(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 140964, 6806]*) (*NotebookOutlinePosition[ 141858, 6839]*) (* CellTagsIndexPosition[ 141814, 6835]*) (*WindowFrame->Normal*) 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, TextAlignment->Center, AspectRatioFixed->True], 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", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], 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", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["The Black-Scholes Model"], "Section", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], 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, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["CallPayoff[price_,strike_] = Max[0,price-strike]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ Max[0, price - strike]\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ " 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", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Plot[CallPayoff[x,60],{x,0,120}]"], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.0238095 0.00793651 0.0147151 0.00981006 [ [(20)] .18254 .01472 0 2 Msboxa [(40)] .34127 .01472 0 2 Msboxa [(60)] .5 .01472 0 2 Msboxa [(80)] .65873 .01472 0 2 Msboxa [(100)] .81746 .01472 0 2 Msboxa [(120)] .97619 .01472 0 2 Msboxa [(10)] .01131 .11282 1 0 Msboxa [(20)] .01131 .21092 1 0 Msboxa [(30)] .01131 .30902 1 0 Msboxa [(40)] .01131 .40712 1 0 Msboxa [(50)] .01131 .50522 1 0 Msboxa [(60)] .01131 .60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g 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provides the value of an \ option at a boundary--the moment the option expires at the close of trading \ on the expiration date. Determining the value of the option prior to \ expiration can be reduced to taking the current price of the stock, which is \ presumably known, and projecting it forward in time to the expiration date. \ Because of the many factors that can affect the price of a stock, this \ projection must be made probabilistically rather than deterministically. In \ the Black-Scholes model, it is assumed that the percentage change in the \ stock price (its rate of return when no dividends are paid) follows a Weiner \ process (i.e., a random walk) with a known drift and standard deviation, so \ that on the expiration date, the stock price has a lognormal distribution \ with a known mean and variance. "], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "\nAlthough the lognormality assumption of the Black-Scholes model cannot be \ expected to hold in any but the most contrived situations, the model (and its \ variants) can be used to predict option prices with great accuracy under a \ broad range of real-world situations. Indeed, most traders on the floors of \ options exchanges carry computerized \"crib sheets\" with them that give the \ theoretical value of the options in which they trade so that they may \ identify options whose prices are temporarily mispriced relative to the \ model.\n"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Under the further assumptions that the stock pays no dividends and will only \ be exercised, if at all, upon expiration, the Black-Scholes model yields a \ closed-form equation for the value of the call option that is easily modelled \ in Mathematica. Although this derivation was a notable technical feat when it \ was first derived around 1970, it is now a standard application of stochastic \ calculus. (The use of Mathematica to do stochastic calculus appears elsewhere \ in this book.)"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Because it is relatively complex, the Black-Scholes formula is best \ expressed in terms of an auxiliary function. Both the Black-Scholes formula \ and the auxiliary functions take five arguments as follows:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "p = current price of the stock\nk = exercise price of the option\nsd = \ volatility of the stock (standard deviation of annual rate of return)\nr = \ continuously compounded risk-free rate of return, e.g., the return on\n \ U.S. Treasury bills with very short maturities\nt = time (in years) until \ the expiration date"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "The critical feature of the Black-Scholes and related option valuation \ models is that the value of the option depends only on the standard deviation \ of the stock's rate of return and not upon its expected value. That is \ because the model is based on an arbitrage argument in which any risk premium \ above the risk-free rate of return is cancelled out."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Using the five variables given above, we can define the auxiliary function, \ AuxBS, and the Black-Scholes valuation function, BlackScholes, as follows:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "\nAuxBS[p_,k_,sd_,r_,t_] = (Log[p/k]+r t)/\n (sd \ Sqrt[t])+\n .5 sd Sqrt[t]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ p r t + Log[-] k 0.5 sd Sqrt[t] + ------------ sd Sqrt[t]\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[CellGroupData[{Cell[TextData[ "BlackScholes[p_,k_,sd_,r_,t_] =\n p Norm[AuxBS[p,k,sd,r,t]]- \n \ k Exp[-r t] (Norm[AuxBS[p,k,sd,r,t]-sd Sqrt[t]])"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ p r t + Log[-] k k Norm[-0.5 sd Sqrt[t] + ------------] sd Sqrt[t] -(--------------------------------------) + r t E p r t + Log[-] k p Norm[0.5 sd Sqrt[t] + ------------] sd Sqrt[t]\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "The function Norm is the cumulative normal distribution function and can be \ defined for numerical arguments as follows:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData["\nNorm[z_?NumberQ]:= N[0.5 + 0.5 Erf[z/Sqrt[2]]]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[TextData[ "A separate definition of the derivative of Norm used for symbolic valuation \ will be given in the following section on risk management."