Options and Volatility

Transcription

1 Opions and Volailiy Peer A. Abken and Saika Nandi Abken and Nandi are senior economiss in he financial secion of he Alana Fed s research deparmen. V olailiy is a measure of he dispersion of an asse price abou is mean level over a fixed ime inerval. Careful modeling of an asse s volailiy is crucial for he valuaion of opions and of porfolios conaining opions or securiies wih implici opions (for example, callable Treasury bonds) as well as for he success of many rading sraegies involving opions. The problem of pricing opions is one confroning no only opion raders bu also, increasingly, a broad specrum of invesors. In paricular, insiuional invesors porfolios frequenly conain opions or securiies wih embedded opions. More and more, riskmanagemen pracices of financial insiuions as well as of oher corporae users of derivaives require frequen valuaion of securiies porfolios o deermine curren value and o gauge porfolios sensiiviies o marke risk facors, including changes in volailiy (see Peer A. Abken 1994). A model of volailiy is needed for managing porfolios conaining opions (including derivaives and oher securiies conaining opions) for which marke quoes are no readily available and ha consequenly mus be marked o model (ha is, valued by model) raher han marked o marke. Accurae assessmens of volailiy are also key inpus ino he consrucion of hedges, which limi risk exposures, for such porfolios. Because of he cenral role ha volailiy plays in derivaive valuaion and hedging, a subsanial lieraure is devoed o he specificaion of volailiy and is measuremen. Modeling volailiy is challenging because volailiy in financial and commodiy markes appears o be highly unpredicable. There has been a proliferaion of volailiy specificaions since he original, simple consan-volailiy assumpion of he famous opion pricing model developed by Fischer Black and Myron S. Scholes (1973). This aricle gives an overview of differen specificaions of asse price volailiy ha are widely used in opion pricing models. Federal Reserve Bank of Alana Economic Review 21

2 The Effec of Volailiy A simple example will illusrae he imporance of volailiy for opions. Consider a call opion ha gives he holder of he opion he righ o buy one uni of a sock a a fuure dae T a a paricular price (also called he srike or he exercise price of he opion). Le he srike price of he opion, denoed by K, be equal o $50. Noe ha he value of a call opion a mauriy is given by max(s K, 0), where S denoes he sock price a he mauriy of he opion. Thus, he call opion a is mauriy has a value equal o he difference beween he sock price a mauriy and he srike price if S > K and zero oherwise. If he sock price a mauriy of he opion is less han he srike price, he opionholder would raher buy he sock from he marke han exercise he opion and pay he higher srike price K for he sock. Now consider he following wo sock price scenarios in which an opion exiss ha has a srike price of $50: High-Volailiy Scenario Sock Price $30 $40 $50 $60 $70 Opion Payoff $10 $20 Low-Volailiy Scenario Sock Price $40 $45 $50 $55 $60 Opion Payoff $ 5 $10 The average sock price (across he five saes of he world) is $50 in boh scenarios, bu volailiy is higher in he firs scenario because of he wider dispersion of possible sock prices. In conras o he consan average sock price in each scenario, he average opion payoff is $7.50 in he low-volailiy scenario and $15.00 in he high-volailiy scenario. The reason is ha he downside of he payoff o he opionholder is limied o zero because he opion does no have o be exercised if he sock price a mauriy is less han he srike price. The opionholder merely loses he price paid o he opion wrier (or seller) for he purchase of he opion. However, he opionholder gains if he sock price a mauriy is greaer han he srike price. The higher he volailiy, he higher is he probabiliy ha he opion payoff a mauriy will be greaer han he srike price and consequenly will be of value o he opionholder. Very high sock prices can increase he value of he call opion a mauriy wihou limi. However, very low sock prices canno make he value of he opion payoff a mauriy less han zero. Thus he asymmery of he payoffs due o he naure of he conrac implies ha volailiy is of value o he opionholder a he mauriy of he opion. In general, since he price of he opion prior o is mauriy is he expecaion of he opion payoff a mauriy (discouned a an appropriae rae), an increase in he volailiy of he underlying asse increases he expecaion and consequenly he price of he opion oday. In general, fuure volailiy is difficul o esimae. While he hisorical volailiy of an asse reurn is readily compued from observed asse reurns (see Box 1), his measure may be an inaccurae esimae of he fuure volailiy expeced o prevail over he life of an opion. The fuure volailiy is unobservable and may differ from he hisorical volailiy. Hence, unlike he oher parameers ha are imporan for pricing opions (namely, he curren asse price, he srike price, he ineres rae, and ime o mauriy), he volailiy inpu has o be modeled. For example, he Black- Scholes opion pricing model is a simple formula involving hese five variables ha prices European opions. (Such opions can be exercised only upon mauriy. See John C. Cox and Mark Rubinsein 1985 for an exposiion of he Black-Scholes formula.) The Black-Scholes model assumes ha volailiy is consan, he simples possible approach. However, a preponderance of evidence (see Tim Bollerslev, Ray Y. Chou, and Kenneh F. Kroner 1992) poins o volailiy being ime-varying. In addiion, ha variaion may be random or, equivalenly, sochasic. Randomness means ha fuure volailiy canno be readily prediced using curren and pas informaion. Before proceeding wih an overview of he various approaches for reaing ime-varying volailiy, he discussion examines a frequenly documened phenomenon known as he volailiy smile o moivae he consideraion of differen volailiy specificaions. The exisence of he smile is an indicaion of he inadequacy of he consan-volailiy Black-Scholes model. A common feaure of all ime-varying volailiy models reviewed below is ha hey have he poenial o give prices ha are free of he Black-Scholes biases, such as he smile. For he Black-Scholes model, he only inpu ha is unobservable is he fuure volailiy of he underlying asse. One way o deermine his volailiy is o selec a value ha equaes he heoreical Black-Scholes price of he opion o he observed marke price. This value is ofen referred o as he implied (or implici) volailiy of he opion. Under he Black-Scholes model, implied volailiies from opions should be he same regardless 22 Economic Review December 1996

