We exploit the information in the options market to study the variations of return risk and market prices

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1 MANAGEMENT SCIENCE Vol. 56, No. 12, December 2010, pp issn eissn informs doi /mnsc INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period Gurdip Bakshi Smih School of Business, Universiy of Maryland, College Park, Maryland 20742, gbakshi@rhsmih.umd.edu Liuren Wu Zicklin School of Business, Baruch College, Ciy Universiy of New York, New York, New York 10010, liuren.wu@baruch.cuny.edu We exploi he informaion in he opions marke o sudy he variaions of reurn risk and marke prices of differen sources of risk during he rise and fall of he Nasdaq marke. We specify a model ha accommodaes flucuaions in boh risk levels and marke prices of differen sources of risk, and we esimae he model using he ime-series reurns and opion prices on he Nasdaq 100 racking sock. Our analysis reveals hree key variaions during he period from March 1999 o March Firs, reurn volailiy increased ogeher wih he rising Nasdaq index level, even hough he wo end o move in opposie direcions. Second, alhough he marke price of diffusion reurn risk averages around 1.82 over he whole sample, he esimaes reached negaive erriory a he end of The esimaes revered back o highly posiive values afer he collapse of he Nasdaq marke. Third, he marke price of jump risk increased wih he rising Nasdaq valuaion, and his increase in marke price coincided wih an increased imbalance in open ineres beween pu and call opions. Key words: Nasdaq; risks; marke prices of risk; diffusion risk; jump risk; open ineres; opions Hisory: Received January 17, 2009; acceped Augus 10, 2010, by Wei Xiong, finance. 1. Inroducion The Nasdaq marke endured an unusual ransformaion a he urn of he 21s cenury. During he one-year period from March 1999 o March 2000, he Nasdaq 100 index, which consiues he vas majoriy of he Nasdaq marke capializaion, rose by 128%. Sock prices sared o decline afer ha, and by March 2001 he level of he Nasdaq 100 index was abou 30% of is peak value in 2000, a phenomenon labeled by many as he Nasdaq bubble. This paper provides a framework for analyzing he behavior of reurn risk and marke prices of differen sources of risk as revealed by opion prices on he Nasdaq 100 racking sock over he rise and fall of he Nasdaq. In conras o earlier bubbles, e.g., Duch ulip mania and he Souh Sea bubble, he Nasdaq bubble segmen incorporaes a unique addiional informaion source in he form of opion quoes on he Nasdaq 100 racking sock. Each day, opion prices of differen srikes and mauriies provide us wih a picure of he marke s percepion of he risk level and he condiional reurn disribuion over differen forwardlooking horizons (Aï-Sahalia and Lo 1998, Birru and Figlewski 2009, Figlewski 2009). Thus, by analyzing he variaions of he opion prices and he underlying index, one can infer how he reurn risk and marke prices of risk have varied during he bubble period. 1 For his purpose, we specify a model ha accommodaes several disinc risk sources: (i) diffusion reurn risk, (ii) reurn volailiy risk, (iii) upside jump risk, and (iv) downside jump risk. We assign a separae marke price for each risk source, and we decompose he variaions in he reurn risk premium ino variaions in he reurn risk level and in he marke prices of differen sources of risk. We also presen an esimaion procedure o exrac he variaions of reurn risk and marke prices of risk from he opion prices and he daily sock reurns, while accouning for he correlaion beween opion errors across opion srikes and mauriies. Our model consrucion and esimaion shows ha reurn risk sared o climb in lae 1999 and spiked afer he collapse of he Nasdaq. Reurn volailiy calmed down during a shor period in he middle of 1 Ever since he classic works by Garber (1989) and Kindleberger (2000), bubbles and burss have been a recurring heme of he financial marke research. See, for insance, Abreu and Brunnermeier (2003), Baalio and Schulz (2006), Brunnermeier and Nagel (2004), Dhar and Goezmann (2005), Griffin e al. (2004), Hong e al. (2006), Lakonishok e al. (2007), LeRoy (2004), Ofek and Richardson (2002, 2003), Pásor and Veronesi (2006), Schwer (2002), and Temin and Voh (2004). 2251

2 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period 2252 Managemen Science 56(12), pp , 2010 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org as he Nasdaq marke experienced a shor-lived recovery, bu volailiy rose again ogeher wih furher deerioraion in marke capializaion. Our esimaion also idenifies significan variaions in he marke prices of differen sources of risk around he bubble period. The esimaes for he marke price for diffusion risk average around 1.82, close o common findings in he lieraure, bu he esimaes urned negaive from Sepember 21, 1999, o January 5, 2000, as he Nasdaq kep increasing. These negaive esimaes reflec a possible increase in he marke s appeie for risk, and/or invesors subjecive beliefs abou fuure cash flows became much rosier han realiy during his period. We find ha he marke price of jump risk rose wih he Nasdaq. The elevaed marke price of jump risk reflecs pu opions becoming more expensive han call opions during his period. This pu call price asymmery is suppored by a corresponding imbalance in he pu call open ineres. On average, he open ineres and rading volume are boh higher for call opions han for pu opions on he Nasdaq 100 racking sock. However, he pu open ineres became much higher han he call open ineres heading ino he collapse of he Nasdaq. Overall, our resuls show ha he opions marke can provide an informaive perspecive abou he developmen and analysis of financial bubbles and crises, and abou how invesors reac o various sources of risk. Along he same lines, Birru and Figlewski (2009) provide insighs on he mos recen financial crisis of 2008 by analyzing he riskneural reurn densiies implied by he S&P 500 index opions. The nex secion formalizes he model srucure ha decomposes he sources of reurn risk premium and characerizes he marke prices of risk. Secion 3 discusses he daa, and 4 oulines he esimaion procedure. Secion 5 presens our esimaion resuls on he behavior of he reurn risk and he marke prices