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Notebook[{
Cell[TextData[{
StyleBox["MathSource",
FontSlant->"Italic"],
" Reviews"
}], "Chapter",
CellTags->{"S0", "0.1"}],
Cell["The Black\[Hyphen]Scholes Equation for European Call Options", "Section",
CellTags->{"S0.0.0", "0.2"}],
Cell["Edited by Matthew M. Thomas", "Subsection",
CellTags->{"S0.0.1", "1.1"}],
Cell[TextData[{
"This review completes the fifth year of ",
StyleBox["MathSource", "TI"],
" Reviews in this journal. It is most timely that Year Five would meet its \
end so near ",
StyleBox["Mathematica", "TI"],
"'s tenth anniversary celebration, held in Chicago during June 1998. During \
that celebratory conference, attendees were told that ",
StyleBox["The Mathematica Journal", "TI"],
" would soon resume publication, this time under the aegis of Wolfram \
Research. In 1990, when the ",
StyleBox["Journal", "TI"],
" debuted under Addison\[Hyphen]Wesley auspices, its premiere issue \
featured at least two papers of particular interest to your reviewer. One of \
those papers was co\[Hyphen]written by your reviewer (thus the interest), and \
dealt with illustrating conductive heat transfer. The second of those papers \
\[Hyphen] and the more important one, for our immediate purposes \[Hyphen] \
deals with pricing options. At first blush, it is hard to fathom any two \
topics that could be more dissimilar: After all, the first is rooted in the \
physical sciences, while the second is rooted in finance. But the two do have \
a point of commonality, in the partial differential equation on which both \
are based. For more on that PDE, continue reading. \nWhy the interest by your \
reviewer in the latter of the two ",
StyleBox["Journal", "TI"],
" papers? Because it is an interest shared by many, as indicated by the \
newest feature on the ",
StyleBox["MathSource", "TI"],
" web site (www.mathsource.com). Our previous review cited the updated \
search features available through said site, which differentiate between old \
(2.2 and before) and new (3.0) ",
StyleBox["MathSource", "TI"],
" material. Since June 1998 and at least through this early\[Hyphen]July \
1998 writing, said site now features a page of links to the twenty most \
popular ",
StyleBox["MathSource", "TI"],
" items from the previous month, with popularity presumably based on number \
of \[OpenCurlyDoubleQuote]hits\[CloseCurlyDoubleQuote]. The custodians of the \
",
StyleBox["MathSource", "TI"],
" web site deserve praise for adding this useful \[OpenCurlyDoubleQuote]Top \
20 List\[CloseCurlyDoubleQuote] feature to the site. Ninth on the May 1998 \
list and fifteenth on the June 1998 list was the Black\[Hyphen]Scholes Option \
Pricing Model, by Ross Miller. This ",
StyleBox["MathSource", "TI"],
" item is the electronic complement of Miller's \
\[OpenCurlyDoubleQuote]Computer\[Hyphen]Aided Financial Analysis: An \
Implementation of the Black\[Hyphen]Scholes Model\[CloseCurlyDoubleQuote] \
[Miller 1990] \[Hyphen] the aforementioned finance\[Hyphen]rooted paper on \
pricing options from the premiere issue of ",
StyleBox["The Mathematica Journal", "TI"],
". Herein, we review this item. "
}], "Text",
CellTags->{"S0.0.1", "1.2"}],
Cell["Black\[Hyphen]Scholes Background", "Section",
CellTags->{"S0.0.0", "0.3"}],
Cell[TextData[{
"Before we attempt a discussion of Ross Miller's ",
StyleBox["MathSource", "TI"],
" item and related paper, we should first review the model upon which that \
item is based. The model, developed by Myron S. Scholes and Fischer Black \