3 of which opion is used o compue he volailiy. However, in pracice, his is usually no he case. Differen opions (in erms of srike prices and mauriies) on he same asse yield differen implied volailiies, oucomes ha are inconsisen wih he Black-Scholes model. The paern of he Black-Scholes implied volailiies wih respec o srike prices has become known as he volailiy smile. The exisence of a smile also means ha if only one volailiy is used o price opions wih differen srikes, pricing errors will be sysemaically relaed o srikes. The smile has also been shown o depend on opions mauriies. The exisence of pricing biases for he Black-Scholes model has been well documened. These biases have varied hrough ime. For example, Rubinsein (1985) repors ha shor-mauriy ou-of-he-money calls on equiies have marke prices ha are much higher han he Black-Scholes model would predic. On he oher hand, since he sock marke crash of 1987, he volailiy smile has had a persisen shape, especially when derived from equiy-index opion prices as he srike price of index-equiy opions increases, heir implied volailiies decrease. Thus, an ou-of-he-money pu (or in-he-money call) opion has a greaer implied volailiy han an in-he-money pu (or ou-of-he-money call) of equivalen mauriy. Buying an ou-of-he-money pu can serve as insurance agains marke declines. The surprising severiy of he marke crash of 1987 increased he cos of crash proecion, as manifesed by a relaively high cos for ou-of-he-money pu opions. Because he opion price, for calls or pus, increases as volailiy rises, higher opion prices are associaed wih higher implied volailiies. Thus, relaively high ou-of-he-money pu prices are mirrored in high implied volailiies for hose opions. The smile in equiy index opions is ofen referred o as a skew because he high implied volailiies for ou-of-he-money pus (or, equivalenly, for in-hemoney calls) progressively decline as pus become furher in-he-money (or as calls become furher ouof-he-money). Box 2 gives more deail abou he volailiy skew in he S&P 500 index-equiy opions. This aricle reviews wo overarching approaches o generalizing he consan-volailiy assumpion of he Black-Scholes model ha have appeared in he opion pricing lieraure. Boh lines of research have developed concurrenly. The firs approach assumes ha variaions in volailiy are deermined by variables known o marke paricipans, such as he level of he asse price. Models of his ype are referred o as deerminisic-volailiy models. This approach conrass wih he second, more demanding one, commonly called sochasic volailiy, in whicππh he source of uncerainy ha generaes volailiy is differen from, alhough possibly correlaed wih, he one ha drives The sandard way o measure volailiy from asse prices is sraighforward. Assuming no inermediae cash flows like dividend paymens, suppose r = (P P 1 )/P 1 represens he reurn of an asse as measured by buying he asse a ime 1 a P 1 and selling i a a P, ha is, over a single period of ime. Furher assume ha hese reurns are no dependen on each oher: he fac ha oday s reurn is high or low reveals nohing abou omorrow s reurn. A saisical descripion of he behavior of reurns is ha hey are jus differen realizaions of a random variable ha is evolving hrough ime. If hese singleperiod asse reurns are calculaed from ime = 1 o ime = T, hen he mean or average reurn over his period, denoed by r, is esimaed by T r (1) r = = 1. T The above equaion is he symbolic represenaion for summing he reurns from = 1 o = T and hen dividing Box 1 Measuremen of Hisorical Volailiy by he lengh of he ime inerval (T) over which he reurns are measured. Having esimaed he mean, he sandard deviaion, a measure of volailiy, is esimaed by σ = T = 1 ( r r) T 1 Variance, denoed by 2, is he square of he above quaniy and is also a measure of volailiy. In oher words, in order o esimae he sandard deviaion, one sums he squared deviaions of he individual reurns from he mean reurn, divides by T 1, and akes he posiive square roo of he resulan quaniy. This measure is usually referred o as hisorical volailiy. The relevan volailiy for pricing opions is no ha which occurred in he pas bu ha which is expeced o prevail in he fuure. However, hisorical volailiy may be useful in forming ha expecaion inasmuch as volailiy is correlaed hrough ime. 2. (2) Federal Reserve Bank of Alana Economic Review 23

4 asse prices. Therefore, knowledge of pas asse prices is no sufficien o deermine volailiy (using discreely observed prices). For reasons ha will be explained more fully below, he firs approach has been he mos popular as a modeling sraegy because of is relaive simpliciy. The models under consideraion in his aricle have been developed for equiy, currency, and commodiy opions. Praciioners have also used he Black-Scholes model o price and hedge such opions. Sochasic volailiy is even more challenging o incorporae ino models of fixed-income securiies because of he complexiies of modeling he erm srucure of ineres raes. The Black-Scholes model has no been he benchmark model for pricing opions in fixed-income markes, and less work has been done on sochasicvolailiy bond pricing models. Thus, his opic is beyond he scope of his aricle. Wihin he firs approach, hree ypes of models have been proposed. These are (1) implied binomial ree models, (2) general auoregressive condiional heeroscedasiciy (GARCH) models, and (3) exponenially weighed momens models. Alhough somewha arcane sounding on firs reading, each will prove o have is own inuiive appeal. Each ype of model also has he poenial o closely or exacly mach model opion prices wih acual marke opion prices. For he second approach involving sochasic volailiy, models may be divided ino hose ha have closedform soluions for opion prices and hose ha do no. Closed-form soluions refer o pricing formulas ha are readily compued, given curren compuer echnology. The disincion is a maer of pracicaliy because he ime i akes o compue prices is relevan o praciioners who rade opions or hedge posiions using opions. Advances in compuer echnology will gradually blur he disincion among curren sochasic-volailiy models as ime-consuming compuaions become less so in he fuure. Deerminisic Volailiy The firs approach ha has been used o address he deficiencies of he benchmark Black-Scholes model and he volailiy smile involves deerminisic-volailiy specificaions. The Black-Scholes formula for pricing European opions is predicaed on he assumpion of consan volailiy. The simples relaxaion of he consan volailiy assumpion is o allow volailiy o depend on is pas in such a way ha fuure volailiy can be perfecly prediced from is hisory and possibly oher observable informaion. As an example, suppose he variance of asse reurns 2 is described by he +1 following equaion: 2 +1 = + 2. Box 2 The Volailiy Skew in S&P 500 Index Opions The char illusraes he skew on four differen days. 1 The skew is compued from S&P 500 index opions ha are raded a he Chicago Board Opions Exchange (CBOE). These are sandard European opions, for which exercise can occur only on he opion expiraion dae, and heir payoffs are deermined by he level of he S&P 500 index on he opion mauriy dae. The Black- Scholes equaion is used o infer he volailiy using he oher opion formula inpus and he quoed opion price. Each char conains implied volailiies from pus and calls ha were raded beween 10:00 A.M. and 2:30 P.M. All opions had fory-five days o mauriy. Diamonds are he implied volailiies derived from individual pu ransacions, and squares are implied volailiies from individual call ransacions. The volailiies are ploed agains he raio of he opion srike o he index level. Thus, a value of one corresponds o pus or calls being a he money. Raios less han one represen srike prices ha are ou-of-he-money for pus and in-he-money for calls. The mos noiceable feaure of each of hese plos is ha he deep-ou-of-he-money pus have implied volailiies subsanially above he volailiies of oher opions. These volailiies decline almos linearly as he srike-index raio increases. Similar smile effecs have been observed in ineres rae opion markes (see Amin and Moron 1994 for Eurodollar fuures opions and Abken and Cohen 1994 for Treasury bond fuures opions) and in foreign exchange markes (Baes 1995). Noe 1. These prices are from a CBOE daa base ha covers he years More recen smiles compued from selemen price daa have he same shape as hose illusraed in he chars. 24 Economic Review December 1996