of risk, specific o he rise and he fall of he Nasdaq. Conclusions are offered in Risk Sources and Marke Prices We firs decompose he index reurn ino several risk sources and hen assign a separae marke price o each risk source. The decomposiion of risk and marke prices of risk, via a model, allows us o invesigae wheher risk levels and/or marke prices of various sources of risk experienced sysemaic variaions during he bubble period The Model Specificaion Fix a filered probabiliy space F F 0, and le S be he level of he Nasdaq 100 racking sock a ime. Assume ha S is governed by he following sochasic differenial equaion: ds = rp S + r q d + V dw + e x 1 + dx d V + x dx d 0+ + e x 1 dx d V x dx d (1) 0 where S denoes he ime- prejump level of he sock price. The insananeous reurn ds /S is decomposed ino four componens. The firs componen includes he ex-dividend riskfree reurn (r q ) and an insananeous reurn risk premium rp. We allow he insananeous ineres rae r and he dividend yield q o evolve deerminisically over ime, and we derive he reurn risk premium based on our risk and marke price specificaions. The second componen capures he innovaion of a coninuous componen, wih dw denoing he change of a sandard Brownian moion and V denoing is insananeous variance rae, which varies sochasically according o he following sochasic differenial equaion: dv = V d+ V dz (2) where dz denoes he change of anoher sandard Brownian moion, correlaed wih he Brownian movemen in he reurn by d = Ɛ dw dz. Given his correlaion, we can decompose he Brownian moion in he sock reurn ino wo componens: dw = dz db (3) where db denoes he Brownian componen in he sock reurn ha is independen of he variance rae movemen. The hird and fourh componens in Equaion (1) capure he conribuions of upside and downside jumps, wih 0+ and 0 denoing he posiive and negaive halfs of he real line excluding zero, respecively. In paricular, + dx d and dx d coun he number of upside and downside jumps of size x a ime, wih V + x dx d and V x dx d being he corresponding compensaors for wo ypes of jumps, respecively. In he model, V governs no only he insananeous variance rae of he Brownian movemen, bu also he arrival rae of upside and downside jumps. Furhermore, deparing from Meron (1976), he model reas upside jumps and downside jumps separaely, wih + x and x conrolling he disribuion of he upside and downside jumps, respecively.

3 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period Managemen Science 56(12), pp , 2010 INFORMS 2253 Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. We assume an exponenial disribuion for boh ypes of jumps: { e + x x>0 + x = 0 x<0 { (4) 0 x>0 x = x<0 e x where conrols he mean arrival inensiy, and + and conrol how fas he number of upside and downside jumps decrease wih increasing jump sizes. The jump process has a sochasic arrival rae of V Condiional on a jump occurring, he jump size x in log reurn has a double-exponenial disribuion as in Kou (2002). Under our specificaion, he insananeous sock reurn variance rae is V , wih he variance conribuion from he diffusion componen being V and he variance conribuion from he jump componen being 2V To link he saisical dynamics o marke prices of various sources of risk, we specify he dynamics of he sae-price deflaor M as dm M = r d B V db Z V dz J + 0+ e x 1 + dx d V + x dxd J 0 e x 1 dx d V x dxd (5) The specificaion assigns a separae marke price for each of he four sources of risk: he independen diffusive reurn risk B, he diffusion variance risk Z, he upside jump risk J +, and he downside jump risk J. The marke prices of differen sources of risk capure he differences beween he risk-neural dynamics and he saisical dynamics as a resul of (i) invesors risk aversions o differen sources of risk and/or (ii) deviaions of invesors subjecive beliefs from saisical realiies. We idenify hese marke prices wihou making a disincion beween he wo explanaions. Our model is in line wih exan lieraure and is designed o address he quesion of how he marke prices of differen risk sources changed during he Nasdaq bubble period. Earlier works, e.g., Bakshi e al. (1997) and Huang and Wu (2004), ofen combine Heson s (1993) sochasic volailiy specificaion wih Meron s (1976) jump diffusion. The doubleexponenial specificaion of Kou (2002) in Equaion (4) provides an aracive alernaive o he Meron (1976) model, while allowing us o consider he disinc pricing of downside and upside jumps The Reurn Risk Premium Decomposiion Under our specificaion of risk dynamics and marke prices of differen sources of risk, we can decompose he reurn risk premium ino wo componens: he reurn risk level as capured by he variance rae V and he risk premium per uni risk. We can furher aribue he risk premium per uni risk o conribuions from diffusion risks and conribuions from jump risks. Theorem 1. Under he price dynamics in (1) (3) and he sae-price deflaor dynamics in (5), he insananeous reurn risk premium is rp = V, where V measures he ime- reurn risk level, and measures he risk premium per uni risk. The per-uni risk premium can be decomposed furher ino conribuions from diffusion risks ( d ) and jump risks ( J ): wih = d + J (6) d = B Z (7) J = J J J J 1 (8) Condiional on a fixed reurn risk level, he diffusion risk premium is given by he negaive of he covariance beween he diffusion innovaions in he reurn and he sae-price deflaor, d 1 V d Ɛ V dw B V db + Z V dz = B Z (9) which capures he conribuion from boh he reurn risk B and he variance risk Z. The jump risk premium is obained by performing he inegraion (e.g., Küchler and Sørensen 1997) J = e x 1 1 e J +x + x dx 0+ + e x 1 1 e J x x dx (10) 0 he magniude of which is deermined by he marke prices of upside and downside jumps J + and J, respecively. The marke price of he diffusion reurn risk, B, generaes a reurn risk premium as in Meron (1976). In addiion, he marke price of diffusion variance risk Z also conribues o he diffusion reurn risk premium d hrough is correlaion wih he reurn innovaion. Furhermore, he marke prices of upside and downside jump risks ( J + and J ) induce addiional risk premiums in he form of J in (8). Finally, sochasic volailiy renders he risk premium sochasic as he risk premium is proporional o he variance rae V.