with considerable input from Robert C. Merton, was built for option \
valuation. Options are contracts which give their buyers/holders the right, \
but not the obligation, to buy or sell shares of the underlying unit at the \
\[OpenCurlyDoubleQuote]strike price\[CloseCurlyDoubleQuote] or \
\[OpenCurlyDoubleQuote]exercise price\[CloseCurlyDoubleQuote]. Options to buy \
are \[OpenCurlyDoubleQuote]call\[CloseCurlyDoubleQuote] options; options to \
sell are \[OpenCurlyDoubleQuote]put\[CloseCurlyDoubleQuote] options. \
\[OpenCurlyDoubleQuote]American\[CloseCurlyDoubleQuote] options can be \
exercised on or before the option expiration date; \
\[OpenCurlyDoubleQuote]European\[CloseCurlyDoubleQuote] options can be \
exercised only on the expiration date. Option contracts are available on \
stocks, indexes, U.S. Treasury rates, non\[Hyphen]U.S. currencies, and other \
units of commerce and finance. American options are traded on exchanges such \
as the New York Stock Exchange, the American Stock Exchange, and the Chicago \
Board Options Exchange. \nOptions let the investor manage the risk from \
movement in the underlying unit's value. For example: Let's say that a stock \
is currently $50/share, and you like its prospects over the next three \
months. To benefit, you can either buy, say, 100 shares today at $50/share, \
or buy a call option on 100 stock shares at $50/share for a premium such as \
$3/share (or $300, given our 100\[Hyphen]share basis). If, in three months, \
the stock price leaps by $10/share to $60/share, you would have a profit of \
$1000 (minus transaction fees) if you buy the stock outright. You would have \
a profit of ",
Cell[BoxData[
\(TraditionalForm\`$1000 - $300 = $700\)], "InlineFormula"],
" (minus transaction fees) if you buy the option and exercise it upon \
expiration. But if, in three months, the stock price falls by $10/share to \
$40/share, you would have a loss of $1000 (plus transaction fees) if you buy \
the stock outright. You would have a loss of only $300 (plus transaction \
fees) if you buy the option and let it expire. Call options limit your loss \
upon unexpected downturns while letting you benefit from expected upturns. \
Put options let you benefit from unexpected upturns, while limiting your loss \
upon expected downturns. \nStandardized, exchange\[Hyphen]listed, and \
government\[Hyphen]regulated options have been available in the U.S. for the \
past quarter\[Hyphen]century. It is not a coincidence that the pioneering \
Black\[Hyphen]Scholes option valuation model was published a \
quarter\[Hyphen]century ago [Black\[Hyphen]Scholes 1973]. The \
Black\[Hyphen]Scholes model uses ",
StyleBox["w[x,t]", "MR"],
" to represent the value of an European stock option call, as a function of \
underlying stock price ",
StyleBox["x", "MR"],
" and time ",
StyleBox["t", "MR"],
". Under ideal conditions (no transaction costs, known and constant short\
\[Hyphen]term interest rate ",
StyleBox["r", "MR"],
", no dividends or other distributions from the stock, etc.), the stock \
price and option value are related by "
}], "Text",
CellTags->{"S0.0.0", "0.4"}],
Cell["\<\
D[w[x,t],t] = r w[x,t] - r x D[w[x,t],x] -
0.5 v^2 x^2 D[w[x,t],{x,2}] (1)\
\>", "Input",
CellLabel->"In[1]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"where ",
Cell[BoxData[
\(TraditionalForm\`v\^2\)], "InlineFormula"],
" is the variance of the annual rate of return on the stock (",
Cell[BoxData[