5 Volailiy Smiles (Opions Quoes from 10:00-14:30) Calls Pus Volailiy 1/31/ Srike/Index 1/31/ Srike/Index 7/31/ Srike/Index 7/31/ Srike/Index Federal Reserve Bank of Alana Economic Review 25

6 The fuure volailiy depends on a consan and a consan proporion of he las period s volailiy. In his case, he consan variance of he asse reurns in he Black-Scholes formula can be replaced by he average variance ha is expeced o prevail from ime unil ime T (he expiraion ime), which is approximaely given by T 1 T 2 u u= σ, and he Black-Scholes formula can coninue o be used. A more general case specifies volailiy as a funcion of oher informaion known o marke paricipans. One alernaive of his kind posis volailiy as a funcion of he level of he asse price: (S). One paricular model of his ype, known as he consan elasiciy of variance (CEV) model, in which volailiy is proporional o he level of he sock price raised o a power, appeared early in he opion pricing lieraure (Cox and Seve Ross 1976). However, he CEV model proved no o be free of pricing biases (David Baes 1994). A more recen variaion on his volailiy specificaion was developed by Rubinsein (1994). Insead of assuming a paricular form of he volailiy funcion, Rubinsein s mehod effecively infers he dependence of volailiy on he level of he asse price from raded opions a all available srike prices. He calls he model implied binomial rees because he implied risk-neural disribuion (which depends on he volailiy) of he asse price a mauriy is inferred from opion prices by consrucing a so-called binomial ree for movemens of he asse price. 1 (See Box 3 for a discussion of risk-neural valuaion.) Relaed models have been proposed by Emanuel Derman and Iraz Kani (1994), Bruno Dupire (1994), and David Shimko (1993). In a recen empirical es of deerminisic-volailiy models, including binomial ree approaches, Bernard Dumas, Jeffrey Fleming, and Rober Whaley (1996) show ha he Black-Scholes model does a beer job of predicing fuure opion prices. The opion dela, which is derived from an opion pricing model and measures he sensiiviy of he opion price o changes in he underlying asse price, can be used o specify posiions in opions ha offse underlying asse price movemens in a porfolio. The auhors demonsrae ha he Black-Scholes model resuled in beer hedges han hose from models based on deerminisic-volailiy funcions. For heir ess based on using S&P 500 index opions prices, hey conclude ha simpler is beer (20). The auhors noe ha one reason for he beer performance of he Black-Scholes model is ha errors, from various sources, in quoed opion prices disor parameer esimaes for deerminisic-volailiy models and consequenly degrade hese models predicions. However, hedging performance, which is a key consideraion for risk managers and raders alike, has no been sysemaically esed across all opion pricing models. As noed below, oher research indicaes ha some versions of sochasic-volailiy models may ouperform he simple Black-Scholes model in erms of hedging. ARCH Models. Auoregressive condiional heeroscedasiciy (ARCH) models for volailiy are a ype of deerminisic-volailiy specificaion ha makes use of informaion on pas prices o updae he curren asse volailiy and have he poenial o improve on he Black-Scholes pricing biases. The erm auoregressive in ARCH refers o he elemen of persisence in he modeled volailiy, and he erm condiional heeroscedasiciy describes he presumed dependence of curren volailiy on he level of volailiy realized in he pas. ARCH models provide a wellesablished quaniaive mehod for esimaing and updaing volailiy. ARCH models were inroduced by Rober F. Engle (1982) for general saisical ime-series modeling. An ARCH model makes he variance ha will prevail one sep ahead of he curren ime a weighed average of pas squared asse reurns, insead of equally weighed squared reurns, as is done ypically o compue variance (see Box 1). ARCH places greaer weigh on more recen squared reurns han on more disan squared reurns; consequenly, ARCH models are able o capure volailiy clusering, which refers o he observed endency of high-volailiy or low-volailiy periods o group ogeher. For example, several consecuive abnormally large reurn shocks in he curren period will immediaely raise volailiy and keep i elevaed in succeeding periods, depending on how persisen he shocks are esimaed o be. Assuming no furher large shocks, he cluser of shocks will have a diminishing impac as ime progresses because more disan pas shocks ge less weigh in he deerminaion of curren volailiy. Some echnical feaures of ARCH models also make hem aracive compared wih many oher ypes of opion pricing models ha allow for ime-varying volailiy. In an ARCH model, he variance is driven by a funcion of he same random variable ha deermines he evoluion of he reurns. 2 In oher words, he random source ha affecs he saisical behavior of reurns and volailiy hrough ime is he same. 26 Economic Review December 1996

7 Box 3 Risk-Neural Valuaion The risk-neural approach o opion valuaion was pioneered by Cox and Ross (1976) and hen developed sysemaically by Harrison and Kreps (1979) and Harrison and Pliska (1981). I was moivaed by he observaion ha he Black-Scholes opion pricing formula does no depend on any parameers ha reflec invesors preferences oward risk ha is, heir risk-reurn rade-offs. The key assumpion is merely ha invesors prefer more wealh o less wealh. In paricular, he opion price does no depend on he expeced reurn of he asse, which is deermined by invesor preferences. Since he opion price does no depend on invesors aiudes oward risk, he same opion price will resul irrespecive of he form of invesor preferences. A very convenien preference is risk neuraliy. A risk-neural invesor cares only abou he average level of wealh ha can be aained by rading in a risky asse and pays no aenion o he associaed risk. If invesors are risk-neural, hen in equilibrium he expeced reurns on all asses in he economy have o equal he risk-free rae; oherwise, invesors would aemp o buy (sell) hose securiies ha have expeced reurns greaer (less) han he reurn on he risk-free rae, driving he expeced reurn o equaliy wih he risk-free rae. Therefore, under risk neuraliy, he dynamics of he reurns process ha is, he saisical behavior of reurns hrough ime has o be adjused o make he mean reurn on he risky asse equal o he risk-free rae. As an example, consider an asse whose reurns process is described by he following equaion: r = µ + 1,, (1) where 1, is a random variable ha is disribued normally wih mean zero and variance of uniy (a uni normal random variable). This equaion is someimes called he law of moion or dynamics for he reurn process. The mean reurn on he asse is µ. The realizaions of he random variable 1, make he reurns r (a ime ) differen from µ, and hese realizaions are referred o as innovaions. The above equaion can be rewrien using a differen normal random variable 1,, wih zero mean and uni variance, r = rf + 1,,, (2) reurn of he asse, equals rf. For opion pricing, he law of moion of he asse reurns ha is relevan is (2) and no (1). Since he mean reurn of he asse under (2) is he risk-free rae, (2) is also known as he law of moion of he asse under he risk-neural disribuion an environmen in which all risky asses have expeced reurns equal o he risk-free rae. One of he key resuls of opion pricing heory is ha he price of an opion, or any financial claim ha has an uncerain fuure payoff, is given by he mahemaical expecaion of is payoff a is mauriy, discouned a he risk-free rae. The compuaion of his expecaion assumes ha he reurns of he asse follow risk-neural dynamics, such as he example given by equaion (2). If here is a second random variable ha affecs he price of he opion, hen, as in he previous example, he mean of ha sae variable is adjused o give he dynamics of he sae variable in a risk-neural world. Suppose he variance 2 follows he random process 2 = ( 2 1 ) + 2,, (3) where 2, is a sandard normal random variable. The above equaion is he discree-ime counerpar of he coninuous-ime variance process given in Heson (1993), in which he variance revers o is long-erm mean a rae, and he volailiy of he variance iself is measured by. A risk-neuralized represenaion of he above process analogous o (2) is 2 = * ( * 2 1 ) + * 2,. (4) The shock * 2, is anoher sandard normal random variable, and * and * are obained from and by a riskadjusmen procedure (see Heson 1993). In his case, he value of he opion is equal o he mahemaical expecaion under he risk-neural disribuion as generaed by (2) and (4), alhough he saisical behavior of reurns and variance in he real world is generaed by (1) and (3). The risk-neural disribuion iself can be inferred from raded opion prices. See Abken (1995) for a basic illusraion and Abken, Madan, and Ramamurie (1996) and Aï-Sahalia and Lo (1995) for advanced approaches. where rf is he risk-free rae. Thus, under he law of moion governed by he innovaion process, 1,, he mean Federal Reserve Bank of Alana Economic Review 27