4 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period 2254 Managemen Science 56(12), pp , 2010 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. Equaion (7) suggess ha a posiive marke price of reurn risk ( B ) generaes a posiive reurn risk premium. Furhermore, when he reurn and he variance rae has a negaive correlaion ( <0), a posiive reurn risk premium can also resul from a negaive marke price of variance risk ( Z < 0). Posiive marke prices for jump risks generae a posiive reurn risk premium, as refleced in Equaion (8). Unlike he radiional focus on he behavior of oal reurn risk premium in some early sudies of bubbles and burss, he risk premium decomposiion in Theorem 1 is imperaive o our analysis. The firs quesion o ask is wheher a risk premium change is due o a change in reurn risk level (V ) or a change in is marke price (he risk premium per uni reurn risk, ). Our explici disincion and separae esimaion of he wo componens are imporan for undersanding how he reurn risk level and he marke price per uni risk had varied separaely during he bubble period. Second, by classifying he differen risk sources and assigning a differen marke price o each risk source, we can go one sep furher in documening he differen variaions of he marke prices on differen risk sources over he Nasdaq bubble period. In his sense, our goals are aligned wih hose of Bollerslev and Todorov (2009) and Sana-Clara and Yan (2010), who sudy he ime variaion in risk premiums, and wih hose of Figlewski (2009), who analyzes he variaion of he risk-neural densiy of he S&P 500 index reurn over he 2008 crisis period The Informaion Conen in Opion Prices Given he sae-price deflaor specificaion in (5), he price dynamics under he risk-neural measure can be wrien as (Küchler and Sørensen 1997) ds /S = r q d+ V dw + e x 1 + dx d V 0+ + x dx d + e x 1 dx d V 0 x dx d (11) where + x and x are exponenial iled versions of heir saisical counerpars, { e + + J + x + x = x>0 0 x<0 x = { 0 e J x x>0 x<0 (12) 2 See also, among ohers, Bliss and Panigirzoglou (2004), Bollen and Whaley (2004), Huang and Wu (2004), Jones (2006), Bakshi e al. (2008), Li e al. (2008), and Chrisoffersen e al. (2008). from which we can define he risk-neural dampening coefficiens as J + and J (13) Posiive marke prices on he wo jump risks increase he dampening on upside jumps bu reduce he dampening on downside jumps. As a resul, he reurn innovaion disribuion becomes more negaively skewed under he risk-neural measure. The risk-neural dynamics of he insananeous variance rae in our seing become dv = V d+ V dz (14) wih + Z and /. A negaive marke price of variance risk reduces he risk-neural meanreversion speed and increases he risk-neural mean of he variance rae. The Fourier ransform of he log reurn under he risk-neural measure is exponenial affine in he variance rae V, u V Ɛ eiu ln S + /S ( + = exp iu r s q s ds a b V ) u (15) where denoes he parameer se, and he coefficiens a and b are funcions of he parameers ha govern he reurn risk dynamics and marke prices of risk, ) (1 ln 1 e 2 a = b = 2 1 e 2 1 e (16) wih + Z iu, , and he risk-neural characerisic exponen of he sandardized log reurn innovaion wih V fixed a uniy: = 1 2 iu + u2 ( + + J + iu J iu + + J J + 1 ) ( J + iu 1 J 1 + iu J J 1 ) (17) Given he Fourier ransform in (15), we can numerically compue he ime- values of opions wih ime o mauriy via fas Fourier inversion, as proposed in Carr and Madan (1999) and Carr and Wu (2004).

5 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period Managemen Science 56(12), pp , 2010 INFORMS 2255 Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. Equaions (15) (17) show ha he opion value depends on he reurn risk level V as well as he parameers ha govern he price and variance rae dynamics and he marke prices of risk. We can use he observed opion prices across differen srikes and mauriies each day o infer he reurn risk level and he marke prices on ha dae The Informaion Conen in Sock Reurns Time-series observaions of reurns provide addiional informaion on he price dynamics, as revealed hrough he likelihood or, equivalenly, he condiional probabiliy densiy of he reurns. To incorporae his informaion in our esimaion, le ln S +h /S denoe he ime- log reurn over a sampling frequency of h. Analogous o (15), he reurn characerisic funcion under measure is where u V Ɛ eiu ln S +h/s = exp iurh a h b h V u (18) a h = 2 2 ) (1 ln 1 e h 2 2 h b 2 1 e h h = 2 1 e h wih iu, , and = iu iu + u2 ( + iu iu ) ( + iu 1 1 (19) + iu ) (20) Condiional on he variance rae level V, we can apply fas Fourier ransform o he reurn characerisic funcion in Equaion (18) o obain he corresponding condiional probabiliy densiy. 3. Daa on he Nasdaq Tracking Sock and Opions We esimae he risk dynamics and marke prices of risk using boh he ime-series reurns and opion prices on he Nasdaq 100 racking sock. The Nasdaq 100 index capures he vas majoriy of he Nasdaq marke capializaion and srongly comoves wih he Nasdaq composie index. For esimaion, we sample he daa daily from March 17, 1999, o February 19, Daily log reurns on he index racking sock are compued based on daily closing dividend-adjused prices. The ime series of excess log reurns is obained by subracing he overnigh London Inerbank Offered Rae (LIBOR) from he daily reurns. Opions on he Nasdaq 100 racking sock are acively raded. We obain he opions daa from he Ivy DB daa se sold by OpionMerics. Because hese opions are American syle, OpionMerics employs a binomial ree approach o adjus for he early exercise feaure in calculaing he implied volailiy from he daily closing midquoe of he opion. Pu call pariy dicaes ha he pu and he call