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Notebook[{
Cell[TextData["\n\n"], "Input",
PageWidth->Infinity,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData["Option Valuation"], "Title",
Evaluatable->False,
CellHorizontalScrolling->False,
TextAlignment->Center,
AspectRatioFixed->True],
Cell[TextData["Ross M. Miller"], "Subtitle",
Evaluatable->False,
CellHorizontalScrolling->False,
TextAlignment->Center,
AspectRatioFixed->True],
Cell[TextData[
"GE Corporate Research & Development\nP.O. Box 8\nSchenectady, NY 12301\n\
Email: millerrm@crd.ge.com"], "Subsubtitle",
Evaluatable->False,
CellHorizontalScrolling->False,
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Cell[CellGroupData[{Cell[TextData["Introduction"], "Section",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"\nAlthough financial computation is often viewed as an exercise in number \
crunching, the emergence of a new breed of financial \"rocket scientists\" \
has expanded the role of computers in finance to include not only numerical \
manipulations, but also structural manipulations. Investment houses now \
routinely \"slice and dice\" securities such as mortgages, government bonds, \
and even the infamous \"junk bonds,\" to engineer their cash flows to meet \
particular risk/return criteria. While spreadsheet programs and traditional \
programming languages (e.g., FORTRAN and C) continue to play an important \
role in financial computation, symbolic programming languages, i.e., \
languages that manipulate both the numbers and the symbols with which \
financial structures are represented, are taking hold as a way of dealing \
with the increasing complexity of the financial world. Indeed, some of the \
more innovative investment houses around the world have been using LISP and \
Smalltalk since the mid-1980's to handle a variety of difficult valuation and \
design problems."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"This chapter contains a redesigned version of a suite of option valuation \
tools that the author originally developed in LISP as part of a comprehensive \
textbook on the application of object-oriented and artificial intelligence \
technology to finance analysis (Miller, 1990a), and some of which were \
included in the premiere issue of the Mathematic Journal (Miller, 1990b). It \
is a testament to Mathematica's flexibility that even the most complex \
LISP-based tool developed in conjunction with that textbook ports easily to \
Mathematica. This chapter contains a brief introduction to option valuation; \
however, a more complete introduction to the topic can be found in any of \
several sources. The original exposition of the Black-Scholes model appears \
in Black and Scholes, 1973 and an excellent adaptation of the \
Cox-Ross-Rubinstein binomial model appears in Cox and Rubinstein, 1985. An \
excellent textbook that covers a wide range of topics in option valuation is \
Hull, 1990."], "Text",
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Cell[TextData[
"This chapter takes a very general approach to the problem of financial \
valuation, utilizing object-oriented design methods to the extent that they \
are possible within Mathematica to achieve this generality. The traditional \
approach that economists have taken to computing has been to create \
completely separate programs or procedures for each model. In the \
object-oriented approach to computing the goal is to create general \
valuation procedures, called methods, that can operate on many different \
types of objects, in this case options. The advantage to the object-oriented \
approach is that the development of new financial instruments does not \
require programming new, ad hoc valuation procedures. Instead, existing \
objects that represent related financial instruments are updated to reflect \
any innovations in the newly-created instrument. Furthermore, once an object \
has been formally defined, other methods can be created to perform other \
functions, such as accounting, without starting from scratch."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"The Mathematica functions developed in this chapter were designed with their \
pedagogical utility foremost in mind, especially when used interactively as a \
Mathematica notebook. Although we have tried to make them as computationally \
efficient as possible, in some instances speed has been sacrificed in favor \
of simplicity or elegance. In particular, the binomial model has been \
developed within a very general framework that is readily extensible to more \
complex valuation problems, but is in no way optimized for the binomial \
model. In addition, the tendency in this chapter is to use Mathematica's \
built-in algorithms even in cases where user-defined alternatives would be \
far more efficient. Finally, the functions developed in this chapter have \
been designed for interpreted rather than compiled use. "], "Text",
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Cell[CellGroupData[{Cell[TextData["The Black-Scholes Model"], "Section",
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Cell[TextData[
"\nThe Black-Scholes model provides a direct way of valuing a call option for \
common stock. A call option is an option to buy stock at a pre-specified \
exercise or strike price prior to a given expiration date. (An option which \
