption  of  W1  for a given option level  \
Soption:"], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(OptionLevel = 
      First[\(Solve[SS == Soption, W] /. W \[Rule] Woption\) /. 
            t \[Rule] t1] /. Log[s_\^2] \[Rule] 2\ Log[s]\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
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  Background->GrayLevel[1]],

Cell[BoxData[
    \(General::"spell1" \( : \ \) 
      "Possible spelling error: new symbol name \"\!\(Woption\)\" is similar \
to existing symbol \"\!\(Soption\)\"."\)], "Message"],

Cell[BoxData[
    \(Solve::"ifun" \( : \ \) 
      "Inverse functions are being used by \!\(Solve\), so some solutions may \
not be found."\)], "Message"],

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    \({Woption \[Rule] 
        \(-\(\(2\ t1\ \[Mu] - t1\ \[Sigma]\^2 - 2\ Log[Soption]\)\/\(2\ 
                \[Sigma]\)\)\)}\)], "Output"]
}, Open  ]],

Cell["\<\
Some coaching is required in the integration procedure, since Mathematica has \
no way of telling what is the sign of  t1  and therefore fails to integrate \
to the limit at plus or minus  Infinity!  Note also that version 3.0 of \
Mathematica warns again about multiple solutions.\
\>", "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(TotalInitial = 
      \[Integral]\_Woption\%bbb\( Soption\ InitialKernel\) 
              \[DifferentialD]W1 + 
          \[Integral]\_\(-aaa\)\%Woption\( S1\ InitialKernel\) 
              \[DifferentialD]W1 /. OptionLevel\)], "Input",
  PageWidth->Infinity,
  FontFamily->"Courier New",
  FontSize->12,
  FontWeight->"Bold",
  FontColor->GrayLevel[0],
  Background->GrayLevel[1]],

Cell[BoxData[
    \(\(-\(1\/2\)\)\ E\^\(\(-t1\)\ \[Alpha]\)\ Soption\ 
        Erf[\(t1\ \((\[Alpha] - \[Mu])\) - bbb\ \[Sigma] + Log[S0]\)\/\(\@2\ 
              \@t1\ \[Sigma]\)] + 
      1\/2\ S0\ Erf[
          \(aaa\ \[Sigma] + t1\ \((\[Alpha] - \[Mu] + \[Sigma]\^2)\) + 
              Log[S0]\)\/\(\@2\ \@t1\ \[Sigma]\)] + 
      1\/2\ E\^\(\(-t1\)\ \[Alpha]\)\ Soption\ 
        Erf[\(t1\ \((\[Alpha] - \[Mu])\) + Log[S0] + 
              1\/2\ \((2\ t1\ \[Mu] - t1\ \[Sigma]\^2 - 2\ Log[Soption])
                  \)\)\/\(\@2\ \@t1\ \[Sigma]\)] - 
      1\/2\ S0\ Erf[
          \(t1\ \((\[Alpha] - \[Mu] + \[Sigma]\^2)\) + Log[S0] + 
              1\/2\ \((2\ t1\ \[Mu] - t1\ \[Sigma]\^2 - 2\ Log[Soption])
                  \)\)\/\(\@2\ \@t1\ \[Sigma]\)]\)], "Output"]
}, Open  ]],

Cell[TextData[
"We now have to make a careful set of simplifications so as to ensure \
Mathematica spots the special values of two of the  Erf  arguments (we also \
add some simplifications for tidying up the answer!)."], "Text",
  Evaluatable->False],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(Simplify[
          \(\(\(TotalInitial /. 
                  aaa \[Rule] 
                    xxx\ \@\(2\ t1\) - 
                      \(Log[S0] + \((alfa - mu)\)\ t1 + sigma\^2\ t1\)\/sigma
                        \) /. bbb \[Rule] 
                  xxx\ \@\(2\ t1\) + 
                    \(Log[S0] + \((alfa - mu)\)\ t1\)\/sigma\) /. 
              \@\(\[Pi]\ t1\) \[Rule] \(\@\[Pi]\ \@\(2\ t1\)\)\/\@2\) /. 
            xxx \[Rule] \[Infinity]] /. 
        Log[s_] - Log[t_] \[Rule] Log[s\/t]\) /. 
      \@t1 \[Rule] \@\(2\ t1\)\/\@2\)], "Input",
  PageWidth->Infinity],

Cell[BoxData[
    \(\(-\(1\/2\)\)\ E\^\(\(-t1\)\ \[Alpha]\)\ 
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          E\^\(t1\ \[Alpha]\)\ S0\ 
            Erf[\(t1\ \((2\ \[Alpha] + \[Sigma]\^2)\) + 2\ Log[S0] - 
                  2\ Log[Soption]\)\/\(2\ \@2\ \@t1\ \[Sigma]\)] + 
          Soption\ Erf[
              \(t1\ \((\(-2\)\ \[Alpha] + \[Sigma]\^2)\) - 2\ Log[S0] + 
                  2\ Log[Soption]\)\/\(2\ \@2\ \@t1\ \[Sigma]\)])\)\)], 
  "Output"]
}, Open  ]]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData["Further Exercise"], "Subsubsection",
  Evaluatable->False,
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Cell[TextData["Compute the amount  S  of share to be held at time  0"], "Text",
  Evaluatable->False]
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