opions a he same srike and mauriy should generae he same implied volailiy. A imes, Opion- Merics generaes wo differen implied volailiies when is assumpions on he ineres rae curve, he dividend schedule, and he borrow cos differ from he marke (see also Lamon and Thaler 2003). To recify his aberraion, we choose, a each mauriy, he srike closes o he spo price and measure he difference beween he OpionMerics implied volailiy esimaes for he call and he pu opions. Then, we add half of he difference o call implied volailiy esimaes and deduc half of he difference o pu implied volailiy esimaes. Through his adjusmen, he implied volailiies corresponding o he pu and he call a he srike closes o he spo price become idenical o each oher. A oher srikes, we ake he adjused implied volailiy of he ou-of-he-money (OTM) opions, i.e., he call opion when he srike is above he spo, and he pu opion when he srike is below he srike. We conver he implied volailiy a each srike and mauriy ino he corresponding OTM European opion value according o he Black and Scholes (1973) pricing formula. The forward price is compued based on he ineres rae curve and he dividend projecion provided by OpionMerics. Similar o exan pracices (e.g., see he deails in Figlewski 2009), we apply hree addiional filers o he daa in consrucing he daily opions sample. Firs, he bid price is greaer han zero. Second, he opion mauriy is no less han en days and no more han one year. Third, he srikes of he opions are confined o wo sandard deviaions of he forward. The procedure generaes 159,218 opion prices over 987 business days. Table 1 displays summary informaion on he ime series of he Nasdaq 100 racking sock and is daily reurns in panel A, and on he opions sample in panel B. Panel A shows ha he average price of he Nasdaq 100 racking sock is $53.32 over our sample period from March 17, 1999, o February 19, The lowes level is $19.55, and he highes level is $ The daily reurn has a sandard deviaion of 3.01% per day, wih he larges single day percenage

6 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period 2256 Managemen Science 56(12), pp , 2010 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. Table 1 Descripive Saisics on Nasdaq 100 Tracking Sock and Is Opions Average SD Min Max A. Nasdaq 100 racking sock Index level ($) Daily reurn (%) B. Opions on he Nasdaq 100 racking sock Number of calls per day Number of pus per day Days o expiraion Call implied volailiy (%) Pu implied volailiy (%) Noes. Enries repor summary saisics on he Nasdaq 100 racking sock and is daily reurns in panel A and he opions sample in panel B over he period from March 17, 1999, o February 19, 2003 (987 business days). Daily log sock reurns on he Nasdaq 100 index are compued based on daily closing dividend-adjused prices. Opions daa are from OpionMerics, which compues he opions implied volailiies based on a binomial ree. The opions sample is seleced according o he following crieria: (i) he bid price is greaer han zero, (ii) he opion mauriy is beween 10 days and one year, (iii) he srikes of he opions are confined o wo sandard deviaions of he forward, and (iv) he ou-of-he-money opion is chosen a each srike and mauriy. The selecion crieria yield a oal of 159,218 observaions, among which 70,185 are calls and 89,033 are pus. reurn drop a 9 06% and he larges single day spike a 15.57%. Panel B shows ha, on average, 71 call opions and 90 pu opions are used per day for he esimaion. Over ime, he number of calls per day varies from zero o 176, and he number of pus per day varies from 20 o 235. We filer he opions o have mauriies of no less han 10 days and no more han 365 days, wih an average mauriy of 129 days. The Black and Scholes (1973) implied volailiies for calls Table 2 Opions Open Ineres, Volume, and Implied Volailiy by Mauriy and Moneyness average a 43.43% and vary from 24.82% o 95.52%. The pu implied volailiies are higher, wih an average of 50.56%, a minimum of 27.67%, and a maximum of %. In Table 2 we divide he opions sample ino six moneyness and four mauriy regions, and repor he average saisics in each of he 24 moneyness/ mauriy regions. Mauriy is measured in number of acual days, and moneyness is in absolue percenage log srike deviaion from forward, ln K/F, where K is he srike price, and F is he forward. Panel A repors he average daily open ineres in each region, and panel B repors he average daily rading volume. Over our sample period, he average open ineres and rading volume on OTM calls are higher han heir pu counerpars. Furhermore, rading volume and open ineres are concenraed in opions wih mauriies wihin six monhs, wih rading aciviies waning hereafer. Panel C of Table 2 repors he average implied volailiy across differen moneyness and mauriy regions. A shor mauriies (10 90 days), he average implied volailiy is U-shaped in moneyness. A longer mauriies, he moneyness relaion becomes downward sloping, wih he average implied volailiies a low srikes higher han he average implied volailiies a high srikes. 4. The Maximum Likelihood Esimaion Our model feaures sochasic volailiy, a separaion of up and down jumps, and sochasically varying risk Nasdaq 100 pus (%) Nasdaq 100 calls (%) ln K/F, % Days o expiraion: > > 10 A. Average daily open ineres (number of conracs) B. Average daily volume (number of conracs) C. Average implied volailiy (%) Noes. Enries repor summary saisics on he Nasdaq 100 opions by mauriy and moneyness over he daily sample period from March 17, 1999, o February 19, 2003 (987 business days). The daa are from OpionMerics. Mauriy is in number of acual days, and moneyness is defined as ln K/F T, where F T is he forward price and K is he srike.