\(TraditionalForm\`v\)], "InlineFormula"],
" is also called the \[OpenCurlyDoubleQuote]volatility\
\[CloseCurlyDoubleQuote]), and the other variables are as defined above. \
Letting ts be the expiration date of the option, and letting ",
Cell[BoxData[
\(TraditionalForm\`c\)], "InlineFormula"],
" be the strike price, one gets the final condition "
}], "Text",
CellTags->{"S0.0.0", "0.5"}],
Cell["\<\
w[x,ts] = x - c for x >= c,
w[x,ts] = 0 for x < c (2)\
\>", "Input",
CellLabel->"In[2]:= ",
CellTags->"S0.0.0"],
Cell["\<\
which notes that the option value is worthless if the stock price \
does not exceed the strike price upon expiration. \
\>", "Text",
CellTags->{"S0.0.0", "0.6"}],
Cell["Solving the Black\[Hyphen]Sholes PDE", "Section",
CellTags->{"S0.0.0", "0.7"}],
Cell["\<\
How do we solve (1) subject to (2)? By substituting variables. In \
their 1973 paper, Black and Scholes recommend the following \
non\[Hyphen]obvious but nonetheless applicable substitution: \
\>", "Text",
CellTags->{"S0.0.0", "0.8"}],
Cell["w[x,t] = Exp[r (t-ts)] y[u[x,t], s[t]] (3)", "Input",
CellLabel->"In[3]:= ",
CellTags->"S0.0.0"],
Cell["where ", "Text",
CellTags->{"S0.0.0", "0.9"}],
Cell["\<\
u[x,t] = (2/v^2) (r - 0.5 v^2) (Log[x/c] -
(r - 0.5 v^2) (t - ts)) (4)\
\>", "Input",
CellLabel->"In[4]:= ",
CellTags->"S0.0.0"],
Cell["s[t] = -(2/v^2) (r - 0.5 v^2)^2 (t - ts) (5)", "Input",
CellLabel->"In[5]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"If a stint in chain\[Hyphen]rule boot camp is your cup of tea, plug (4) \
and (5) into (3), then plug (3) into (1), and simplify (1). If you keep your \
bookkeeping correct (",
StyleBox["Mathematica", "TI"],
" can help), you should emerge from the battle with "
}], "Text",
CellTags->{"S0.0.0", "0.10"}],
Cell["D[y[u,s],s] = D[y[u,s],{u,2}] (6)", "Input",
CellLabel->"In[6]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"This PDE, readers, is the equation for one\[Hyphen]dimensional conductive \
heat transfer, with y starring as temperature, and with u and s in the \
supporting roles of spatial dimension and time, respectively. The premiere \
issue of ",
StyleBox["The Mathematica Journal", "TI"],
" had in fact not one but two papers devoted to this PDE: Mine explicitly \
discussed the 2\[Hyphen]D analogue to the PDE and illustrated a solution to \
that analogue; Ross Miller's ignored the PDE but provided code for a \
particular solution of it. To get that particular solution, we extend the \
variable substitutions (3\[Hyphen]5) to the final condition (2). \
Specifically, we let ",
StyleBox["t -> ts", "MR"],
" in (4) and (5), which yields ",
StyleBox["s[ts] = 0", "MR"],
" from (5), and "
}], "Text",
CellTags->{"S0.0.0", "0.11"}],
Cell["u[x,ts] = (2/v^2) (r - 0.5 v^2) Log[x/c] (7)", "Input",
CellLabel->"In[7]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"from (4). Solving (7) for ",
StyleBox["x", "MR"],
", and substituting that result and ",
StyleBox["s[ts] = 0", "MR"],
" into (2) yields the final condition "
}], "Text",
CellTags->{"S0.0.0", "0.12"}],
Cell["\<\
y[u,0] = 0 for u <= 0, y[u,0] =
c (Exp[(u v^2)/(2 (r-0.5 v^2))]-1)
for u > 0 (8)\
\>", "Input",
CellLabel->"In[8]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"The task is now to solve (6) subject to (8). [Churchill 1963, pp.\
\[NonBreakingSpace]152\[Hyphen]154] and [Carslaw and Jaeger 1959, pp.\
\[NonBreakingSpace]58\[Hyphen]62] address this situation. These works \
describe 1\[Hyphen]D conductive heat transfer in a semi\[Hyphen]infinite \