can only be exercised on its expiration date is known as a European option, \
while one that can be exercised at any time prior to expiration is known as \
an American option. Most exchange-traded stock options in the U.S. are \
American options. Except in special cases, the Black-Scholes model must be \
modified to deal with the possibility of early exercise.)"], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"If the market price of the stock is greater than the exercise price when the \
expiration date arrives, then the value of the option will be equal to the \
payoff that can be created by buying the stock at the exercise price and then \
immediately selling the stock at its market price. Otherwise, it will not \
pay to exercise the option, and so it will expire with zero value. The payoff \
pattern of an option is easily modeled in Mathematica by the function \
CallPayoff as follows:"], "Text",
Evaluatable->False,
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Cell[CellGroupData[{Cell[TextData["CallPayoff[price_,strike_] = Max[0,price-strike]"], "Input",
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" For example, consider the payoff function for a call option that gives the \
holder the option to buy a share of DEC stock at a price of $60 on the third \
Friday in June. If the price of DEC stock is $80 on expiration in June, then \
the option will pay off $20; however, if DEC stock is below $60, the option \
will be worth $0 as it will not pay to exercise it. This payoff function is \
readily plotted as follows:\n"], "Text",
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volatilities the option value is nearly perfectly linear. Nonetheless, it is \
evidence of an important feature of the Black-Scholes model, i.e., that it \
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closed form. The most important parameter that affects the value of an \
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"Delta is a useful measurement of risk for an option because it indicates how \
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stock. For call options, delta can range from 0 to 1, i.e., the option may \
be insensitive to change in the stock price, may track it exactly, or may lie \
somewhere in between. Delta is a particularly useful gauge of the risk \
contained in a portfolio that contains more than one option on a given stock. \
In particular, the theoretical risk associated with the holding of a stock \
will be completely neutralized (in the very short run) if the overall \
dollar-weighted delta of a portfolio of options on that stock is equal to \
zero. This is because the portfolio delta gives change in portfolio value for \
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delta is a partial derivative it assumes that all other variables are held \
constant and the change in stock price is relatively small, which will \
usually not be the case over even a short period of time. Additional \
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.62 .784 .96 r
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.621 .784 .96 r
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.62 .784 .96 r
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.62 .784 .96 r
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.62 .784 .96 r
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.62 .784 .96 r
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.62 .784 .96 r
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.62 .784 .96 r
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.62 .784 .96 r
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0 0 .562 r
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.626 .779 .955 r
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.626 .779 .955 r
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.62 .784 .96 r
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.62 .784 .96 r
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Cell[TextData[
"As was mentioned at the beginning of this section, the partial derivatives \
of the Black-Scholes formula can also be applied to an entire portfolio of \
options to determine the sensitivity of the value of the portfolio to changes \
in any or all of the variables that underlie the formula."], "Text",
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CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"\nOn a related note, in situations where an option's price is believed to be \
an accurate indication of its value, it can be desirable to \"reverse \
engineer\" the volatility of an option from its market price. The function \
ImpliedVolatility uses the built-in function FindRoot to solve numerically \
for the volatility of an option given its price as follows:\n"], "Text",
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Cell[TextData[
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optionprice,{sd,0.2}]\n"], "Input",
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Cell[TextData[
"Hence, if we knew the price of the DEC option given above was 3.34886 we \
could verify that the volatility is 0.29 as follows:"], "Text",
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Cell[CellGroupData[{Cell[TextData["ImpliedVolatility[58.5,60.,0.04,0.3,3.34886]"], "Input",