7 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period Managemen Science 56(12), pp , 2010 INFORMS 2257 Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. premia on each ype of risk. The aim of his secion is o presen a procedure o esimae he risk dynamics V joinly wih he marke prices of risk, based on he daily ime series of reurns and opion prices on he Nasdaq 100 racking sock. We esimae he model parameers by maximizing he join likelihood of he opion prices and daily reurns. To consruc he likelihood funcion, we cas he model ino a sae-space form and exrac he disribuions of he saes a each dae using a filering echnique The Propagaion Equaions Our base model assumes ha all he marke prices of risk coefficiens Z B J + J are consan. In his case, we regard he insananeous variance rae as he sae X = V. The sae propagaion equaion is deermined by he saisical dynamics of he variance rae in Equaion (2). To analyze he reurn risk and risk premium variaion around he rise and fall of he Nasdaq, we exend he base model o accommodae ime-varying marke prices for differen sources of risk. In his exension, we use z Z ln + ln o denoe he addiional sae variables ha we allow o vary over ime, and we use X = V z as he expanded sae vecor. We reain he assumpion of a consan marke price of reurn risk B, because is idenificaion comes mainly from he daily reurns, and opions provide lile informaion on inferring his coefficien. Furhermore, wih ( + ) fixed, we use he variaion of + o reflec he variaion in he marke prices of upside and downside jumps ( J + J, wih J + = + + and J =. The log ransformaion of + and expands he domain of he wo coefficiens o he whole real line. We specify he following auxiliary dynamics for z = Z ln + ln : z = 1 z z + z z 1 + z (21) where z = Z ln + ln denoes he long-run mean vecor. We also use wo auxiliary coefficiens, z and z, o capure he mean reversion speed and variance of z, respecively, wih denoing a sandardized normal vecor. To avoid he complicaion of convexiy erms for opion pricing, we ake he marke prices as deerminisically ime varying and rea Equaion (21) as a condiional forecasing equaion analogous o he generalized auoregressive condiional heeroskedasiciy (GARCH) specificaion for volailiies (Engle 1982, Bollerslev 1986). Hence, he filered updaes on z can be regarded as ime- forecass of fuure marke prices of risk. For boh consan and ime-varying marke prices of risk specificaions, we can wrie he sae propagaion equaion generically as X = A + X (22) where is an independen and idenically disribued (i.i.d.) sandard normal innovaion vecor, and 1 denoes he condiional covariance marix of he innovaion. When X = V, Equaion (22) represens a discreized version of he coninuous-ime dynamics of V in (2), wih A = 1 e h, = e h, 1 = 2 V 1 h, and h = 1/252 denoing he daily ime inerval used in our esimaion. When X = V z, he sae propagaion equaion is expanded accordingly, wih A = 1 e h 1 z z, indicaing a diagonal marix wih he diagonal elemens given by e h z e 3, e 3 denoing a hree-dimensional vecor of ones, and 1 also indicaing a diagonal marix wih he diagonal elemens given by 2 V 1 h 2 z e The Measuremen Equaions The measuremen equaions are consruced based on he observed OTM opion prices, assuming addiive, normally disribued measuremen errors. Le y denoe hese observed opion prices a ime, and le X denoe heir corresponding model values as a funcion of he sae level X and he model parameers. We can wrie he measuremen equaion as y = X + e (23) wih e denoing he measuremen error. Because he exchange-lised opion conracs have fixed srikes and fixed expiraions, he moneyness and ime o mauriy of each conrac varies daily as ime passes by and as he spo level changes. Furhermore, wih old conracs expiring and new conracs being issued, he number of opion conracs, and hence he dimension of he measuremen equaion, varies over ime. These variaions make i difficul o esimae a generic covariance marix for he measuremen errors e in Equaion (23). To make he esimaion seps racable in our conex, we firs represen each OTM opion price as a percenage of he underlying spo level, and hen scale his value by he vega of he opion as in Bakshi e al. (2008). Wih his scaling, we assume ha he scaled measuremen errors on all opion conracs have he same error variance. As opion values vary wih moneyness and mauriy, so do he measuremen errors. The raionale for vega scaling is o make opion values more comparable across moneyness and mauriy, hereby faciliaing he simplifying assumpion of idenical error variance Adjusing for Cross-Dependence and Serial Dependence in Measuremen Errors Measuremen errors on differen opion conracs are likely o be correlaed. We capure his correlaion

8 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period 2258 Managemen Science 56(12), pp , 2010 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. of he measuremen errors beween conracs i and j hrough he following funcional form: ( ij = max 0 1 D ) i D j g d ( max 0 1 ) i j (24) g m where D denoes he Black and Scholes (1973) dela of he opion, and denoes he opion mauriy in years. Inuiively, as he disance beween he wo conracs in erms of moneyness and mauriy increases, he correlaion declines. The wo posiive scaling parameers g d and g m conrol he speed of decline along he wo dimensions, respecively. We rea he wo scaling coefficiens as free parameers and deermine heir value hrough he maximum likelihood esimaion. Furhermore, a posiive measuremen error oday is likely o be followed by a posiive measuremen error omorrow on he same conrac. To recify he impac of serial dependence on our esimaion, we assume he following error srucure for each conrac i: e i = ee i 1 + e i (25) where we posi he same serial correlaion e and error variance