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As a preliminary \ step, the well-known Black-Scholes theoretical price function for a vanilla \ European call is defined. The code for this formula, and many others, is \ available in ", StyleBox[ButtonBox["MathSource", ButtonData:>{ URL[ "http://www.mathsource.com/Content/Applications/EconomicsFinance/0209-\ 270"], None}, ButtonStyle->"Hyperlink"], FontSlant->"Italic"], " or for purchase in application packages such as ", StyleBox[ButtonBox["Derivatives Expert", ButtonData:>{ URL[ "http://www.internal.wolfram.com/products/applications/derivatives/"], None}, ButtonStyle->"Hyperlink"], FontSlant->"Italic"], "." }], "Text"], Cell[BoxData[ \(\(normalcdf[x_] := 1\/2\ Erf[x\/\@2] + 1\/2;\)\)], "Input", CellLabel->"In[1]:="], Cell[BoxData[{ \(\(d1[s_, x_, \[Sigma]_, T_, r_] := \(\((\[Sigma]\^2\/2 + r)\)\ T + Log[s\/x]\)\/\(\[Sigma]\ \@T\ \);\)\), "\[IndentingNewLine]", \(\(d2[s_, x_, \[Sigma]_, T_, r_] := d1[s, x, \[Sigma], T, r] - \[Sigma]\ \@T;\)\)}], "Input", CellLabel->"In[2]:="], Cell[BoxData[ \(\(bsCall[s_, x_, \[Sigma]_, T_, r_] := s\ normalcdf[ d1[s, x, \[Sigma], T, r]] - \(x*normalcdf[d2[s, x, \[Sigma], T, r]]\)\/\ \[ExponentialE]\^\(r\ T\);\)\)], "Input", CellLabel->"In[4]:="], Cell[TextData[{ "Now, to take an example from Hull (2000), we can find the price for \ writing a European call option on 100,000 shares, given the following \ parameters:\n\n\t\tCurrent stock price", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(s = 49\)\)\)], FormatType->StandardForm], "\n\t\tStrike price", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(x = $50\)\)\)], FormatType->StandardForm], "\n\t\tStock volatility", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(\[Sigma] = 20 %\)\)\)], FormatType->StandardForm], "\n\t\tOption time to maturity", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(T = 20\ weeks\)\)\)], FormatType->StandardForm], "\n\t\tMarket rate of interest", Cell[BoxData[ \(TraditionalForm\`\(\(=\)\(r = 5 %\)\)\)], FormatType->StandardForm] }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bsCall[49, 50, 0.2, 20/52, 0.05]*100000\)], "Input", CellLabel->"In[5]:="], Cell[BoxData[ \(240052.73232717137`\)], "Output", CellLabel->"Out[5]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ "Symbolic Calculus with ", StyleBox["Mathematica", FontSlant->"Italic"], ": Deriving the Delta Function" }], "Section"], Cell[TextData[{ "To create a delta hedge, ", Cell[BoxData[ \(TraditionalForm\`\[Delta]C\/\(\(\[Delta]s\)\(\ \)\)\)], FormatType->StandardForm], " shares of the underlying stock are purchased, where ", StyleBox["C", FontSlant->"Italic"], " is the call price. As a first pass, a static, one-time hedge is \ considered. Here, ", StyleBox["Mathematica", FontSlant->"Italic"], "'s symbolic calculus capabilities are used to derive the complicated \ definition of the delta function; then numeric parameters are fed into this \ new equation to find the specific hedging ratio." }], "Text"], Cell[BoxData[ \(\[Delta][s_, x_, \[Sigma]_, T_, r_] := D[bsCall[s1, x, \[Sigma], T, r], s1] /. s1 \[Rule] s // FullSimplify\)], "Input", CellLabel->"In[6]:="], Cell[CellGroupData[{ Cell[BoxData[ \(\[Delta][s, x, \[Sigma], T, r]\)], "Input", CellLabel->"In[7]:="], Cell[BoxData[ \(1\/2\ \((1 + Erf[\(T\ \((2\ r + \[Sigma]\^2)\) + 2\ Log[s\/x]\)\/\(2\ \@2\ \@T\ \ \[Sigma]\)])\)\)], "Output", CellLabel->"Out[7]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\[Delta][49, 50, 0.2, 20/52, 0.05]*100000\)], "Input", CellLabel->"In[8]:="], Cell[BoxData[ \(52160.46610663959`\)], "Output", CellLabel->"Out[8]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(52160*49. // AccountingForm\)], "Input", CellLabel->"In[9]:="], Cell[BoxData[ TagBox[ InterpretationBox["\<\"2555840.\"\>", .255584*^7, AutoDelete->True], AccountingForm]], "Output", CellLabel->"Out[9]//AccountingForm="] }, Open ]], Cell["\<\ So, to properly hedge this call, 52,160 shares of the underlying stock are \ bought at a cost of $49 each, for a total of $2,555,840. To finance this \ purchase, the capital is borrowed at the market rate of interest (which has \ been defined as 5%). \ \>", "Text"] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica ", FontSlant->"Italic"], "as a Calculator: Analyzing the Results with Numbers" }], "Section"], Cell["\<\ Now, suppose, after one week the underlying stock price rises to $49.50. The \ value of the option position has grown from $240,053 to $258,422. If the \ writer of this call had not hedged the position with a purchase of the \ underlying stock, the small price movement would have created a loss of over \ $18,000.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bsCall[49.5, 50, 0.2, 19/52, 0.05]*100000. \)], "Input", CellLabel->"In[10]:="], Cell[BoxData[ \(258422.22189471932`\)], "Output", CellLabel->"Out[10]="] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(240053 - 258422\)], "Input", CellLabel->"In[11]:="], Cell[BoxData[ \(\(-18369\)\)], "Output", CellLabel->"Out[11]="] }, Open ]], Cell["\<\ However, with the hedge, the net outcome for the writer of the call is the \ following: net gain = Original price received for the call - price of the call one \ week later + \t\tprice received for the underying stock one week later - repayment of \ principal and interest used to buy the underlying stock \t\t In this case, it turns out to be a profit of $5,229.95.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(bsCall[49, 50, 0.2, 20/52, 0.05]*100000. - \[IndentingNewLine]bsCall[ 49.5, 50, 0.2, 19/52, 0.05]*100000. + \[IndentingNewLine]52160*49.5 - \ \[IndentingNewLine]2555863. *\((1 + 0.05/52)\)\)], "Input", CellLabel->"In[12]:="], Cell[BoxData[ \(5229.949855528772`\)], "Output", CellLabel->"Out[12]="] }, Open ]] }, Closed]], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Mathematica ", FontSlant->"Italic"], "as a Visualization Tool: Analyzing Relationships with Graphics" }], "Section"], Cell[TextData[{ "A unique feature of ", StyleBox["Mathematica", FontSlant->"Italic"], " as a programming language is its integrated graphics capabilities. \ Viewing these relationships graphically, we can definitevely see how marginal \ changes in the underlying stock price have a mitigated effect on returns, \ when the position is hedged." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(<< Graphics`FilledPlot`\), "\[IndentingNewLine]", \(plot1 = Plot[bsCall[49, 50, 0.2, 20/52, 0.05]*100000. - 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