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Cell[TextData[
"Implied volatility is an extremely useful way of looking at options; indeed, \
some options on foreign currencies and other financial instruments are \
frequently quoted in terms of their implied volatility rather than by price, \
much as bonds are quoted by yield rather than price. There are many trading \
strategies that are designed to go either long or short volatility while \
limiting risk using the techniques described above."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"\nAs with any other group of related functions in Mathematica, the functions \
associated with the Black-Scholes model can be collected into a single \
Mathematica package that accompanies this book. The public part of this \
package, which appears before the Begin[\"`private`\"] statement, provides \
access to these functions, as well as to a small database described later. \
This package includes not only the functions defined in this chapter but \
additional functions for popular measures of option value sensitivity--theta, \
kappa, rho, gamma, and (stock price) elasticity.\n"], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True]}, Open]],
Cell[CellGroupData[{Cell[TextData["Options as Objects"], "Section",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"\nMathematica provides now only numerical and symbolic manipulation \
capabilities for formulas, it also provides the basic tools needed to treat \
options and option-based securities as self-contained objects and to make the \
action of function depend on the type of object to which it is applied. \
Although the object manipulation capabilities of Mathematica fall short of \
those provided by object-oriented design toolkits, the core facilities for \
object creation and data abstraction are contained within Mathematica. This \
section will develop the tools for representing options as objects and the \
following section will exploit this representation to develop the technology \
for having a single Value function that is capable of evaluating a variety of \
options and other securities."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"We will start by showing how the DEC call option (and its underlying DEC \
stock) can be represented as objects in Mathematica. The distinguishing \
features of a computational object are its properties. For example, the \
properties of the DEC option, DECFL, are that it is a call option on DEC \
stock, has an exercise price of $60 and expires in June, which we have \
assumed to be 0.3 of a year away. In Mathematica, we can link these \
properties to the symbol DECFL as follows:"], "Text",
Evaluatable->False,
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AspectRatioFixed->True],
Cell[TextData[
"Type[DECFL] ^= \"call\" ;\nAsset[DECFL] ^= DEC ;\nExercisePrice[DECFL] ^= \
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Cell[TextData[
"The operator ^= is the known as UpSet and is used to make sure that \
Mathematica associates the value assignment with the \"upvalue\" DECFL rather \
than the head of the left-hand side of the assignments as it normally would. \
It is easy to check that these properties have been associated with DECFL as \
follows:"], "Text",
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Cell[CellGroupData[{Cell[TextData["?DECFL"], "Input",
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\>", "\<\
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Cell[OutputFormData["\<\
No Input Form Generated
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\>", "\<\
Asset[DECFL] ^= DEC
ExercisePrice[DECFL] ^= 60.
ExpirationTime[DECFL] ^= 0.3
Type[DECFL] ^= \"call\"\
\>"], "Print",
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Cell[TextData[
"Global`DECFL\n\nAsset[DECFL] ^= DEC\n \nExercisePrice[DECFL] ^= 60.\n \n\
ExpirationTime[DECFL] ^= 0.3\n \nType[DECFL] ^= \"call\""], "Info",
PageWidth->Infinity,
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Cell[TextData[
"As a convenience we will define an object constructor function, ConsObj, \
that takes a symbol and property list as its arguments:"], "Text",
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Cell[TextData[
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propname[obj]^=propval],\n {i,Length[proplist]/2}]\n"], "Input",
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Cell[TextData["The object, DECFL, can now be constructed as follows:"], "Text",
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Cell[TextData[
"ConsObj[DECFL,{Type,\"call\",\n Asset,DEC,\n \
ExercisePrice,60.,\n ExpirationTime,0.3}]"], "Input",
PageWidth->Infinity,
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Cell[TextData[
"Notice that not all of the information needed to value the DECFL option is \
contained in its object description. In particular, the price and volatility \
of DEC stock are properties that must be \"inherited\" from the object \
representation of the stock. This object, DEC, can be constructed as \
follows:"], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData[
"ConsObj[DEC,{Type,\"stock\",\n Price,58.5,\n\t\t\t \
Volatility,0.29}]\n"], "Input",
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Cell[OutputFormData["\<\
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\>", "\<\