8 As a resul, volailiy can be esimaed direcly from he ime series of observed reurns on an asse. In conras, he direc esimaion of volailiy from he reurns process is very difficul using sochasic-volailiy models. There are many differen ypes of ARCH models ha have a wide variey of applicaions in macroeconomics and finance. In finance, he wo mos popular ARCH processes are generalized ARCH (GARCH) (Bollerslev 1986) and exponenial GARCH (EGARCH) (Daniel B. Nelson 1991). The echnical disincions are beyond he scope of his aricle; however, researchers have ended mosly o use he GARCH process and is variaions for opion pricing. 3 Alhough GARCH capures he evoluion of he variance process of asse reurns quie well, i urns ou ha here is no easily compuable formula, like he Black-Scholes formula, for European opion pricing under a GARCH volailiy process. Insead, compuer-inensive mehods are used o simulae he reurns and he volailiy under he risk-neural disribuion in order o compue European opion prices and hedge raios. (Recen examples include Kaushik Amin and Vicor Ng 1993 and Jin C. Duan 1995.) Owing o he lack of efficien pricing and hedging formulas for GARCH models, praciioners and some researchers ofen subsiue he expeced average variance from a GARCH model for he variance inpu in he Black-Scholes formula (see Engle, Alex Kane, and Jaesun Noh 1994). However, he Black- Scholes formula does no hold if he variance of asse reurns follows a GARCH process; such a subsiuion is heoreically inconsisen bu may work in pracice. Anoher problem wih using he exan GARCH opion pricing models is ha hey do no value American opions, which accoun for mos of all raded opions. American opions can be exercised a any ime before mauriy, and consequenly heir prices equal or exceed he prices of comparable European opions by he value of his exra flexibiliy, ermed he early-exercise premium. A simple approximaion is achieved by adding an esimae of he early-exercise premium o he European price derived from a GARCH model. (There are numerical mehods, such as Mone-Carlo simulaions, ha can value American opions, bu hese mehods are currenly impracical because of he enormous number of compuaions required.) The value of he early-exercise premium is ofen evaluaed using he Barone-Adesi-Whaley (1987) formula for he Black-Scholes model. An early es of a GARCH opion pricing model is Engle and Chowdhury Musafa (1992), who examined S&P 500 index opions. Their resuls show ha he GARCH pricing model canno accoun for all of he pricing biases observed in he opion marke. Engle, Kane, and Noh (1994) compared he rading profis resuling from a paricular rading rule by using wo alernaives for he variance forecass needed for Black-Scholes: he variance forecas from a GARCH model and he variance forecas in he form of he Black-Scholes implied volailiy from a previous period. As noed above, plugging a GARCH forecas ino he Black-Scholes formula is ad hoc; however, in an experimen using S&P 500 index opions, Engle, Kane, and Noh produced greaer hypoheical rading profis using he GARCH volailiy forecas han hey did using he Black-Scholes implied volailiy. To summarize, alhough GARCH is a good descripion of he evoluion of he variance process of he asse reurns, opion pricing models based on GARCH are compuaionally demanding and may no be very useful for many praciioners given curren compuing echnology. In addiion, only a limied number of empirical ess have been done o dae on GARCH opion pricing models; as a consequence, i is hard o say how well he model does in pricing opions and evaluaing hedge raios. 4 Exponenially Weighed Momens Models. David G. Hobson and L.C.G. Rogers (1996) propose a new ype of opion pricing model for ime-varying volailiy ha also has he poenial o mach he observed volailiy smile. Their mahemaical specificaion allows pas asse-price movemens o feed back ino curren volailiy. This characerisic has some of he flavor of a GARCH model in erms of a similar feedback effec; however, he ype of feedback can be much more general han encounered in sandard GARCH models. Also like GARCH, bu unlike sandard sochasic-volailiy models, here is only one source of uncerainy ha drives boh he asse price and is volailiy. 5 The Hobson-Rogers model capures pas asse price volailiy hrough a so-called offse funcion. The feedback relaionship is primarily embodied in he funcional dependence of he volailiy on he offse funcion. The inuiion behind he offse funcion is apparen from is form: S ( ) = λe u ( Z Z ), m u= 1 λ where S (m) is he value of he funcion a ime and m is he order of he funcion. 6 This funcion simply weighs deviaions of a ransformed curren price Z (a discouned logarihm of he price) from is value u u m 28 Economic Review December 1996

