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Article to review - Pricing of Options and Corporate Liability

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Fin 5560 Article Review format (two or three pages maximum) The review in Times New Roman double spaced format Authors: In APA format example: Anderson, R.C., & Reeb, D.M. (2003a). Founding-family ownership and firm performance: Evidence from the S&P 500. Journal of Finance, 58(3), 1301-1327. Abstract: One paragraph Literature Review: a brief no more than one page discussion of the important (top three articles) literature and the findings of that literature. Data/Sample: example S&P 500 index if an empirical article, and if a theoretical article there is no data. Author's hypotheses: what are the author's testing/ or proposing? Authors Findings/conclusions: perhaps three paragraphs Implications for Practitioners/ Finance professionals: 1 The Pricing of Options and Corporate Liabilities Author(s): Fischer Black and Myron Scholes Source: Journal of Political Economy, Vol. 81, No. 3 (May - Jun., 1973), pp. 637-654 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/1831029 Accessed: 06-06-2015 17:53 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. The University of Chicago Press is collaborating with JSTOR to digitize, preserve and extend access to Journal of Political Economy. http://www.jstor.org This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:53:42 UTC All use subject to JSTOR Terms and Conditions The Pricing of Options and Corporate Liabilities FischerBlack Universityof Chicago MyronScholes MassachusettsInstituteof Technology If optionsare correctlypricedin the market,it shouldnot be possible to make sure profitsby creatingportfoliosof long and shortpositions in options and theirunderlyingstocks. Using this principle,a theoreticalvaluationformulafor optionsis derived.Since almost all corporateliabilitiescan be viewedas combinationsof options,the formula and the analysisthatled to it are also applicableto corporateliabilities such as commonstock, corporatebonds, and warrants.In particular, the formulacan be used to derive the discountthat shouldbe applied to a corporatebond because of the possibilityof default. Introduction An option is a securitygiving the right to buy or sell an asset, subject to certain conditions,withina specifiedperiod of time. An "American option" is one that can be exercised at any time up to the date the option expires. A "European option" is one that can be exercised only on a specified future date. The price that is paid for the asset when the option is exercisedis called the "exercise price" or "strikingprice." The last day on which the option may be exercised is called the "expiration date" or "maturitydate." The simplest kind of option is one that gives the right to buy a single share of common stock. Throughout most of the paper, we will be discussing this kind of option, which is often referredto as a "call option." Received for publication November 11, 1970. Final version received May 9, 1972. The inspirationfor this work was provided by Jack L. Treynor (1961a, 1961b). We are grateful for extensive comments on earlier drafts by Eugene F. Fama, Robert C. Merton, and Merton H. Miller. This work was supported in part by the Ford Foundation. 637 This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:53:42 UTC All use subject to JSTOR Terms and Conditions 638 JOURNAL OF POLITICAL ECONOMY In general,it seems clear that the higherthe price of the stock,the greaterthe value of the option.When the stock price is much greater thantheexerciseprice,theoptionis almostsureto be exercised.The currentvalue of the optionwill thus be approximately equal to the priceof the stockminusthe priceof a pure discountbond that matureson the samedate as theoption,witha facevalue equal to thestriking priceof the option. On the otherhand, if the price of the stock is much less than the exerciseprice,the optionis almostsure to expirewithoutbeingexercised, so its value willbe nearzero. If the expirationdate of the optionis veryfarin the future,thenthe priceof a bond thatpays the exercisepriceon the maturitydate will be verylow, and the value of the optionwill be approximately equal to the priceof the stock. On the otherhand,if the expirationdate is verynear,the value of the optionwill be approximately equal to the stock price minusthe exercise price,or zero,if the stockpriceis less than the exerciseprice.Normally, thevalueof an optiondeclinesas its maturity date approaches,if thevalue of thestockdoes not change. These generalpropertiesof the relationbetweenthe optionvalue and thestockpriceare oftenillustrated in a diagramlikefigure1. Line A representsthemaximumvalue of theoption,sinceit cannotbe worthmorethan the stock.Line B represents the minimumvalue of the option,since its value cannotbe negativeand cannotbe less thanthestockpriceminusthe exerciseprice.Lines T1, T2, and T3 representthe value of the optionfor shortermaturities. successively Normally,thecurverepresenting the value of an optionwill be concave upward.Since it also lies below the 45 line, A, we can see that the $40 /A // $3 // TI * ~~~~~~~/ ._ // c$20 oa/~~/ $10 - /0l-o* / /-* * .'