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Possible spelling error: new symbol name \"Price\"
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Cell[TextData[
"The properties of the DEC object are stored in Mathematica as follows:"],
"Text",
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Cell[CellGroupData[{Cell[TextData["?DEC"], "Input",
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Cell[OutputFormData["\<\
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\>", "\<\
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Cell[OutputFormData["\<\
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Price[DEC] ^= 58.5
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Cell[TextData[
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Evaluatable->False,
AspectRatioFixed->True]}, Open]],
Cell[TextData[
"Of course, these properties of the DECFL option that come from the \
underlying stock are not inherited automatically; Mathematica needs \
assignment rules to facilitate this inheritance as follows:"], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"AssetPrice[option_] := (AssetPrice[option] ^=\n \
Price[Asset[option]]);\nAssetVolatility[option_] := (AssetVolatility[option] \
^=\n Volatility[Asset[option]])"], "Input",
PageWidth->Infinity,
InitializationCell->True,
AspectRatioFixed->True],
Cell[TextData[
"This form for the assignment ensures that Mathematica associates the value \
with the option rather than the function. We can now test that these two new \
properties of the DECFL option have been properly inherited:"], "Text",
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CellHorizontalScrolling->False,
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Cell[CellGroupData[{Cell[TextData["AssetPrice[DECFL]"], "Input",
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Cell[OutputFormData["\<\
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58.5\
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Cell[CellGroupData[{Cell[TextData["AssetVolatility[DECFL]"], "Input",
PageWidth->Infinity,
InitializationCell->True,
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0.29\
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Cell[TextData[
"Checking the values now associated with DECFL, we can see that these two new \
properties are listed:"], "Text",
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Cell[CellGroupData[{Cell[TextData["?DECFL"], "Input",
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\>", "\<\
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Asset[DECFL] ^= DEC
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ExercisePrice[DECFL] ^= 60.
ExpirationTime[DECFL] ^= 0.3
Type[DECFL] ^= \"call\"\
\>"], "Print",
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Cell[TextData[
"Global`DECFL\n\nAsset[DECFL] ^= DEC\n \nAssetPrice[DECFL] ^= \n \
OptionValue`private`Price[OptionValue`private`Ass\\\n et[DECFL]]\n \n\
AssetVolatility[DECFL] ^= \n \
OptionValue`private`Volatility[OptionValue`private`Ass\\\n et[DECFL]]\n \n\
ExercisePrice[DECFL] ^= 60.\n \nExpirationTime[DECFL] ^= 0.3\n \nType[DECFL] \
^= \"call\""], "Info",
PageWidth->Infinity,
Evaluatable->False,
AspectRatioFixed->True]}, Open]],
Cell[TextData[
"As noted above, Mathematica does not provide a full object-oriented design \
environment so these new \"properties\" are not generated automatically, but \
become part of the object description after their first use.\nThe final piece \
of information necessary to evaluate DECFL is the risk-free rate of return. \
Because this rate is assumed to be constant and can be applied to all \
options, it makes sense to define it as a global variable as follows:"],
"Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData["RiskFreeRate = 0.04"], "Input",
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Cell[OutputFormData["\<\
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\>", "\<\
0.04\
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Cell[TextData[
"It is now a simple matter to define a value function that takes an option's \
symbol as its argument and retrieves the necessary information to apply the \
Black-Scholes function defined above:"], "Text",
Evaluatable->False,
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Cell[CellGroupData[{Cell[TextData[
"Value[option_] := BlackScholes[AssetPrice[option],\n \
ExercisePrice[option],\n \
AssetVolatility[option],\n RiskFreeRate,\n \
ExpirationTime[option]]"], "Input",
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InitializationCell->True,
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Cell[OutputFormData["\<\
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Possible spelling error: new symbol name \"Value\"
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\>"], "Message",
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Cell[TextData[
"We can now demonstrate that Value actually works when applied to DECFL:"],
"Text",
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Cell[CellGroupData[{Cell[TextData["Value[DECFL]"], "Input",
PageWidth->Infinity,
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No Input Form Generated
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\>", "\<\
3.34886\
\>"], "Output",
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Evaluatable->False]}, Open]],