9 periods ago, (Z Z u ), raised o he power m. The power applied o he deviaion, or order of he offse funcion, is echnically he saisical momen of he offse ha is employed. For example, a firs-order offse funcion (m = 1) considers he deviaion iself, whereas a second-order offse funcion akes he squares of hose deviaions and herefore consiss of a measure relaed o he variances of hose deviaions. The weighing is done by an exponenial funcion ha hrough he parameer places more or less imporance on he pas relaive o he presen. A high value for implies ha recenly experienced changes in he asse price have a much greaer impac on volailiy (and he drif) han more disan pas shocks. This weighing is similar o he reamen of pas reurn shocks in ARCH modeling. A low gives relaively more weigh o he pas shocks. The persisence of pas shocks can be esimaed indirecly from opions prices. The feedback mechanism in his model works primarily hrough he asse price volailiy, which can ake any number of funcional forms. Hobson and Rogers consider one simple form in deail in heir paper. They show ha even a simple version of he offse funcion, wih m = 1, can give opion prices ha when subsiued ino he Black-Scholes equaion generae a volailiy smile in implied Black-Scholes volailiies evaluaed a differen srike prices, mimicking he smile observed in acual markes. The impac of he Hobson-Rogers assumpion abou he volailiy specificaion and he persisence of volailiy on opion prices needs o be evaluaed empirically o see how i compares wih Black-Scholes or any oher model. The model s abiliy o race ou a smile is suggesive and may indicae he model s poenial o mach acual prices well; an empirical evaluaion of his model has no been performed o dae. Sochasic Volailiy Sochasic volailiy implies ha he fuure level of he volailiy canno be perfecly prediced using informaion available oday. The populariy of sochasic volailiy in opion pricing grew ou of he fac ha disribuions of he asse reurns exhibi faer ails han hose of he normal disribuion (Benoi Mandelbro 1963 and Eugene F. Fama 1965). In oher words, he observed frequency of exreme asse reurns is much higher han would occur if reurns were described by a normal disribuion. Sochasic-volailiy models can be consisen wih fa ails of he reurn disribuion. The occurrence of fa ails would imply, for example, ha ou-of-he-money opions would be underpriced by he Black-Scholes model, which assumes ha reurns are normally disribued. However, he fa-ailed asse reurn disribuions can also come from ARCHype volailiy as well as from jumps in he asse reurns (Rober C. Meron 1976). Sochasic-volailiy models could also be an alernaive explanaion for skewness of he reurn disribuion. Despie he relaive complexiy of sochasic-volailiy models, hey have been popular wih researchers, and addiional jusificaion for hese models has recenly come o ligh in he lieraure on asymmeric informaion abou he fuure asse price and is impac on raded opions. 7 In a sochasic-volailiy model, volailiy is driven by a random source ha is differen from he random source driving he asse reurns process, alhough he wo random sources may be correlaed wih each oher. In conras o a deerminisic-volailiy model in which he invesor incurs only he risk from a randomly evolving asse price, in a sochasic-volailiy environmen, an invesor in he opions marke bears he addiional risk of a randomly evolving volailiy. In a deerminisicvolailiy model, an invesor can hedge he risk from he asse price by rading an opion and a risk-free asse based on a risk exposure compued using an opion pricing formula (see Cox and Rubinsein 1985). (Equivalenly, he opion s payoff can be replicaed by rading he underlying asse and a risk-free asse.) However, wih a random-volailiy process, here are wo sources of risk (he risk from he asse price and he volailiy risk); a risk-free porfolio canno be creaed as in he Black-Scholes model. Afer hedging, here is a residual risk ha sems from he random naure of he volailiy process. Since here is no raded asse whose payoff is a known funcion of he volailiy, volailiy risk canno be perfecly hedged. In order o bear his volailiy risk, raional invesors would demand a volailiy risk premium, which has o be facored ino opion prices and hedge raios. 8 A feaure of sochasic-volailiy models ha is no shared by deerminisic-volailiy models is ha he price of an opion can change wihou any change in he level of he asse price. The reason is ha he opion price is driven by wo random variables: he asse price and is volailiy. In sochasic-volailiy models, hese wo variables may no be perfecly correlaed, implying ha he expeced volailiy over he life of he opion may change wihou any change in he asse price. The change in volailiy alone can cause he opion price o change. Federal Reserve Bank of Alana Economic Review 29

10 The modeling of volailiy and is dynamics is a difficul ask because he pah of volailiy during he life of an opion is highly unpredicable. Mos sochasic-volailiy models assume ha volailiy is mean revering. Tha is, alhough volailiy varies from day o day, here is a presumed long-run level oward which volailiy seles in he absence of addiional shocks. Marke paricipans refer o his feaure as regressing o he mean of he volailiy. (The evidence for his phenomenon is especially srong in markes for ineres rae derivaives. See, for example, Rober Lierman, Jose Scheinkman, and Laurence Weiss 1991 and Amin and Andrew Moron 1994.) Sochasic-volailiy models can be classified ino wo broad caegories: hose ha lack closed-form soluions for European opions and hose ha have closedform soluions. 9 Even if a model s parameers are known, mos sochasic-volailiy opion pricing models are compuaionally demanding for pricing European opions and especially so for pricing American opions. A noable excepion is he model of Seven Heson (1993) ha gives closed-form soluions for prices and hedge raios of European opions. All oher models compue opion prices eiher by numerically solving a complicaed parial differenial equaion or by Mone Carlo simulaion. However, many key parameers are no readily esimaed from daa, paricularly hose of he volailiy process, because, unlike he reurns process of he underlying asse, he volailiy process is no direcly observable. Since parameer esimaion is ofen ime-consuming, he lack of readily compued soluions for opion prices in many sochasicvolailiy models can compound he difficulies of esimaion. Alhough sochasic-volailiy pricing models give only closed-form soluions for European opions, a good approximaion for he price of an American opion can be obained by adding an early exercise premium using he Barone-Adesi-Whaley approximaion in he same way as for ARCH models. Examples of his pracice are in Hans J. Knoch (1992) and Baes (1995). A presen, he only oher way o price American opions under sochasic volailiy is by solving a second-order parial differenial equaion (Angelo Melino and Suar Turnbull 1992), which is exremely compuaionally burdensome. Sochasic-Volailiy Opion Models wihou Closed-Form Soluion. John C. Hull and Alan Whie (1987), Louis O. Sco (1987), and James B. Wiggins (1987) were among he firs o develop opion pricing models based on sochasic volailiy. Hull and Whie as well as Sco made he quesionable assumpion ha he risk premium of volailiy is zero ha is, he volailiy risk is no priced in he opions marke and ha volailiy is uncorrelaed wih he reurns of he underlying asse. Wiggins, who also assumed a zero-volailiy risk premium, found ha he esimaed opion values under sochasic volailiy were no significanly differen from Black-Scholes values, excep for long mauriy opions. For equiy opions, Chrisopher Lamoureux and William Lasarapes (1993) offer evidence agains he assumpion of a zero-volailiy risk premium. For currency opions, Melino and Turnbull (1992) found ha a random-volailiy model yields opion prices ha are in closer agreemen wih he observed opion prices han hose of he Black-Scholes model. While he numerical mehods and compuers currenly available allow compuaion of hese sochasicvolailiy opion prices, hey are sill largely impracical for deermining hedge raios, which are vial o markemakers, dealers, and ohers. As a resul, hese sochasic-volailiy models may no currenly be useful for praciioners. Neverheless, developmen of sochasicvolailiy models coninues as researchers aemp o find more racable models. Sochasic-Volailiy Models wih Closed-Form Soluions. Elias M. Sein and Jeremy C. Sein (1991) develop a European opion pricing model under sochasic volailiy ha is somewha easier o evaluae han he models described above. 10 Alhough less compuaionally expensive han he oher models, he auhors make he unrealisic assumpion of zero correlaion beween he volailiy process and he reurns of he underlying asse. Heson (1993) was he firs o develop a sochasicvolailiy opion pricing model for European equiy and currency opions ha can be easily implemened, is compuaionally inexpensive, and allows for any arbirary correlaion beween asse reurns and volailiy. 11 The model gives closed-form soluions no only for opion prices bu also for he hedge raios like he 30 Economic Review December 1996