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Notice that because BlackScholes is defined using the immediate assignment \ operator, =, it contains the full Black-Scholes formula with the actual \ values from AuxBS appropriately substituted as shown by the output given \ above."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "We are now ready to compute the dollar value for stock options by calling \ the function BlackScholes with numerical arguments. For example, to continue \ the above dialog, we might wish to find the value of a call option on DEC \ stock with an exercise price of 60 assuming that the current price of DEC is \ 58 1/2, the time until expiration is 0.3 years, the volatility of DEC stock \ is 29%, and the continuously compounded risk-free rate of return is 4% as \ follows:\n"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["BlackScholes[58.5,60.,0.29,0.04,0.3]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 3.34886\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "Using the graphic capabilities of Mathematica one can easily explore the \ various qualititative properties of the Black-Scholes formula and even go so \ far as to animate them. 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For example, although the Black-Scholes model \ assumes the volatility of the stock is constant, it will tend to fluctuate \ over the life of the option. The sensitivity of the value of an option to \ changes in volatility can be graphed as follows:\n"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Plot[BlackScholes[60.,60.,x,0.04,0.3],{x,0.2,0.4}]"], "Input", PageWidth->Infinity, AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .61803 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations -0.928571 4.7619 -0.662326 0.2273 [ [(0.25)] .2619 .01957 0 2 Msboxa [(0.3)] .5 .01957 0 2 Msboxa [(0.35)] .7381 .01957 0 2 Msboxa [(0.4)] .97619 .01957 0 2 Msboxa [(3.5)] .01131 .13322 1 0 Msboxa [(4)] .01131 .24687 1 0 Msboxa [(4.5)] .01131 .36052 1 0 Msboxa [(5)] .01131 .47417 1 0 Msboxa [(5.5)] .01131 .58782 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 .61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w .2619 .01957 m .2619 .02582 L s P [(0.25)] .2619 .01957 0 2 Mshowa p .002 w .5 .01957 m .5 .02582 L s P [(0.3)] .5 .01957 0 2 Mshowa p .002 w .7381 .01957 m .7381 .02582 L s P [(0.35)] .7381 .01957 0 2 Mshowa p .002 w .97619 .01957 m .97619 .02582 L s P [(0.4)] .97619 .01957 0 2 Mshowa p .001 w .07143 .01957 m .07143 .02332 L s P p .001 w .11905 .01957 m .11905 .02332 L s P p .001 w .16667 .01957 m .16667 .02332 L s P p .001 w .21429 .01957 m .21429 .02332 L s P p .001 w .30952 .01957 m .30952 .02332 L s P p .001 w .35714 .01957 m .35714 .02332 L s P p .001 w .40476 .01957 m .40476 .02332 L s P p .001 w .45238 .01957 m .45238 .02332 L s P p .001 w .54762 .01957 m .54762 .02332 L s P p .001 w .59524 .01957 m .59524 .02332 L s P p .001 w .64286 .01957 m .64286 .02332 L s P p .001 w .69048 .01957 m .69048 .02332 L s P p .001 w .78571 .01957 m .78571 .02332 L s P p .001 w .83333 .01957 m .83333 .02332 L s P p .001 w .88095 .01957 m .88095 .02332 L s P p .001 w .92857 .01957 m .92857 .02332 L s P p .002 w 0 .01957 m 1 .01957 L s P p .002 w .02381 .13322 m .03006 .13322 L s P [(3.5)] .01131 .13322 1 0 Mshowa p .002 w .02381 .24687 m .03006 .24687 L s P [(4)] .01131 .24687 1 0 Mshowa p .002 w .02381 .36052 m .03006 .36052 L s P [(4.5)] .01131 .36052 1 0 Mshowa p .002 w .02381 .47417 m .03006 .47417 L s P [(5)] .01131 .47417 1 0 Mshowa p .002 w .02381 .58782 m .03006 .58782 L s P [(5.5)] .01131 .58782 1 0 Mshowa p .001 w .02381 .0423 m .02756 .0423 L s P p .001 w .02381 .06503 m .02756 .06503 L s P p .001 w .02381 .08776 m .02756 .08776 L s P p .001 w .02381 .11049 m .02756 .11049 L s P p .001 w .02381 .15595 m .02756 .15595 L s P p .001 w .02381 .17868 m .02756 .17868 L s P p .001 w .02381 .20141 m .02756 .20141 L s P p .001 w .02381 .22414 m .02756 .22414 L s P p .001 w .02381 .2696 m .02756 .2696 L s P p .001 w .02381 .29233 m .02756 .29233 L s P p .001 w .02381 .31506 m .02756 .31506 L s P p .001 w .02381 .33779 m .02756 .33779 L s P p .001 w .02381 .38325 m .02756 .38325 L s P p .001 w .02381 .40598 m .02756 .40598 L s P p .001 w .02381 .42871 m .02756 .42871 L s P p .001 w .02381 .45144 m .02756 .45144 L s P p .001 w .02381 .4969 m .02756 .4969 L s P p .001 w .02381 .51963 m .02756 .51963 L s P p .001 w .02381 .54236 m .02756 .54236 L s P p .001 w .02381 .56509 m .02756 .56509 L s P p .001 w .02381 .61055 m .02756 .61055 L s P p .002 w .02381 0 m .02381 .61803 L s P P 0 0 m 1 0 L 1 .61803 L 0 .61803 L closepath clip newpath p p .004 w .02381 .01472 m .06349 .03922 L .10317 .06373 L .14286 .08825 L .18254 .11278 L .22222 .1373 L .2619 .16184 L .30159 .18637 L .34127 .21091 L .38095 .23544 L .42063 .25998 L .46032 .28452 L .5 .30905 L .53968 .33359 L .57937 .35812 L .61905 .38266 L .65873 .40719 L .69841 .43172 L .7381 .45624 L .77778 .48076 L .81746 .50528 L .85714 .5298 L .89683 .55431 L .93651 .57882 L .97619 .60332 L s P P % End of Graphics MathPictureEnd \ \>"], "Graphics", PageWidth->Infinity, Evaluatable->False, ImageSize->{300, 185}, ImageMargins->{{100, Inherited}, {Inherited, 0}}, ImageCacheValid->False], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ -Graphics-\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "This graph is not particularly exciting because over a broad range of \ 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 \ provides an excellent approximation to the value of an option with variable \ volatility as long as the mathematical expectation of the volatility is \ known."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Although Mathematica's powerful and convenient graphics provide a \ qualitative insight into the Black-Scholes, the quantitative application of \ the model, especially to the area of risk management, is facilitated by \ applying the symbolic manipulation capabilities of Mathematica directly to \ the Black-Scholes formula. Such risk management applications are the focus \ of the next section."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True]}, Open]], Cell[CellGroupData[{Cell[TextData["Risk Management using the Black-Scholes Model"], "Section", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "\nThe Black-Scholes model is useful not only because it provides a \ closed-form expression for the value of an option, but also because the \ sensitivity of the model to changes in its parameters, as represented by \ (partial) derivatives of the option valuation formula, can be expressed in \ closed form. The most important parameter that affects the value of an \ option is the price of the underlying stock. The partial derivative of \ option value with respect to stock price is known as delta ."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Delta is a useful measurement of risk for an option because it indicates how \ much the price of the option will respond to a $1 change in the price of the \ 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 \ a $1 change in stock price, which is zero in this case. However, because \ 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 \ measures of risk that either are derived later in this section or are \ contained in the accompanying Mathematica package can help the reader to get \ a handle on other changes that can influence option value."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Using the derivative function, D, that is built into Mathematica, it is easy \ to calculate symbolically any derivative of the BlackScholes function, \ including delta. Before these derivatives can be defined, however, it is \ first necessary to supply Mathematica with the derivative of the Norm \ function, since only the numerical value of Norm was provided above and its \ derivative frequently figures into derivatives of BlackScholes. In \ Mathematica, the derivative of Norm is defined as follows:\n"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["\nNorm'[z_] = N[(1/Sqrt[2 Pi])] Exp[-z^2/2]\n"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 0.398942 -------- 2 z /2 E\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "The leading constant in the definition of this derivative is expressed as a \ number rather than being left in terms of Pi to ensure that full numerical \ conversion occurs when derivatives of the Black-Scholes formula are evaluated \ for a particular set of parameter values."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "\nThen, the function that computes delta, Delta, becomes:\n"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "\nDelta[p_,k_,sd_,r_,t_] = D[BlackScholes[p,k,sd,r,t],p]\n"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 0.398942 / (Power[E, (0.5 sd Sqrt[t] + 2 (r t + Log[p/k])/(sd Sqrt[t])) /2] sd Sqrt[t]) - (0.398942 Power[E, -(r t) - (-0.5 sd Sqrt[t] + 2 (r t + Log[p/k])/(sd Sqrt[t])) /2] k) / (p sd Sqrt[t]) + Norm[0.5 sd Sqrt[t] + p r t + Log[-] k ------------] sd Sqrt[t]\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "\nThe assignment statement that defines Delta, like the one that defined \ BlackScholes, generates a formula for delta directly. It should be noted \ that the evaluation mechanism in Mathematica does not always generate \ formulas in their simplest form and, indeed, the expression output for Delta \ (and some other derivatives of BlackScholes) could be further simplified in \ the interests of computational efficiency as it is actually equal to \ Norm[AuxBS[p,k,sd,r,t]."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "When Delta is applied to the situation given above, we get the following:\n\ "], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["\nDelta[58.5,60.,0.29,0.04,0.3]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 0.498235\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "\nHence, for a $1 increase in DEC stock, the price of this call option will \ 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worthless. For very high stock prices, i.e., \ when the option is very much in the money, delta approaches one because the \ increase in the stock price dollar for dollar translates into an expected \ increase in the value of the option at expiration."], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Other aspects of option value are as easy to investigate as delta using the \ symbolic math features of Mathematica. 