e 2 on all opion conracs, wih i denoing an i.i.d sandard normal variable. The wo coefficiens e and e are esimaed wihin our maximum likelihood procedure. Wih hese assumpions, we can build he covariance marix of he measuremen errors R 1 a each dae The Filering Procedure Le X, X, ȳ, y denoe he ime- 1 forecass of ime- values of he sae, he covariance marix of he sae, he measuremen series, and he covariance marix of he measuremen series, respecively, and le X and X denoe he ex pos updaes on he sae vecor and he sae covariance based on observaions y a ime. Given he Gaussian linear srucure of he saepropagaion equaion in (22), we generae he sae predicions analyically as X = A + X 1 X = X (26) In he case of linear measuremen equaions of he form y = HX + e, he prediced values of he measuremens and heir covariances can also be compued analyically: ȳ = H X y = H X H + R 1 Xy = (27) X H The filering of he saes can be performed according o he classic Kalman (1960) filer, X = X + K y ȳ X = X K y K (28) where K is he Kalman gain, given by K = Xy y 1. In our applicaion, he measuremen equaion in (23) is nonlinear. We use he unscened Kalman filer (Wan and van der Merwe 2001) o handle his nonlineariy. Under his approach, we firs approximae he disribuion of he sae vecor using a se of deerminisically chosen sigma poins and hen propagae hese sigma poins hrough he nonlinear measuremen equaion. Specifically, le p be he number of saes, >0bea conrol parameer, and i be he ih column of a marix. A se of 2p + 1 sigma vecors i are generaed based on he condiional mean and covariance forecass on he sae vecor according o he equaions 0 = X i = X ± p + X j (29) j = 1 p i= 1 2p wih corresponding weighs w i given by w 0 = / p + w i = 1/ 2 p + j = 1 2p (30) We can regard hese sigma vecors as forming a discree disribuion, aking w i he corresponding probabiliies. Given he sigma poins, we can compue he prediced values of he measuremens and heir covariances as y = 2p i=0 ȳ = 2p i=0 w i i w i i ȳ i ȳ +R 1 Xy = 2p i=0 w i i X i ȳ (31) Wih he condiional mean and covariance compued from he sigma poins, we can apply he Kalman (1960) filer in Equaion (28) o updae he saes and heir covariances The Join Log-Likelihood from Opions and Index Reurns The unscened Kalman filer is applied o he opions daa sequenially from he firs day o he las day of he observaion. Wih he filering resuls, we consruc he log-likelihood for each day s opion observaions by assuming ha he forecasing errors are normally disribued: l O = 1 2 log y 1 2 y ȳ y 1 y ȳ (32)

9 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period Managemen Science 56(12), pp , 2010 INFORMS 2259 Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. Furhermore, condiional on he sae vecor filered from he opions, we compue he condiional probabiliy densiy of he log reurn, f x V = 1 0 e iux u V du, hrough he fas Fourier inversion of he characerisic funcion ( u V )in Equaion (18). We firs compue he probabiliy densiy numerically on a fine grid of reurn values x, and hen map he grid o he realized reurn on ha day, ln S +1 /S, o obain he condiional probabiliy densiy of he realized reurn, f ln S +1 /S V. Then, we compue he condiional log-likelihood of he daily reurn as l s = ln f ln S +1/S V (33) Finally, assuming condiional independence beween he opions forecasing errors and he daily reurns, we consruc he join daily log-likelihood as he sum of he log-likelihoods from he wo daa sources in (32) and (33), respecively. We choose model parameers o maximize he join log-likelihoods from boh opions and daily reurns, max T l O + ls (34) =1 where T = 987 denoes he number of days in our sample. Table 3 Model Parameer Esimaes Under Time-Varying Marke Prices of Risk A. Variance risk dynamics Coef saisic B. Jump risk dynamics Coef saisic C. Marke prices of diffusion and jump risks Z ln + ln B z 2 z Coef saisic D. Cross-secional and serial dependence in measuremen errors g d g m e 2 e Coef saisic Risk and Marke Price Variaions During he Bubble Period Through model esimaion, we exrac he variaion of he reurn variance rae (V ), as a measure of he reurn risk level, and he marke prices of differen sources of risk on he Nasdaq 100 racking sock. We documen how hey vary during he rise and fall of he Nasdaq marke. We have esimaed models wih boh consan and ime-varying marke prices of risk. The wo models generae similar parameer esimaes and reurn variances, confirming he robusness of our specificaions and esimaion mehodology. For ease of exposiion, we focus on he model wih ime-varying marke prices of risk, wih parameer esimaes repored in Table 3. We also provide he parameer esimaes under consan marke prices in he appendix. The parameer esimaes in panel D in boh ables show he relevance of accouning for cross-correlaions and serial correlaions among he measuremen errors on he opions Reurn Volailiy Increased wih Nasdaq 100 Valuaion Panel A of Table 3 presens he esimaes of parameers governing he variance rae dynamics in (2), wih he corresponding -saisics in parenhesis. The Noes. Enries repor he maximum likelihood esimaes of he model parameers wih ime-varying marke prices of risk. The -saisics are shown in parenheses. The esimaion is based on daily ime-series reurns and opion prices on Nasdaq 100 racking sock over he sample period from March 17, 1999, o February 19, Likelihoods on opions are consruced by assuming normally disribued opions forecasing errors, wih he mean and covariance of he forecass obained from he unscened Kalman filer. Likelihoods on daily reurns are compued by invering he characerisic funcion of he daily reurn, condiional on he variance rae exraced from he opions. The join likelihood is obained by assuming independence beween opions forecasing errors and daily sock reurns.