solid, with the initial temperature a function of only the one spatial \
dimension, and with the one face of the solid maintained at zero temperature. \
In our situation, that translates to ",
StyleBox["u >= 0", "MR"],
", ",
StyleBox["y[u,0] = f[u]", "MR"],
", and ",
StyleBox["y[0,s] = 0", "MR"],
". We've already seen how, in the ",
StyleBox["u,s", "MR"],
" domain, ",
StyleBox["y[u,0]", "MR"],
" represents not an initial condition but a final condition (",
StyleBox["y[u,s]", "MR"],
" upon option expiration, in particular). Note from (4), however, that ",
StyleBox["u -> 0", "MR"],
" only when 1) ",
StyleBox["r = 0.5 v^2", "MR"],
", or 2) ",
StyleBox["x = c", "MR"],
" and ",
StyleBox["t = ts", "MR"],
". In either case, (5) shows that ",
StyleBox["s -> 0", "MR"],
" as well: Thus, ",
StyleBox["y[0,s] = y[0,0] = 0", "MR"],
", which is consistent with (8). The second case makes eminent sense: If \
stock price ",
StyleBox["x", "MR"],
" only equals strike price ",
StyleBox["c", "MR"],
" on expiration date ",
StyleBox["ts", "MR"],
", then option value ",
StyleBox["Exp[r (ts - ts)] y[0,0] = w[x,ts]", "MR"],
" should be zero. The first case shows why the conditions for ",
StyleBox["u", "MR"],
" are what they are in (8). Were they ",
StyleBox["u < 0", "MR"],
" and ",
StyleBox["u >= 0", "MR"],
" (as [Black\[Hyphen]Scholes 1973, p.\[NonBreakingSpace]644, eqn.\
\[NonBreakingSpace]11] suggests but as [Churchill 1963, \
p.\[NonBreakingSpace]152, eqn.\[NonBreakingSpace]3] does not), ",
StyleBox["r = 0.5 v^2", "MR"],
" would lead to an indeterminate ",
StyleBox["Exp[0/0]", "MR"],
" expression for ",
StyleBox["y[u,0]", "MR"],
" in (8). Note how, in the ",
StyleBox["u,s", "MR"],
" domain, the boundary condition ",
StyleBox["y[0,s] = 0", "MR"],
" cannot be met without also meeting the final condition ",
StyleBox["y[u,0] = 0", "MR"],
" \[Hyphen] such is the nature of (6) in an option valuation context. \nThe \
machinations in the preceding paragraph let us solve the option valuation \
problem using a conductive heat transfer template. [Carslaw and Jaeger 1959, \
p.\[NonBreakingSpace]58] suppose their semi\[Hyphen]infinite solid continued \
in each direction of the one spatial dimension, to maintain the one solid \
face at zero temperature. Were we to adopt this supposition, we would have to \
write ",
StyleBox["y[u',0] = - f[u']", "MR"],
" for ",
StyleBox["-u' (u' > 0)", "MR"],
". From (8), though, we treat ",
StyleBox["y[u',0] = 0", "MR"],
" for ",
StyleBox["-u'", "MR"],
" as defined. This treatment allows us to zero the integration from ",
StyleBox["-Infinity", "MR"],
" to zero in [Carslaw and Jaeger 1959, p.\[NonBreakingSpace]59, eqn.\
\[NonBreakingSpace]1], or in [Churchill 1963, p.\[NonBreakingSpace]153, eqn.\
\[NonBreakingSpace]10]. We are left with "
}], "Text",
CellTags->{"S0.0.0", "0.13"}],
Cell["\<\
y[u,s] = (2 Sqrt[Pi s])^1 *
Integrate[f[u'] Exp[- (u - u')^2/(4 s)],
{u', 0, +Infinity}] (9)\
\>", "Input",
CellLabel->"In[9]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"With ",
StyleBox["f[u'] = y[u',0]", "MR"],
" from (8), and with either 1) the ",
StyleBox["u' = u + 2 q Sqrt[s]", "MR"],
" substitution from the heat transfer references, or 2) the ",
StyleBox["u' = u + q Sqrt[2 s]", "MR"],
" substitution from [Black and Scholes 1973], we can rewrite (9) in terms \
of ",
StyleBox["q", "MR"],
". Using ",
StyleBox["u' = u + q Sqrt[2 s]", "MR"],
", ",
StyleBox["du' = Sqrt[2 s] dq", "MR"],
", ",
StyleBox["(u - u')^2 = 2 s q^2", "MR"],