Cell[TextData[
"Of course, the Black-Scholes formula only applies to a limited number of \
options. In the next section we will look at other option valuation methods \
that can be applied to options in general, including put options, and see how \
the Value function can be appropriately extended."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True]}, Open]],
Cell[CellGroupData[{Cell[TextData["Valuing Options with Financial Decision Trees \n"], "Section",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"\nWhen considering options for which the Black-Scholes model is not \
designed, e.g., American put options (options to sell that can be exercised \
early), there is usually no closed-form solution and closed-form \
approximations are frequently inadequate. The source of the problem is that \
the continuous-time techniques used to derive the Black-Scholes formula no \
longer apply when the option valuation path can be disrupted by early \
exercise. Nonetheless, the symbol manipulation features of Mathematica can \
still be used to great advantage. This section will contain a brief survey \
of these techniques and how they might be implemented in Mathematica. These \
methods are sufficiently general that they may not only be applied to \
virtually any kind of option, but also apply to securities with embedded \
options, such as callable and convertible bonds and many kinds of \
mortgage-backed securities. Less ambitious extensions, such as options on \
dividend-paying stocks, can be readily incorporated into this framework. A \
more detailed survey of these techniques is contained in the author's book \
Computer-Aided Financial Analysis, where they were originally developed in \
LISP (Miller, 1990a).\n"], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"The key to solving general financial valuation problems that is introduced \
in this chapter is what the author has called financial decision trees. \
These trees are a generalization of the decision trees used in traditional \
decision analysis. The key extension of decision trees that is introduced in \
this section is dynamic discounting that is applied as one traverses the \
tree. A further extension (Miller, 1990a) also handles cash flows that occur \
at any node in the decision tree. The advantage that financial decision \
trees hold over traditional decision trees is both representational and \
computational. Imbedding discounting and cash flows in the tree itself \
rather than imputing them to terminal nodes, which is the only way to take \
them into account in the traditional approach, minimizes the amount of \
computation required to both represent and evaluate the decision tree that \
represents a given option or financial instrument."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"In this section we will focus on an American put option on DEC stock with \
identical properties to those of the DECFL call option presented earlier \
except for the fact that it is a put option and has the symbol, DECQL. \
Recall that a put option is an option to sell stock at a given exercise \
price, in this case $60. As we saw earlier, the payoff function for a put is \
the opposite of that for a call; it is zero for prices above the exercise \
price and it has a slope of -1 for prices below the exercise price. The \
ConsObj function can be used to create the object DECQL as follows:"], "Text",\
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData[
"ConsObj[DECQL,{Type,\"put\",\n Asset,DEC,\n \
ExercisePrice,60.,\n ExpirationTime,0.3}]"], "Input",
PageWidth->Infinity,
InitializationCell->True,
AspectRatioFixed->True],
Cell[OutputFormData["\<\
No Input Form Generated
\
\>", "\<\
General::spell1:
Possible spelling error: new symbol name \"DECQL\"
is similar to existing symbol \"DECFL\".\
\>"], "Message",
PageWidth->Infinity,
Evaluatable->False]}, Open]],
Cell[TextData[
"Of course, if we were to apply the Value function in its current form to \
DECQL it would have no way of dealing with the fact that it was a put and not \
a call; therefore, it is good to create a property for options that affects \
how they are valued. We will use ExerciseFunction to store the payoff \
function and define it as follows: "], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"ExerciseFunction[option_] := CallPayoff /;\n \
Type[option]==\"call\"\nExerciseFunction[option_] := PutPayoff /;\n \
Type[option]==\"put\""], "Input",
PageWidth->Infinity,
InitializationCell->True,
AspectRatioFixed->True],
Cell[TextData["Hence, for our new put option we have:"], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[CellGroupData[{Cell[TextData["ExerciseFunction[DECQL]"], "Input",
PageWidth->Infinity,
InitializationCell->True,
AspectRatioFixed->True],
Cell[OutputFormData["\<\
No Input Form Generated
\
\>", "\<\
PutPayoff\
\>"], "Output",
PageWidth->Infinity,
Evaluatable->False]}, Open]],
Cell[TextData[
"With an American put option it is quite possible that the value of the \
underlying stock can drop low enough that the natural upward drift of the \
stock price will make it profitable to exercise the option early. Indeed, a \
significant component of the put option's value can be associated with the \
potential for early exercise. The simplest way to model the option that \