11 delas and he vegas of opions. (Dela and vega measure he sensiiviy of he opion price o changes in he asse price and o changes in he volailiy, respecively. Knowledge of hese measures enables he consrucion of hedges for opions or for porfolios conaining embedded opions.) In his model, he asse reurns r and he variance are assumed o evolve hrough ime as 2 and r = µ + 1, 2 = ( 2 1 ) + 2,, respecively, where 1, and 2, are wo sandard normal random variables ha could be correlaed wih each anoher, eiher posiively or negaively, wih a correlaion coefficien,. Equivalenly, his coefficien also measures he correlaion beween he reurn of he asse and he volailiy process. In his model, he variance evolves hrough ime in such a way ha is long-run average level is measured by and he speed wih which i is pulled oward his long-run mean is measured by, also known as he mean-reversion coefficien. The variable is a measure of he volailiy of variance. If is zero, he model simplifies o a ime-varying deerminisic-volailiy model. In he finance lieraure, his process for he volailiy is also known as a square-roo volailiy process. The paricular naure of he process ensures ha volailiy reflecs away from zero: if volailiy ever becomes zero, hen he nonzero ensures ha volailiy will become posiive. Noe ha 2 in his model is no direcly comparable o he implied variance from he Black-Scholes model. The reason is ha 2 represens he insananeous variance (a ime ), whereas he implied variance in he Black-Scholes model is he average expeced variance hrough he life of an opion and need no equal he insananeous variance if he model is no rue. In Heson s model, he average expeced variance during he life of an opion is a funcion of he insananeous variance, he long-run average variance, he speed wih which he insananeous variance adjuss, and he ime o expiraion of he opion. The opion price and hedge raios in Heson s model are funcions no only of he parameers ha appear in he Black-Scholes formula bu also of,,,, and an addiional parameer,. The parameer is a consan such ha 2 measures he risk premium of volailiy. The volailiy risk premium is assumed o be direcly proporional o he level of he volailiy. The need for an assumpion abou he form of he volailiyrisk premium is a weakness of any sochasic-volailiy model because he form of he volailiy-risk premium canno be deduced from he weak assumpion ha all invesors prefer more wealh o less wealh, as discussed in Box 3, bu requires assumpions on invesor olerance oward risk ha in general are difficul o jusify. In his model, he form of he volailiy-risk premium is crucial because i enables he derivaion of he closed-form soluions for opion prices and hedge raios. However, i should no be inerpreed as a weakness of his model vis-à-vis oher sochasicvolailiy models of opion prices because ohers make he even sronger and less plausible assumpion ha he risk premium of volailiy is zero. The parameers and are very imporan for deermining he form of he risk-neural disribuion of he asse price a he ime of he opion s expiraion (he erminal asse price) and hence he curren opion price. In oher words, hey may be imporan for accouning for he smile effecs seen in he char. For example, consider he probabiliy ha a European call opion will finish in he money. Ceeris paribus, an increase in (an increase in he volailiy of volailiy) makes he ails of he risk-neural disribuion faer: he occurrence of exreme reurns is more likely. 12 The sign and magniude of deermines he sign and exen of skewness in he risk-neural disribuion of he erminal asse price. Posiive correlaion implies ha an increase in he reurns of he underlying asse is associaed wih an increase in he volailiy, ending o make he righ ail of he disribuion hicker and he lef ail hinner han hose of a normal disribuion of asse reurns. In oher words, he frequency of exreme posiive oucomes is higher and he frequency of exreme negaive oucomes is lower han in he Black- Scholes model ha is, he reurns have posiive skewness. As a resul, prices of ou-of-he-money calls, which benefi from his scenario of posiive skewness, are higher in he sochasic-volailiy model han corresponding Black-Scholes call prices, and hose of ou-of-he-money pus (ha lose under his scenario) are lower. On he oher hand, a negaive correlaion implies ha a decrease in he reurns of he underlying asse is associaed wih an increase in he variance. Therefore, he lef ail would be hicker and he righ ail hinner han assumed for he Black-Scholes model. Since ou-of-he-money pus benefi from a hicker lef ail, marke prices for hese opions would be higher han in he Black-Scholes model (underpricing by he Black-Scholes model), and, similarly, ou-of-he-money Federal Reserve Bank of Alana Economic Review 31

12 calls ha lose from a hicker lef ail would be overpriced by he Black-Scholes model. This las scenario is consisen wih observaions in he marke for S&P 500 index opions since he crash of As noed earlier, ou-of-he-money pus have ended o command much higher prices han can be explained by he Black-Scholes model, whereas ou-ofhe-money calls are overpriced by he Black-Scholes model. According o he sochasic-volailiy model, he underpricing of he ou-of-he-money pus and overpricing of ou-of-he-money calls by he Black- Scholes model he volailiy skew could be he resul of a negaive correlaion beween index reurns and a random volailiy process. A likely cause of financial marke volailiy is he arrival of informaion and is subsequen incorporaion ino asse prices hrough rading. The empirical work done on Heson s model includes ha by Knoch (1992), Saika Nandi (1996), and Baes (1995). In order o ake ino accoun he possibiliy of sudden large price movemens, such as he crash of 1987, Baes generalizes Heson s model by allowing for jumps in asse prices. While Knoch and Baes sudy he pricing issues of his model for opions on foreign currencies, Nandi examines boh pricing and hedging issues using he S&P 500 index opions. All of hese sudies find ha Heson s model is able o generae prices ha are in closer agreemen wih marke opion prices han hose of he Black-Scholes model. However, i is no he case ha his model is able o explain all biases of he Black-Scholes model. While i is rue ha he remaining pricing biases are of smaller magniude han hose of he Black-Scholes model, Nandi finds ha here are sill subsanial biases for ou-of-he-money pus and calls in he S&P 500 index opions marke. In paricular, he model underprices ou-of-he-money pus and overprices ou-of-he-money calls. I is possible ha he square-roo volailiy process and herefore he model iself are misspecified. This misspecificaion would be unforunae because he paricular form of he volailiy process is wha makes his sochasic-volailiy model racable. If he Black-Scholes assumpion of consan volailiy were rue, a hedge porfolio (hedged agains he risk from he asse price) would simply earn he riskfree rae of reurn. Such a porfolio would ypically consis of a posiion in he underlying asse and an opion. The posiion would be alered hrough ime by rading, based on he formulas for hedge raios deermined by he Black-Scholes model (see Cox and Rubinsein 1985) or oher opion pricing models. When volailiy is sochasic, as i probably is in he real world, hedging using he Black-Scholes model does no resul in risk-free posiions. A sochasic-volailiy model may do a beer job of hedging agains price and volailiy risks. Nandi (1996) finds ha for S&P 500 index opions he reurns of a hedge porfolio consruced using Heson s sochasic-volailiy model come closer o maching a risk-free reurn hrough ime beer han hedge porfolio reurns obained using he Black-Scholes model. Volailiy Jumps. All he ime-varying volailiy models ha have been discussed so far assume ha he volailiy of he underlying asse as well as is price evolves smoohly, hough randomly, hrough ime: here are no jumps in he volailiy process. However, a likely cause of financial marke volailiy is he arrival of informaion and is subsequen incorporaion ino asse prices hrough rading. To he exen ha informaion news arrives in discree lumps, i is possible ha volailiy shifs beween episodes of low and high volailiy. For example, uncerainy abou an impending news release (concerning some macroeconomic variable, like an anicipaed change in he fed funds rae by he Federal Open Marke Commiee) may cause he volailiy of an asse price o rise. However, afer a few rounds of rading, wih he informaion having been incorporaed ino asse prices, volailiy may rever back o is previous level. To accoun for jumps like hose in he example, Vasanlilak Naik (1993) develops a pricing model for European opions in which volailiy swiches beween low and high levels. Each level or regime is expeced o las for a cerain period of ime ha is no known a priori. One racable version of his model assumes ha he risk from he volailiy jumps is no priced by marke paricipans. The model akes he same parameers ha ener he Black-Scholes formula as well as addiional parameers such as he probabiliies of jumps from one regime o anoher regime, given ha volailiy is currenly in a paricular regime. Naik finds 32 Economic Review December 1996