", ",
StyleBox["u' = 0 -> q = -u/Sqrt[2 s]", "MR"],
", ",
StyleBox["u' = +Infinity -> q = +Infinity", "MR"],
", and (4) and (5), we get "
}], "Text",
CellTags->{"S0.0.0", "0.14"}],
Cell["\<\
f[u + q Sqrt[2s]] =
x Exp[(r - 0.5 v^2) (ts - t)] Exp[q v
Sqrt[ts - t]] - c (10)\
\>", "Input",
CellLabel->"In[10]:= ",
CellTags->"S0.0.0"],
Cell["we can rewrite (9) as ", "Text",
CellTags->{"S0.0.0", "0.15"}],
Cell["\<\
y[x,t] = (1/Sqrt[2 Pi]) *
Integrate[x Exp[(r - 0.5 v^2) (ts -
t)] Exp[q v Sqrt[ts - t]] Exp[-q^2/2] -
c Exp[-q^2/2],
{q, -u[x,t]/Sqrt[2 s[t]], +Infinity}] (11)\
\>", "Input",
CellLabel->"In[11]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"Solving (11) in the ",
StyleBox["x,t", "MR"],
" domain (",
StyleBox["Mathematica", "TI"],
" can help), simplifying, and plugging the simplification into (3) finally \
yields the option valuation expression we've been seeking: "
}], "Text",
CellTags->{"S0.0.0", "0.16"}],
Cell["\<\
w[x,t] = x/2 +
(x/2) Erf[(Log[x/c] + (r +
0.5 v^2) (ts - t))/
(v Sqrt[2] Sqrt[ts - t])] -
(c/2) Exp[r (t - ts)] -
((c/2) Exp[r (t - ts)] *
Erf[(Log[x/c] + (r - 0.5 v^2) (ts -
t))/(v Sqrt[2] Sqrt[ts - t])]) (12)\
\>", "Input",
CellLabel->"In[12]:= ",
CellTags->"S0.0.0"],
Cell[TextData[{
"[Black and Scholes 1973, p.\[NonBreakingSpace]644, \
eqn.\[NonBreakingSpace]13] present a less intimidating version of (12) that \
uses the cumulative normal density function ",
StyleBox["N[d] = 0.5 + (0.5 * Erf[d/", "MR"],
" ",
StyleBox["Sqrt[2]])", "MR"],
". "
}], "Text",
CellTags->{"S0.0.0", "0.17"}],
Cell["Solution Significance", "Section",
CellTags->{"S0.0.0", "0.18"}],
Cell[TextData[{
"Why go through the trouble of illustrating this derivation? There are a \
number of reasons. For one: Your reviewer is admittedly smitten by the \
elegant parallel between 1\[Hyphen]D heat conduction and European stock call \
option valuation. For another: The derivation as presented by [Black and \
Scholes 1973] is a bit spotty, and its vague reference to [Churchill 1963, p.\
\[NonBreakingSpace]155] is a bit bemusing: Eqn.\[NonBreakingSpace]6 of p.\
\[NonBreakingSpace]155 of [Churchill 1963] would suffice as the template \
integral, ",
StyleBox["except", "TI"],
" for a ",
StyleBox["-Infinity", "MR"],
" lower limit of integration that does not correspond to the required \
finite lower limit. Using Churchill's eqn. 6 ... ",
StyleBox["-Infinity", "MR"],
" lower limit and all ... appears to yield ",
StyleBox["w[x,t] = x - c Exp[r (t - ts)]", "MR"],
", which satisfies (2) but erroneously implies that volatility plays no \
role in option valuation. The ",
StyleBox["main", "TI"],
" reason, however, is that (12) is of enormous importance. Primarily for \
their work to derive (12), Myron S. Scholes and Robert C. Merton received the \
Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel, 1997, \
from the Royal Swedish Academy of Sciences. \nIf you read the announcement \
regarding 1997 Nobel laureates Scholes and Merton (at www.nobel.se/ \
announcement\[Hyphen]97/economy97.html, for one), you will also find \
accolades for Fischer Black. There is no question that Black contributed as \
much to (12) as Scholes and Merton did. In fact, [Black 1989] wrote an \
informal essay on the 1973 paper he and Scholes co\[Hyphen]authored, \
describing how (12) came to be. In the essay, he notes that, despite his Ph. \
D. in applied mathematics, he lacked familiarity with standard PDE solutions. \
He also notes that, despite his A.B. in physics, he didn't recognize the \