enables one to consider the possibility of early exercise explicitly is the \
binominal model. The binomial model divides the time until expiration into a \
number of equal time periods and over each time segment considers two \
possibilities, that the stock move either up in price by a fixed proportion \
or down in price by a fixed proportion. The size of the up and down \
movements as well as their probability can be chosen so that in the limit as \
the number of periods approaches infinity, the distribution of prices will \
converge to the lognormal distribution used in the Black-Scholes model. The \
derivation of these up and down movements and their probabilities is given in \
Hull (1989) and are used here within further discussion."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"As with the Black-Scholes model, the value of the option is known at \
expiration and can be determined recursively by working backwards until the \
present is reached. The difference is that at each point in time the \
potential advantage of exercising the option immediately must be \
considered."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"The approach that we will take to computing the binomial model is to embed \
it in a more general framework that can handle far more complex options. \
This framework casts the process by which stock prices change and the option \
holder considers his or her alternatives at each point in time as a financial \
decision tree. Although every option valuation method may generate a \
different financial decision tree, every tree can be evaluated using the same \
TreeValue function. Hence, the process for valuing an option will be to \
convert it into a tree and then evaluate that tree."], "Text",
Evaluatable->False,
CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"The conversion of an American option into a financial decision tree is \
performed with the MakeAmerTree function which is defined as follows: "],
"Text",
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CellHorizontalScrolling->False,
AspectRatioFixed->True],
Cell[TextData[
"MakeAmerTree[option_,n_:4] := \n Block[{ChanceSymbol = \
Unique[\"cnode\"],\n DecisionSymbol = Unique[\"dnode\"],\n \
s = AssetPrice[option],\n k = ExercisePrice[option],\n\t \
ex = ExerciseFunction[option],\n\t t = ExpirationTime[option],\n \
sd= AssetVolatility[option] Sqrt[t/n] // N,\n a = \
Exp[RiskFreeRate t/n] // N,\n u = Exp[sd],\n d = 1/u},\n\
ConsObj[Evaluate[ChanceSymbol],\n \
{Type,\"chance\",\n Dfactor,1/a,\n Upamt,u,\
\n Downamt,d,\n Upprob,(a-d)/(u-d) // N,\n \
ExercisePrice,k,\n ExerciseFunction,ex,\n \
Succsym,DecisionSymbol}];\n \
ConsObj[Evaluate[DecisionSymbol],\n {Type,\"decision\",\n \
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"This rather imposing function is essentially a list of ingredients needed to \
build a financial decision tree for an American option. It starts by using \
the built-in Mathematica function Unique to create symbols for the two types \
of nodes in the tree, chance nodes, which reflect the up and down movement in \
the stock price, and decision nodes, which reflect the ability of the \
optionholder to choose whether or not to exercise the option. The ConsObj \
function is then used to assign the properties needed by chance and decision \
nodes to their symbols, ChanceSymbol and DecisionSymbol, respectively. \
Finally, it creates the seed of the tree as an expression with node as its \
head and the beginning state of the financial decision tree, including the \
stock price and period remaining as its body. For expository purposes, \
MakeAmerTree defaults to four periods, which is enough to understand what it \
is doing, but not enough for an accurate valuation, which can require ten or \
more periods."], "Text",
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"Hence, we start the tree at a chance node called cnode1 at a stock price of \
58.5 with 4 periods to go and a \"weight\" of 1. (The weight only provides \
useful information at decision nodes, as we shall see.) To find out where we \
can proceed from this root node, we need a Successors function that derives a \
list of successors to both chance and decision nodes. Here is the definition \
of that function for each type of node:"], "Text",
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"Successors[node[symbol_,price_,left_,weight_]] := \n{node[Succsym[symbol],\n \
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Successors[node[symbol_,price_,left_,weight_]] := \n{node[Succsym[symbol],\n \
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"The successors of this chance node are two decision nodes, both with the \
symbol dnode1, but reflecting different states of nature. The first decision \
node corresponds to a DEC stock price of 63.3355, which will occur with a \
probability/weight of 0.499051. The second decison node corresponds to a DEC \
stock price of 54.0336 which will occur with a probability/weight of \
0.5000949. The \"rolling of the dice\" associated with the chance node also \
consumed one period, leaving three to go."], "Text",
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"At a decison node, we are given a choice of continuing to hold the option, \
leaving periods left set to three, or exercising the option by setting the \
periods left to zero."], "Text",