13 ha shor-mauriy opions are much more sensiive o volailiy shifs han long-mauriy opions. The reason is ha, over a long period of ime, expeced upward and downward jumps in volailiy are canceled by each oher, resuling in a volailiy ha is close o he normal level. This model has no been empirically esed and herefore canno ye be evaluaed agains oher sochasicvolailiy models. In general, jump models can be difficul o verify empirically because jumps occur infrequenly. The parameers of such models may be imprecisely esimaed using relaively small hisorical daa series on opion prices or underlying asse prices. Conclusion Since volailiy of he underlying asse price is a criical facor affecing opion prices, he modeling of volailiy and is dynamics is of vial ineres o raders, invesors, and risk managers. This modeling is a difficul ask because he pah of volailiy during he life of an opion is highly unpredicable. Clearly, he Black- Scholes assumpion of consan volailiy can be improved upon by incorporaing ime variaion in volailiy. While deerminisic-volailiy models can capure he dynamics of he volailiy reasonably well, many of hese opion pricing models, such as ARCH models, are compuaionally expensive, especially for American opions. Deerminisic-volailiy opion pricing models have he advanage ha mos parameers can be esimaed direcly from he observable ime series of reurns daa. However, superior hedging performance of such models relaive o ha of he Black-Scholes model has no been demonsraed. On he oher hand, here is evidence ha some sochasic-volailiy opion pricing models provide beer hedges han Black- Scholes, alhough for sochasic-volailiy opion pricing models and volailiy-jump models, parameer esimaion is ypically demanding and problemaic. The developmen of racable sochasic-volailiy models as well as more efficien mehods of model parameer esimaion are currenly an area of inensive research. For boh academic researchers and marke praciioners, no consensus exiss regarding he bes specificaion of volailiy for opion pricing. Alhough a number of alernaive approaches can accoun, a leas parially, for he pricing deficiencies of he Black-Scholes model, none dominaes as a clearly superior approach for pricing opions. Noes 1. Insead of aking a wide range of values as in he real world, a binomial ree resrics sock price movemens a any momen in ime o be eiher up wih one probabiliy or down wih anoher (see Cox and Rubinsein 1985). 2. Alhough here is one source of uncerainy ha drives boh he asse reurns and he volailiy in a GARCH model, which is a special case of ARCH, he asse reurns are disribued coninuously ha is, one ou of an infinie number of possible uncerain reurns will be realized over he nex period. Therefore, wih discree rading (as in a GARCH model), i is no possible o replicae all possible uncerain reurns oucomes (see Duffie and Huang 1985) by rading in he opion and a risk-free asse (or, equivalenly, a unique risk-free porfolio canno be creaed by rading in he underlying asse and an opion). Hence, a risk premium associaed wih he reurns of he underlying asse is required in a GARCH model. 3. The NGARCH of Engle and Ng (1993) is one such variaion. 4. GARCH can capure he volailiy smile. In a GARCH model, such as Duan s (1995), he price of an opion, besides being a funcion of he variables ha appear in he Black-Scholes formula, is also a funcion of variables ha describe he ime variaion in volailiy as well as a variable ha accouns for he risk premium of he asse reurns, ha is, he excess reurn over a risk-free asse. Since he risk premium summarizes invesor preferences, he GARCH opion pricing model is no preference-free a key aribue of he Black-Scholes model. Duan shows ha under he riskneural disribuion, he value of he GARCH variance a a poin in ime is negaively correlaed wih pas asse reurns if he risk premium of he asse is greaer han zero. Such a negaive correlaion can give rise o negaive skewness in he risk-neural disribuion, which seems o be a feaure of he empirical daa, as discussed in Baes (1995). GARCH models can herefore poenially generae opion prices ha are consisen wih he observed volailiy skew. 5. The Hobson-Rogers model is also preference-free. This model, unlike GARCH, is se in coninuous ime. There being a single source of uncerainy and coninuous rading, all possible uncerain reurns oucomes of he underlying risky asse over he nex period can be replicaed by rading in an opion and a risk-free asse (Duffie and Huang 1985), and here is no need for any risk premium of reurns. Federal Reserve Bank of Alana Economic Review 33