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portfolio to changes \ in any or all of the variables that underlie the formula."], "Text", Evaluatable->False, 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", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "\nImpliedVolatility[p_,k_,r_,t_,optionprice_] :=\n sd /. \ FindRoot[BlackScholes[p,k,sd,r,t]==\n \ optionprice,{sd,0.2}]\n"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], 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", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["ImpliedVolatility[58.5,60.,0.04,0.3,3.34886]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 0.29\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], 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, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "Type[DECFL] ^= \"call\" ;\nAsset[DECFL] ^= DEC ;\nExercisePrice[DECFL] ^= \ 60. ;\nExpirationTime[DECFL] ^= 0.3 ;"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], 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", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?DECFL"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ Global`DECFL\ \>"], "Print", PageWidth->Infinity, Evaluatable->False], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ Asset[DECFL] ^= DEC ExercisePrice[DECFL] ^= 60. ExpirationTime[DECFL] ^= 0.3 Type[DECFL] ^= \"call\"\ \>"], "Print", PageWidth->Infinity, Evaluatable->False]}, Open]], 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, Evaluatable->False, AspectRatioFixed->True]}, Open]], Cell[TextData[ "As a convenience we will define an object constructor function, ConsObj, \ that takes a symbol and property list as its arguments:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "ConsObj[obj_,proplist_] := \n Do[Block[{propname=proplist[[2*i-1]],\n \ propval=proplist[[2*i]]},\n \ propname[obj]^=propval],\n {i,Length[proplist]/2}]\n"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[TextData["The object, DECFL, can now be constructed as follows:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[TextData[ "ConsObj[DECFL,{Type,\"call\",\n Asset,DEC,\n \ ExercisePrice,60.,\n ExpirationTime,0.3}]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], 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", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ General::spell: Possible spelling error: new symbol name \"Price\" is similar to existing symbols {price, Prime}.\ \>"], "Message", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "The properties of the DEC object are stored in Mathematica as follows:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?DEC"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ Global`DEC\ \>"], "Print", PageWidth->Infinity, Evaluatable->False], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ Price[DEC] ^= 58.5 Type[DEC] ^= \"stock\" Volatility[DEC] ^= 0.29\ \>"], "Print", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "Global`DEC\n\nPrice[DEC] ^= 58.5\n \nType[DEC] ^= \"stock\"\n \n\ Volatility[DEC] ^= 0.29"], "Info", PageWidth->Infinity, 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", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["AssetPrice[DECFL]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 58.5\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[CellGroupData[{Cell[TextData["AssetVolatility[DECFL]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 0.29\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "Checking the values now associated with DECFL, we can see that these two new \ properties are listed:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["?DECFL"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ Global`DECFL\ \>"], "Print", PageWidth->Infinity, Evaluatable->False], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ Asset[DECFL] ^= DEC AssetPrice[DECFL] ^= 58.5 AssetVolatility[DECFL] ^= 0.29 ExercisePrice[DECFL] ^= 60. ExpirationTime[DECFL] ^= 0.3 Type[DECFL] ^= \"call\"\ \>"], "Print", PageWidth->Infinity, Evaluatable->False]}, Open]], 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", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 0.04\ \>"], "Output", PageWidth->Infinity, Evaluatable->False]}, Open]], 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, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Value[option_] := BlackScholes[AssetPrice[option],\n \ ExercisePrice[option],\n \ AssetVolatility[option],\n RiskFreeRate,\n \ ExpirationTime[option]]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ General::spell1: Possible spelling error: new symbol name \"Value\" is similar to existing symbol \"ValueQ\".\ \>"], "Message", PageWidth->Infinity, Evaluatable->False]}, Open]], Cell[TextData[ "We can now demonstrate that Value actually works when applied to DECFL:"], "Text", Evaluatable->False, CellHorizontalScrolling->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData["Value[DECFL]"], "Input", PageWidth->Infinity, InitializationCell->True, AspectRatioFixed->True], Cell[OutputFormData["\<\ No Input Form Generated \ \>", "\<\ 3.34886\ \>"], "Output", PageWidth->Infinity, 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