likes of (6) (even though [Bird et al 1960, p.\[NonBreakingSpace]353] call \
(6) \[OpenCurlyDoubleQuote]one of the most worked\[Hyphen]over equations of \
theoretical physics\[CloseCurlyDoubleQuote]). He notes that, had laureate \
Merton not overslept and missed his and Scholes' presentation at a summer \
1970 conference, their three\[Hyphen]way collaboration would have reached \
fruition much more rapidly. He credits the Univ. of Chicago's Merton Miller \
and Eugene Fama with helping to get the seminal 1973 paper published. And, in \
his essay's sub\[Hyphen]title, he notes that \[OpenCurlyDoubleQuote]like many \
great inventions, [the option valuation formula] started with tinkering and \
ended with delayed recognition\[CloseCurlyDoubleQuote]. How poignant these \
words would prove. In 1995, six years after he published his essay, and two \
years before the ultimate recognition of his great invention, Fischer Black \
died. He was in his mid\[Hyphen]fifties. "
}], "Text",
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Cell["MathSource Item BlackScholes.m", "Section",
CellTags->{"S0.0.0", "0.20"}],
Cell[TextData[{
StyleBox["MathSource", "TI"],
" item www.mathsource.com/MathSource/Applications/EconomicsFinance/0209\
\[Hyphen]270/BlackScholes.m (5 kB) by Ross M. Miller is the ",
StyleBox["Mathematica", "TI"],
" package corresponding to Miller's 1990 paper in ",
StyleBox["The Mathematica Journal", "TI"],
". Specifically, ",
StyleBox["BlackScholes.m", "MR"],
" is the code on p.\[NonBreakingSpace]77 of [Miller 1990]. Both this ",
StyleBox["MathSource", "TI"],
" item and Miller's 1990 paper complement his book ",
StyleBox["Computer\[Hyphen]Aided Financial Analysis", "TI"],
" (1990, Addison\[Hyphen]Wesley). Schooled in mathematics and economics, \
Miller was plying his trade in the General Electric Corporate Research and \
Development Center at the time of his 1990 publications. As his paper \
reveals, Miller is a champion of symbolic approaches to financial analysis, \
so it comes as no surprise that ",
StyleBox["Mathematica", "TI"],
" is a potent weapon in his arsenal. \nMiller's paper predicts heavy use of \
spreadsheet front\[Hyphen]ends with ",
StyleBox["Mathematica", "TI"],
" kernels among financial users. As has been noted, the paper avoids (1) \
and (2), but begins with the Black\[Hyphen]Scholes version of (12) and \
proceeds from there. It provides a symbolic treatment of the stock option \
valuation formula. It includes a thorough assessment of ",
StyleBox["w[x,t]", "MR"],
" sensitivity to its parameters (",
StyleBox["x", "MR"],
", ",
StyleBox["r", "MR"],
", ",
StyleBox["v", "MR"],
", ",
StyleBox["ts", "MR"],
") \[Hyphen] these partial derivatives, in fact, are the core of ",
StyleBox["BlackScholes.m", "MR"],
". The paper also alludes to extensions of the Black\[Hyphen]Scholes model, \
as described in the financial works referenced by the paper. In retrospect, \
it is a credit to the Addison\[Hyphen]Wesley handlers of ",
StyleBox["The Mathematica Journal", "TI"],
" that they chose such a significant topic for their premiere issue. Any \
periodical can deem a subject worthwhile after that subject lands Nobel \
Prizes for those involved. It takes foresight to arrange subject coverage \
several years before the Prizes are awarded. \nOther works, both within and \
outside ",
StyleBox["MathSource", "TI"],
", deal with both the Black\[Hyphen]Scholes formula and the efforts of Ross \