10 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period 2260 Managemen Science 56(12), pp , 2010 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. parameers of he variance dynamics are esimaed wih a high degree of saisical significance, suggesing ha he esimaion procedure is successful in idenifying he evoluion of reurn volailiy. The esimae for he variance of variance coefficien is large a , suggesing ha he reurn variance iself is highly volaile. The esimae for he speed of mean-reversion coefficien is also large a , implying quick mean reversion in he Nasdaq volailiy variaion. The long-run mean ( ) esimae of represens an average volailiy level of 34.68%, compued as Las, he diffusion componen of he reurn innovaion and he variance innovaion have a srongly negaive correlaion a = , which conribues o he implied volailiy skew observed a moderae o long mauriies (Table 2). To assess how reurn volailiy varied around he rise and fall of he Nasdaq, Figure 1 plos he ime series of he reurn volailiy V, represened in volailiy percenage poins, and conrass i wih he ime series of he Nasdaq 100 racking sock from March 1999 o April The solid line represens he volailiy ime series, wih unis displayed on he lef side of he y-axis, and he dashed line represens he sock price ime series, wih unis displayed on he righ side of he y-axis. During he wo-year period, he Nasdaq 100 racking sock rose from $50 in March 1999 o $115 in March 2000 and hen fell o $33 in April During his process, he volailiy sared a a relaively low level in March 1999, a abou 33%, bu seadily increased as he Nasdaq 100 index climbed, and reached 49% a he peak of he Nasdaq 100 valuaion in March Figure 1 reveals a posiive comovemen beween he level of he Nasdaq 100 and is reurn volailiy prior o Figure 1 Nasdaq 100 volailiy (%) Reurn Volailiy V vs. Nasdaq Valuaion Volailiy Price Noes. The solid line plos he daily ime series of he reurn variance rae V, represened in volailiy percenage poins ( V 100) and wih unis on he lef-hand side of he y -axis. The dashed line plos he ime series of he daily closing price of he Nasdaq 100 racking sock, wih unis on he righ-hand side of he y -axis Nasdaq 100 price ($) he collapse of he Nasdaq marke. Under normal marke condiions, equiy volailiy ends o decline wih rising equiy values. The volailiy coninued rising as he Nasdaq 100 sared o fall, and peaked a 78% on April 14, The volailiy remained high afer he collapse of he Nasdaq in March Reurn volailiy subsided for a few monhs when he Nasdaq marke sabilized, bu rended upward hereafer as he Nasdaq index coninued is downfall. The rising Nasdaq volailiy during he bubble period has also been observed by Schwer (2002) in a differen conex. Several heoreical models explain how volailiy can increase as a bubble builds. For example, Scheinkman and Xiong (2003) propose an equilibrium model where overconfidence generaes disagreemens among agens regarding asse fundamenals. These disagreemens, combined wih shor-sale consrains, can push up boh he asse price and is volailiy Marke Price of Diffusion Risk Declined wih Rising Nasdaq Valuaion Panel C of Table 3 repors he coefficien esimaes relaed o he marke prices and heir variaions. In accordance wih Equaion (7) of Theorem 1, he marke price of diffusion risk can be compued as d = B Z. The esimae for he marke price of diffusion reurn risk B is posiive a , and he esimae for he long-run mean of he diffusion variance risk Z is Given he negaive correlaion beween reurn and variance, boh esimaes conribue o an average posiive diffusion reurn risk premium d of 1.82 per uni risk. This average esimae is in line wih he common findings of he lieraure (e.g., Bliss and Panigirzoglou 2004, Bakshi e al. 2008). By allowing Z o vary over ime, he marke price of diffusion reurn risk d varies accordingly. Figure 2 conrass he variaion of d wih he Nasdaq 100 valuaion during he wo-year period from March 1999 o April The marke price of diffusion risk sared close o is average value a 1.88, and began o decline as he Nasdaq 100 price wen up. From Sepember 21, 1999, o January 5, 2000, he marke price of diffusion risk reached negaive erriory. This period of negaive marke price coincided wih he rising Nasdaq 100 valuaion. The marke price of diffusion risk sared o increase hereafer. Following he collapse of he Nasdaq marke, he marke price reached hisorically high values in May 2000, which were abou wice as high as he sample average, and sayed high for a period of four monhs. The marke price of diffusion risk reflecs eiher or boh of (i) invesors aversion o diffusion risk and

11 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period Managemen Science 56(12), pp , 2010 INFORMS 2261 Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. Figure 2 Marke price of diffusion risk ( d ) Marke Price of Diffusion Risk d Variaion vs. Nasdaq Valuaion d Price Noes. The solid line plos he daily ime series of he marke price of diffusion risk d, wih unis on he lef-hand side of he y -axis. The dashed line plos he ime series of he daily closing price of he Nasdaq 100 racking sock, wih unis on he righ-hand side of he y -axis. The marke price of diffusion risk is compued as d = B Z, where ( B ) are model parameers, and Z is allowed o vary over ime. (ii) deviaions of invesors subjecive beliefs from saisical realiies abou fuure cash flows. The negaive marke price esimaes before 2000 highligh a combinaion of he marke s appeie for risk and an exaggeraed view of fuure cash flows ha coincided wih he rending Nasdaq valuaion. On he oher hand, he highly posiive marke price esimaes afer he Nasdaq collapse reflec invesors renewed aversion o diffusive risk and a realigned projecion abou fuure cash flow Marke Price of Jump Risk Rose wih Nasdaq 100 Valuaion Combining Equaions (8) and (13), we can express he marke price of jump risk as J = Nasdaq 100 price ($) (35) Table 3 repors he parameer esimaes on he jump characerisics ( + ) in panel B and he average of he risk-neural counerpars ln + ln + in panel C. These esimaes imply an average jump risk premium of per uni risk, considerably smaller han he average esimae for he diffusion risk. Based on he exraced ime series on (ln + ln ), we compue he ime series on he marke price of jump risk J via (35). Figure 3 conrass is variaion wih he rise and fall of he Nasdaq 100 