14 6. The Hobson-Rogers equaion acually is wrien wih an inegral raher han a summaion. 7. Back (1993) shows how sochasic volailiy migh be inroduced endogenously in asse markes due o asymmeric informaion abou he fuure price of an underlying asse on which an opion is raded. 8. In an ARCH opion pricing model he risk premium ha eners is he risk premium of asse reurns and no he risk premium of volailiy. 9. For American opions, a closed-form soluion in a sochasicvolailiy model has no ye been derived. 10. Their model requires he numerical evaluaion of a wodimensional inegral (ha is compuaionally easier) raher han he soluion of a second-order parial differenial equaion. However, he volailiy process is allowed o become negaive, an undesirable feaure. 11. Heson s (1993) paper gives he closed-form soluion for prices of call opions. The price of a pu opion can be easily obained using he pu-call pariy for European opions. 12. A ail of a probabiliy disribuion is he area under he disribuion ha assigns probabiliies o exreme oucomes. For example, in he ypical bell-shaped normal disribuion, here are wo ails, he righ ail and he lef ail, ha slowly aper off. References Abken, Peer A. Over-he-Couner Financial Derivaives: Risky Business? Federal Reserve Bank of Alana Economic Review 79 (March/April 1994): Using Eurodollar Fuures Opions: Gauging he Marke s View of Ineres Rae Movemens. Federal Reserve Bank of Alana Economic Review 80 (March/April 1995): Abken, Peer A., and Hugh Cohen. Generalized Mehod of Momens Esimaion of Heah-Jarrow-Moron Models of Ineres-Rae Coningen Claims. Federal Reserve Bank of Alana Working Paper 94-8, Augus Abken, Peer A., Dilip B. Madan, and Sailesh Ramamurie. Esimaion of Risk-Neural and Saisical Densiies by Hermie Polynomial Approximaion: Wih an Applicaion o Eurodollar Fuures Opions. Federal Reserve Bank of Alana Working Paper 96-5, June Aï-Sahalia, Yacine, and Andrew W. Lo. Nonparameric Esimaion of Sae-Price Densiies Implici in Financial Asse Prices. Naional Bureau of Economic Research Working Paper 5351, Amin, Kaushik, and Andrew Moron. Implied Volailiy Funcions in Arbirage-Free Term Srucure Models. Journal of Financial Economics 35 (1994): Amin, Kaushik, and Vicor Ng. ARCH Processes and Opion Valuaion. Universiy of Michigan Working Paper, Back, Kerry E. Asymmeric Informaion and Opions. Review of Financial Sudies 6 (1993): Barone-Adesi, Giovanni, and Rober Whaley. Efficien Analyic Approximaion of American Opion Values. Journal of Finance 42 (1987): Baes, David. The Skewness Premium: Opion Pricing under Asymmeric Processes. Universiy of Pennsylvania Working Paper, Jumps and Sochasic Volailiy: Exchange Rae Processes Implici in Deuschemark Opions. Review of Financial Sudies 9 (1995): Black, Fischer, and Myron S. Scholes. The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy 81 (May/June 1973): Bollerslev, Tim. Generalized Auoregressive Condiional Heeroscedasiciy. Journal of Economerics 31 (1986): Bollerslev, Tim, Ray Y. Chou, and Kenneh F. Kroner. ARCH Modeling in finance: A Review of he Theory and Empirical Evidence. Journal of Economerics 5 (1992): Cox, John C., and Seve Ross. The Valuaion of Opions for Alernaive Sochasic Processes. Journal of Financial Economics 3 (1976): Cox, John C., and Mark Rubinsein. Opions Markes. Englewood Cliffs, N.J.: Prenice Hall, Derman, Emanuel, and Iraz Kani. Riding on he Smile. Risk 7 (February 1994): Duan, Jin C. The GARCH Opion Pricing Model. Mahemaical Finance 5 (1995): Duffie, Darrell, and Chi-Fu Huang. Implemening Arrow- Debreu Equilibria by Coninuous Trading of Few Long- Lived Securiies. Economerica 53, no. 6 (1985): Dumas, Bernard, Jeffrey Fleming, and Rober Whaley. Implied Volailiy Funcions: Empirical Tess. Rice Universiy Working Paper, Dupire, Bruno. Pricing wih a Smile. Risk 7 (January 1994): Engle, Rober F. Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of U.K. Inflaion. Economerica 50 (1982): Engle, Rober, Alex Kane, and Jaesun Noh. Forecasing Volailiy and Opion Prices of he S&P 500 Index. Journal of Derivaives (1994): Engle, Rober F., and Chowdhury Musafa. Implied ARCH Models from Opion Prices. Journal of Economerics 52 (1992): Engle, Rober, and Vicor Ng. Measuring and Tesing he Impac of News on Volailiy. Journal of Finance 48 (1993): Fama, Eugene F. The Behavior of Sock Marke Prices. Journal of Business 38 (1965): Economic Review December 1996

15 Harrison, J. Michael, and David Kreps. Maringales and Arbirage in Muliperiod Securiies Markes. Journal of Economic Theory 20 (1979): Harrison, J. Michael, and Sanley Pliska. Maringales and Sochasic Inegrals in he Theory of Coninuous Trading. Sochasic Processes and Their Applicaions 11 (1981): Heson, Seven. A Closed-Form Soluion for Opions wih Sochasic Volailiy wih Applicaions o Bond and Currency Opions. Review of Financial Sudies 6 (1993): Hobson, David G., and L.C.G. Rogers. Complee Models wih Sochasic Volailiy. Universiy of Bah, School of Mahemaical Sciences. Unpublished paper, Hull, John C., and Alan Whie. The Pricing of Opions on Asses wih Sochasic Volailiies. Journal of Finance 42 (1987): Knoch, Hans J. The Pricing of Foreign Currency Opions wih Sochasic Volailiy. Ph.D. disseraion, Yale School of Organizaion and Managemen, Lamoureux, Chrisopher, and William Lasarapes. Forecasing Sock-Reurn Variance: Toward an Undersanding of Sochasic Implied Volailiies. Review of Financial Sudies 6 (1993): Lierman, Rober, Jose Scheinkman, and Laurence Weiss. Volailiy and he Yield Curve. Journal of Fixed Income Research (June 1991): Mandelbro, Benoi. The Variaion of Cerain Speculaive Prices. Journal of Business 36 (1963): Melino, Angelo, and Suar Turnbull. The Pricing of Foreign Currency Opions wih Sochasic Volailiy. Journal of Economerics 45 (1992): Meron, Rober C. Opion Pricing wih Disconinuous Reurns. Journal of Financial Economics 3 (1976): Naik, Vasanlilak. Opion Valuaion and Hedging Sraegies wih Jumps in he Volailiy of Asse Reurns. Journal of Finance 48 (1993): Nandi, Saika. Pricing and Hedging Index Opions under Sochasic Volailiy: An Empirical Examinaion. Federal Reserve Bank of Alana Working Paper 96-9, Augus Nelson, Daniel B. Condiional Heeroscedasiciy in Asse Reurns: A New Approach. Economerica 59 (1991): Rubinsein, Mark. Nonparameric Tess of Alernaive Opion Pricing Models Using All Repored Trades and Quoes on he 30 Mos Acive Opion Classes from Augus 23, 1976, hrough Augus 31, Journal of Finance 40 (1985): Implied Binomial Trees. Journal of Finance 49 (1994): Sco, Louis O. Opion Pricing When he Variance Changes Randomly: Theory, Esimaion, and Applicaion. Journal of Financial and Quaniaive Analysis 22 (1987): Shimko, David. Bounds on Probabiliy. Risk 6 (April 1993): Sein, Elias M., and Jeremy C. Sein. Sock Price Disribuions wih Sochasic Volailiy: An Analyic Approach. Review of Financial Sudies 4 (1991): Wiggins, James B. Opion Values under Sochasic Volailiies. Journal of Financial Economics 19 (1987): Federal Reserve Bank of Alana Economic Review 35