M. Miller. Miller's discussion of option pricing appears in Hal R. Varian's ",
StyleBox["Microeconomic Analysis, 3rd ed.", "TI"],
" (1992, W. W. Norton and Company): See the notebooks within the \
www.mathsource.com/cgi\[Hyphen]bin/MathSource/ \
Publications/BookSupplements/Varian\[Hyphen]1992/0202\[Hyphen]419/ directory. \
Miller also contributed the optvalue.m package to the Varian\[Hyphen]edited ",
StyleBox["Economic and Financial Modeling with Mathematica", "TI"],
" (1993, TELOS/Springer\[Hyphen]Verlag), which covers Black\[Hyphen]Scholes \
and other option pricing models: See the compressed files in the \
www.mathsource.com/cgi\[Hyphen]bin/MathSource/Publications/BookSupplements/\
Varian\[Hyphen]1993/0205\[Hyphen]399/ directory. A recent arrival, William \
Shaw's ",
StyleBox["Modelling Financial Derivatives Using Mathematica", "TI"],
" (1998, Cambridge University Press), expands upon Miller's spreadsheets\
\[Hyphen]",
StyleBox["Mathematica", "TI"],
" discussion (spreadsheets vex Shaw). Shaw's work also presents the Black\
\[Hyphen]Scholes option valuation formula, and \[Hyphen] in Chapter 1 \
\[Hyphen] promises a ",
StyleBox["Mathematica", "TI"],
"\[Hyphen]assisted solution of (1) and (2): See \
http://www.mathsource.com/MathSource/Publications/BookSupplements/Shaw\
\[Hyphen]1998/0209\[Hyphen]023/Chapter1.nb. \nOn the World\[Hyphen]Wide Web, \
there are a vast number of pages devoted to the Black\[Hyphen]Scholes effort, \
as one would expect. Your reviewer's favorite was a page dated 21 March 1997. \
Labeled \[OpenCurlyDoubleQuote]Ashwin's Applet \
Factory\[CloseCurlyDoubleQuote], this work was brought forth by one Ashwin \
Kapur, then an undergraduate at Illinois Wesleyan University pursuing a \
double\[Hyphen]major in mathematics and economics. As of this writing, the \
URL for Kapur's page was www.iwu.edu/~akapur/java/toc.html. The page offers a \
number of option pricers and the like, but it is the Black Scholes European \
Stock Options Pricer that we shall inspect. The page \
www.iwu.edu/~akapur/java/bssapplet.html offers computations involving Black \
Scholes European ",
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test case to this ",
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StyleBox["t", "MR"],
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statements in the package): "
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StyleBox["BlackScholes.m", "MR"],
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StyleBox["OptionKappa[98.5, 100., 0.23, 0.08, 0.3]", "MR"],
" = $21.3522. \nAre these values correct? In his equation for the Black\
\[Hyphen]Scholes stock option call valuation formula, Miller uses ",
StyleBox["(1+r)^(t-ts)", "MR"],
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StyleBox["Exp[r (t-ts)]", "MR"],
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the basis for Black and Scholes' formulation of (1). Shaw's 1998 work \
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unexplained use of simple (in lieu of continuously compounding) interest, \
Miller has written a ",
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reminds one of the promise ",
StyleBox["Mathematica", "TI"],
" held back when publications about it were just getting under way. Once \
corrected, ",
StyleBox["BlackScholes.m", "MR"],
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StyleBox["Mathematica", "TI"],
" plotting functions to generate graphs superior to those from the applet. \