valuaion from March 1999 o April Given he daa noise on deep OTM opions, he daily esimaes for he marke price of jump risk appear o be noisy. To obain a clearer Figure 3 Marke price of jump risk ( J ) Marke Price of Jump Risk J Variaion vs. Nasdaq Valuaion Noes. The solid line plos he daily ime series of he smoohed marke price of jump risk J, wih unis on he lef-hand side of he y -axis. The dashed line plos he ime series of he daily closing price of he Nasdaq 100 racking sock, wih unis on he righ-hand side of he y -axis. The marke price of jump risk is compued as J J Price = where ( + ) are consan model parameers, and ( + + ) are allowed o vary over ime. paern for he ime variaion, we apply an exponenial smoohing on he daily esimaes, J = b J b J, where we se he smoohing coefficien o b = 0 97, corresponding o a half-life of abou a monh. The solid line in Figure 3 represens his smoohed version of he marke price of jump risk. The marke price of jump risk rose sharply from lae 1999 o early 2000 wih he Nasdaq marke, reaching is highes esimae of 0.50 in March Then, righ before he collapse of he Nasdaq, he marke price of he jump risk fell sharply and sayed a a relaively low level during an exended period from middle o lae Risk-Neural Tail Asymmery and Nasdaq Valuaions Wih he jump srucure under he saisical measure ( + ) held invarian, he variaions of he marke price of jump risk J are driven by he relaive variaions of he risk-neural jump dampening coefficiens +. A higher esimae for + han for implies ha he lef ail of he risk-neural reurn innovaion disribuion is heavier han he righ ail. Therefore, we can measure he risk-neural ail asymmery by he difference in he dampening coefficiens, NSKEW +, which is anoher way o assess how expensive OTM pu opions are relaive o OTM call opions of comparable moneyness. A higher difference ranslaes ino a more negaively skewed risk-neural reurn innovaion disribuion Nasdaq 100 price ($)

12 Bakshi and Wu: The Behavior of Risk and Marke Prices of Risk Over he Nasdaq Bubble Period 2262 Managemen Science 56(12), pp , 2010 INFORMS Copyrigh: INFORMS holds copyrigh o his Aricles in Advance version, which is made available o subscribers. The file may no be posed on any oher websie, including he auhor s sie. Please send any quesions regarding his policy o permissions@informs.org. Figure 4 Tail asymmery ( Q + Q ) Variaion of Risk-Neural Tail Asymmery vs. Nasdaq Valuaion Q + Q Price Noes. The solid line plos he daily ime series of he smoohed risk-neural ail asymmery measure, NSKEW = +, wih unis on he lef-hand side of he y -axis. The dashed line plos he ime series of he daily closing price of he Nasdaq 100 racking sock, wih unis on he righ-hand side of he y -axis. and + are he dampening coefficiens under he riskneural measure of upside and downside jumps, respecively. A posiive esimae for NSKEW indicaes a negaive skewness for he risk-neural reurn innovaion disribuion. Figure 4 plos he ime series of NSKEW esimaes and conrass is variaion wih ha of he Nasdaq 100 racking sock. As wih he marke price of jump risk, we smooh he daily NSKEW esimaes o remove he emporal noise. The difference NSKEW has sayed posiive during he whole sample period, indicaing ha OTM pus have been more expensive han he corresponding OTM calls, and ha he lef ail of he reurn innovaion has been heavier han he righ ail. Wha may be more informaive, however, is he observaion ha his price asymmery increased as rapidly as he price rise of he Nasdaq 100 racking sock. OTM pu opions became increasingly expensive as he underlying sock price increased. Because OTM pu opions are naural insrumens for hedging agains marke crashes, i is possible ha as he Nasdaq 100 rose, invesors became increasingly worried abou is susainabiliy and sared o buy OTM pu opions o hedge agains a poenial collapse. The buying pressure raised he OTM pu prices more han i did he OTM call prices, hus generaing a more negaively skewed risk-neural reurn disribuion Supporing Evidence from Open Ineres on Pus and Calls To gauge wheher he observed risk-neural ail asymmery is suppored by he demand imbalance for differen ypes of opions conracs, we rerieve he daily aggregae open ineres daa from OpionMerics for calls and pus on he Nasdaq 100 racking sock. Figure 5 plos he ime series of he open ineres on calls (OPEN call ) and pus (OPEN pu Nasdaq 100 price ($) ), where he wo series Figure 5 Log call open ineres Pu and Call Open Ineres Variaion Call open ineres Pu open ineres Noes. The solid line plos he daily ime series of call open ineres (OPEN call, in logarihms), wih unis on he lef-hand side of he y -axis. The dashed line plos he daily ime series of he pu open ineres (OPEN pu,in logarihms), wih unis on he righ-hand side of he y -axis. are expressed in naural logs of he aggregae number of conracs. The plo shows a seady increase of daily open ineres for boh calls and pus. From he sar of he sample in March 1999 o mid 2000, he open ineres for pus are higher han ha for calls. Afer ha, we observe a reversal in he demand, wih more open ineres for calls han for pus. Over he whole sample period, he average open ineres for calls are higher han for pus. The same average imbalance paern holds for he seleced opions sample used for model esimaion (Table 2). To highligh he ime variaion of he imbalance beween he call and pu open ineress, we define a pu call open ineres imbalance measure as IMBAL OPENpu OPEN pu OPEN call + OPEN call Log pu open ineres (36) Figure 6 plos he ime series of he open ineres imbalance measure and conrass i wih he rise and fall of he Nasdaq 100 racking sock. The imbalance sared negaive, bu quickly urned posiive and reached exreme posiive levels before he collapse of he marke. The imbalance became as high as 74.73% a he peak of he Nasdaq 100 valuaion on March 10, On ha dae, call opions had an open ineres of 54,150 conracs, whereas pu opions had an open ineres of 374,452 conracs, close o seven imes as much as he call open ineres. The open ineres imbalance declined rapidly afer he peak of he Nasdaq. I reduced o abou 30% a monh afer he peak, urning negaive by Ocober of 2000 and saying mosly negaive hereafer. The variaions of he open ineres imbalance are largely consisen wih he variaions of he riskneural ail asymmery and suppor our conjecure on he hedging moive of invesors. While here is a