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"The function TreeValue automates the process of generating successors and \
simultaneously computes the mathematical expectation for each chance node and \
chooses the maximum expected payoff for each decision node. This function \
and its two auxiliary functions, Prob and Expectation, are defined as \
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"TreeValue[node[symbol_,price_,left_,weight_]] :=\n\
(TreeValue[node[symbol,price,left,weight]]\n = Which[left==0,\n \
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Max[TreeValue[Successors[node[symbol,price,left,weight]]]]]);\n\n\
SetAttributes[TreeValue,Listable]\n \nProb[node_] := node[[4]] ; \
SetAttributes[Prob,Listable] \n\nExpectation[nodelist_] := \
Dot[Prob[nodelist], \n TreeValue[nodelist]] "],
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"The TreeValue function uses Which to distinguish three situations, terminal \
nodes (left==0), chance nodes, and decision nodes. TreeValue is designed to \
remember old values so that dynamic programming is employed to reduce the \
number of evaluations that are required. The downside of this approach is \
that without additional memory management, memory will tend to fill with the \
results of old option valuations."], "Text",
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generated as follows:"], "Text",
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"Finally, we can return to the Value function introduced at the end of the \
previous section and extend its definition to handle all American options on \
stock that do not pay dividends as follows:"], "Text",
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Cell[TextData[
"Value[option_] := BlackScholes[AssetPrice[option],\n \
ExercisePrice[option],\n \
AssetVolatility[option],\n RiskFreeRate,\n \
ExpirationTime[option]] /;\n \
Type[option]==\"call\"\n \nValue[option_] := \
TreeValue[MakeAmerTree[option,8]] /;\n \
Type[option]==\"put\""], "Input",
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"This function will now simply apply the Black-Scholes formula to call \
options and will grow and evaluate a binomial tree for put options. Hence, \
we can apply it to the two DEC options as follows:"], "Text",
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"This chapter has provided an extensive introduction to closed-form and \
numerical approaches to option valuation using Mathematica. Because of its \
versatile nature, Mathematica is well-suited to the large variety of \
techniques that may be needed to value options and securities with option \
components to them. The motivated reader will find it easy to use \
Mathematica to extend the basic methods developed here to the wide range of \
option valuation techniques that have been developed in the financial \
literature. Elsewhere (Miller, 1991) the author has addressed the problem of \
additional ways of using rule-based methods to extend and build a more \
\"intelligent\" object-oriented valuation function."], "Text",
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Corporate Liabilities,\" Journal of Political Economy, 81, 637-659."], "Text",\
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Cell[TextData[
"Cox, John C. and Mark Rubinstein. 1985. Options Markets: Englewood Cliffs \
(NJ), Prentice-Hall."], "Text",
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Cell[TextData[
"Hull, John. 1989. Options, Futures and Other Derivative Securities: \
Englewood Cliffs (NJ), Prentice-Hall."], "Text",
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Cell[TextData[
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Addison-Wesley."], "Text",
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"Miller, Ross M. 1991. \"An Intelligent Approach to Financial Valuation,\" \
Proceedings of the First International Conference on Artificial Intelligence \
Applications on Wall Street, IEEE Press."], "Text",
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"Ross M. Miller received his B.S. in mathematics from the California \
Institute of Technology and his A.M. and Ph.D. in economics from Harvard \
University. He served on the faculties of Boston University, the California \
Institute of Technology, and the University of Houston before taking his \
present position as a senior member of the technical staff of the GE \
Corporate Research & Development Center. Dr. Miller is author of the book \
Computer-Aided Financial Analysis and has published extensively on \
information transfer in financial markets and on the design of advanced \
electronic market systems."], "Text",
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Cell[TextData[
"This paper is a virtual duplicate of the paper appearing in the book, \
Economic and Financial Modeling with Mathematica edited by Hal Varian and \
published in 1993 by TELOS/Springer-Verlag. Permission to reproduce this \
paper for noncommercial purposes has been reserved by the author and by the \
General Electric Company. Springer-Verlag, General Electric, and the author \
bear no responsibility for the accuracy of the information contained within \
this paper nor for any losses incurred through its use."], "Text",
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*)
(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)