A deeper knowledge of such functions can bring forth all sorts of instructive \
2\[Hyphen]D plots, 3\[Hyphen]D plots, and animations to explain (or at least \
illustrate) European stock option call pricing phenomena. And, of course, ",
StyleBox["Mathematica", "TI"],
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concepts such as implied volatility and elasticity \[Hyphen] a point that \
Miller himself [Miller 1990] takes care to make. To make these observations \
is not, mind you, to disparage Ashwin Kapur's applet: This product of \
Illinois Wesleyan has put forth a fine effort for all the world to use. But \
Java just does not offer the platform for symbolic financial analysis that ",
StyleBox["Mathematica", "TI"],
" offers. \nOnce in a while, it is good to reflect upon what has occurred \
over the five years since the start of these Reviews ... the several years \
since the introduction of ",
StyleBox["The Mathematica Journal", "TI"],
" ... the decade since the debut of ",
StyleBox["Mathematica", "TI"],
". Things change: Ross M. Miller's paper describes spreadsheets such as \
Lotus 1\[Hyphen]2\[Hyphen]3, Microsoft Excel, and Wingz (Wingz!) as \
computationally inferior to ",
StyleBox["Mathematica", "TI"],
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thinks to run alongside ",
StyleBox["Mathematica", "TI"],
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do well to pay heed to lessons learned along the way. When those lessons \
unveil hidden delights \[Hyphen] commonality between two very different \
phenomena, a paper presaging significant recognition for a great \
accomplishment, a tool and its attendant publications enduring and even \
thriving \[Hyphen] we are reminded, as per Don Quixote, that the road is \
indeed better than the inn. "
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Cell[TextData[{
"Bird, R. Byron, Stewart, Warren E, and Edwin N. Lightfoot. ",
StyleBox["Transport Phenomena", "TI"],
". John Wiley and Sons, New York (1960). \nBlack, Fischer. \
\[OpenCurlyDoubleQuote]How we came up with the option formula\
\[CloseCurlyDoubleQuote], ",
StyleBox["Journal of Portfolio Management", "TI"],
", ",
StyleBox["15", "TB"],
"(2), 4\[Hyphen]8 (1989). \nBlack,Ffischer and Scholes, Myron. \
\[OpenCurlyDoubleQuote]The Pricing of Options and Corporate Liabilities\
\[CloseCurlyDoubleQuote], ",
StyleBox["Journal of Political Economy", "TI"],
", ",
StyleBox["81", "TB"],
"(3), 637\[Hyphen]654 (1973). \nCarslaw, H. S. and Jaeger, J. C. ",
StyleBox["Conduction of Heat in Solids, 2nd ed.", "TI"],
" Oxford University Press, London (1959). \nChurchill, Ruel V. ",
StyleBox["Fourier Series and Boundary Value Problems", "TI"],
". McGraw\[Hyphen]Hill, New York (1963). \nMiller, Ross M. \
\[OpenCurlyDoubleQuote]Computer\[Hyphen]Aided Financial Analysis: An \
Implementation of the Black\[Hyphen]Scholes Model\[CloseCurlyDoubleQuote], ",
StyleBox["The Mathematica Journal", "TI"],
", ",
StyleBox["1", "TB"],
"(1), 75\[Hyphen]79 (1990). "
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Washington University in St. Louis, in December 1995. The batch chemical \
process quality control aspects of that work are meticulously explained in \
the October 1997 issue of the ",
StyleBox["AIChE Journal", "TI"],
", with a related news item in the ",
StyleBox["Chemical Engineering Progress", "TI"],
" issue from the same month. \nthomas@wuche2.wustl.edu "
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(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)