~~~* 48* Add**;/ . /' r- /0 T * . $10 $20 1 $30 $40 Stock Price (ExercisePrice = $20) FiG. 1 -The relationbetweenoptionvalue and stock price This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:53:42 UTC All use subject to JSTOR Terms and Conditions OPTIONS 639 AND LIABILITIES option will be more volatile than the stock. A given percentage change in the stock price,holding maturityconstant,will result in a largerpercentage change in the option value. The relative volatility of the option is not constant,however. It depends on both the stock price and maturity. Most of the previous work on the valuation of options has been expressed in termsof warrants.For example, Sprenkle (1961), Ayres (1963), Boness (1964), Samuelson (1965), Baumol, Malkiel, and Quandt (1966), and Chen (1970) all produced valuation formulas of the same general form.Their formulas,however, were not complete, since they all involved one or more arbitraryparameters. For example. Sprinkle's formulaforthe value of an option can be written as follows: kxN(b1) - lnkx/c + by 1 _v2(t* 2 t) vv/(t*-t) lnkx c- b2 k cN(b2) 2 v2(t* tAv(t* t) t) In this expression,x is the stock price, c is the exercise price, t* is the maturitydate, t is the currentdate, v2 is the variance rate of the returnon the stock,' In is the natural logarithm,and N(b) is the cumulative normal density function.But k and k* are unknown parameters. Sprenkle (1961) definesk as the ratio of the expected value of the stock price at the time the warrantmatures to the currentstock price, and k* as a discount factor that depends on the risk of the 'stock. He tries to estimate the values of k and k> empirically,but finds that he is unable to do so. More typically, Samuelson (1965) has unknown parameters x and (, where x is the rate of expected returnon the stock, and ( is the rate of expected returnon the warrant or the discount rate to be applied to the warrant.2He assumes that the distributionof possible values of the stock when the warrant matures is log-normal and takes the expected value of this distribution,cuttingit offat the exercise price. He then discounts this expected value to the present at the rate (. Unfortunately,there seems to be no model of the pricingof securitiesunder conditions of capital market 1 The variance rate of the return on a securityis the limit, as the size of the intervalof measurementgoes to zero, of the variance of the returnover that interval divided by the length of the interval. 2The rate of expected returnon a securityis the limit,as the size of the interval of measurementgoes to zero, of the expected return over that interval divided by the lengthof the interval. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:53:42 UTC All use subject to JSTOR Terms and Conditions 640 JOURNAL OF POLITICAL ECONOMY equilibrium thatwouldmakethisan appropriate procedurefordetermining thevalueof a warrant. In a subsequentpaper,Samuelsonand Merton(1969) recognizethe fact thatdiscounting the expectedvalue of the distribution of possiblevalues of the warrantwhenit is exercisedis not an appropriateprocedure.They advancethe theoryby treatingthe optionpriceas a functionof the stock price.They also recognizethat the discountratesare determined in part by the requirement thatinvestorsbe willingto hold all of the outstanding amountsof both the stockand the option.But theydo not make use of thefactthatinvestors mustholdotherassetsas well,so thattheriskof an optionor stockthat affectsits discountrate is only that part of the risk thatcannotbe diversified away. Their finalformuladependson the shape of theutilityfunctionthattheyassumeforthe typicalinvestor. One of theconceptsthatwe use in developingour modelis expressedby Thorp and Kassouf (1967). They obtain an empiricalvaluationformula forwarrants by fitting a curveto actual warrantprices.Then theyuse this formulato calculatetheratioof sharesof stockto optionsneededto create a hedgedpositionby goinglong in one securityand shortin the other. Whattheyfailto pursueis thefactthatin equilibrium, theexpectedreturn on such a hedgedpositionmustbe equal to the returnon a risklessasset. What we show below is that this equilibriumconditioncan be used to derivea theoretical valuationformula. The Valuation Formula In derivingour formulaforthevalue of an optionin termsof the priceof the stock,we will assume "ideal conditions"in the marketforthe stock and fortheoption: a) The short-term interestrateis knownand is constantthroughtime. b) The stockprice followsa randomwalk in continuoustime with a variancerateproportional to the square of the stockprice.Thus the distribution of possiblestock pricesat the end of any finiteintervalis lognormal.The variancerateof the returnon the stockis constant. c) The stockpays no dividendsor otherdistributions. d) The option is "European," that is, it can only be exercisedat maturity. e) There are no transactioncosts in buyingor sellingthe stockor the option. f) It is possibleto borrowany fractionof the price of a securityto buy it or to hold it, at the short-term interestrate. g) There are no penaltiesto shortselling.A sellerwho does not own a securitywillsimplyacceptthepriceof thesecurityfroma buyer,and will agree to settle with the buyer on some futuredate by payinghim an amountequal to thepriceof the securityon thatdate. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:53:42 UTC All use subject to JSTOR Terms and Conditions OPTIONS AND LIABILITIES 64I Under these assumptions,the value of the option will depend only on the price of the stock and time and on variables that are taken to be known constants. Thus, it is possible to create a hedged position, consistingof a long position in the stock and a short position in the option, whose value will not depend on the price of the stock, but will depend only on time and the values of known constants. XWriting w(x, t) for the value of the option as a functionof the stock price x and time t, the number of options that must be sold short against one share of stock long is: 1/wI(x,t). (1) In expression (1), the subscript refersto the partial derivative of w(x,t) with respect to its firstargument. To see that the value of such a hedged position does not depend on the price of the stock, note that the ratio of the change in the option value to the change in the stock price, when the change in the stock price is small, is w1(x,t). To a first approximation, if the stock price changes by an amount Ax, the option price will change by an amount w1(xt) Ax, and the number of options given by expression (1) will change by an amount Ax. Thus, the change in the value of a long position in the stock will be approximately offsetby the change in value of a short position in 11w options. As the variables x and t change, the number of options to be sold short to create a hedged position with one share of stock changes. If the hedge is maintained continuously,then the approximationsmentionedabove become exact, and the return on the hedged position is completely independent of the change in the value of the stock. In fact, the returnon the hedged position becomes certain.3 To illustrate the formationof the hedged position, let us refer to the solid line (T.) in figure1 and assume that the price of the stock starts at $15.00, so that the value of the option starts at $5.00. Assume also that the slope of the line at that point is 1112.This means that the hedged position is created by buying one share of stock and selling two options short. One share of stock costs $15.00, and the sale of two options brings in $10.00, so the equity in this position is $5.00. If the hedged position is not changed as the price of the stock changes, then there is some uncertaintyin the value of the equity at the end of a finiteinterval.Suppose that two options go from$10.00 to $15.75 when the stock goes from$15.00 to $20.00, and that they go from $10.00 to $5.75 when the stock goes from $15.00 to $10.00. Thus, the equity goes from $5.00 to $4.25 when the stock changes by $5.00 in either direction. This is a $.75 decline in the equity for a $5.00 change in the stock in either direction.4 3 This was pointed out to us by Robert Merton. 4 These figuresare purely for illustrativepurposes.They correspondroughlyto the way figure1 was drawn, but not to an option on any actual security. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:53:42 UTC All use subject to JSTOR Terms and Conditions 642 JOURNAL OF POLITICAL ECONOMY In addition,the curve shifts (say fromT2 to T3 in fig. 1) as the maturityof the optionschanges.The resultingdecline in value of the optionsmeansan increasein theequityin thehedgedpositionand tendsto offsetthepossiblelossesdue to a largechangein thestockprice. Note that the declinein the equityvalue due to a large changein the stockpriceis small. The ratio of the declinein the equity value to the magnitudeof the changein the stockpricebecomessmalleras the magnitudeof thechangein thestockpricebecomessmaller. Note also that the directionof the changein the equityvalue is independentof the directionof the changein the stockprice.This meansthat underour assumptionthat the stock price followsa continuousrandom walk and that the returnhas a constantvariancerate, the covariance betweenthe returnon theequityand the returnon the stockwill be zero. If the stockpriceand the value of the "marketportfolio"followa joint continuousrandomwalk withconstantcovariancerate,it means that the covariancebetween the returnon the equity and the returnon the marketwill be zero. Thus the riskin the hedgedpositionis zero if the shortpositionin the optionis adjusted continuously. If the positionis not adjusted continuously,theriskis small,and consistsentirelyof riskthatcan be diversified away by forming a portfolioof a largenumberof such hedgedpositions. In general,since the hedgedpositioncontainsone share of stock long and 11wi optionsshort,thevalue of theequityin the positionis: x-w/w1. (2) The changein thevalue of theequityin a shortintervalAt is: Ax-Aw/w1. (3) Assumingthat the shortpositionis changedcontinuously, we can use stochasticcalculus to expandAw, whichis w(x + Ax,t + At) - w(x,t), as follows: Aw- w1Ax+ ?2 W11V2X2At + w2At. (4) In equation (4), the subscriptson w referto partialderivativesand v2 is thevariancerateof the returnon thestock. Substituting fromequation (4) intoexpression(3), we findthatthechangein thevalue of theequity in thehedgedpositionis: . -t-w1 1v-x2+ w2V At/w1 ((5) Since the returnon the equityin the hedgedpositionis certain,the returnmustbe equal to rAt. Even if the hedgedpositionis not changed 5For an expositionof stochasticcalculus,see McKean (1969). B1See footnote1. This content downloaded from 137.52.76.29 on Sat, 06 Jun 2015 17:53:42 UTC All use subject to JSTOR Terms and Conditions OPTIONS 643 AND LIABILITIES continuously, its riskis small and is entirelyriskthat can be diversified away,so theexpectedreturnon the hedgedpositionmustbe at the short terminterestrate.7If this werenot true,speculatorswould tryto profit by borrowing largeamountsof moneyto createsuchhedgedpositions,and wouldin theprocessforcethe returnsdownto theshortterminterestrate. Thus the changein the equity (5) mustequal the value of the equity (2) timesrAt. - w11v2x2+ w2 At/w1 - (x - w/wi)rAt. (6) Droppingthe At fromboth sides,and rearranging, we have a differential equationforthevalue of the option. W2 -rw - rxwl - - 1 2 22 v2x2w1l. (7) Writingt* forthematurity date of theoption,and c fortheexerciseprice, we knowthat: w(xt*) X>' C x-c C, (8) -O x