Estimation of security excess returns from derivative prices and testing for risk-neutral pricing

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1 Uiversity of Widsor Scholarship at UWidsor Odette School of Busiess Publicatios Odette School of Busiess 2001 Estimatio of security excess returs from derivative prices ad testig for risk-eutral pricig Gurupdesh S. Padher, Uiversity of Widsor Follow this ad additioal works at: Part of the Busiess Commos Recommeded Citatio Padher,, Gurupdesh S ). Estimatio of security excess returs from derivative prices ad testig for risk-eutral pricig. Ecoometric Theory, 17 4), This Article is brought to you for free ad ope access by the Odette School of Busiess at Scholarship at UWidsor. It has bee accepted for iclusio i Odette School of Busiess Publicatios by a authorized admiistrator of Scholarship at UWidsor. For more iformatio, please cotact scholarship@uwidsor.ca.

2 Ecoometric Theory, 17, 2001, Prited i the Uited States of America+ ESTIMATION OF EXCESS RETURNS FROM DERIVATIVE PRICES AND TESTING FOR RISK NEUTRAL PRICING GURUPDESH S. PANDHER DePaul Uiversity This paper develops a ecoometric framework for ~i! estimatig excess returs of the security process from high frequecy derivative prices, ~ii! testig for risk eutral pricig, ad ~iii! measurig premiums outside the o-arbitrage pricig model+ The estimator is costructed by applyig quasi-likelihood ad Feyma Kac theory to the risk eutral cotiget claims pricig model to geerate the optimal orthogoality restrictio+ The strog cosistecy ad asymptotic ormality of the estimator are established i the cotext of a ostatioary uderlyig state process+ These results further imply that the estimator is robust to distributioal assumptios o the uderlyig asset process+ The proposed approach is applicable to ay arbitrary derivative security, does ot require estimatio of the risk eutral probability measure, ad has applicatio to spot rate bod pricig models+ A cotrolled diagostic study based o geeratig the S&P500 idex ad calls verifies the ability of the estimators to correctly estimate security excess returs ad test for risk eutral pricig+ The estimator is ivariat to call strikes, ad larger samples costructed by cyclig over shorter maturity optios ca be used to reduce its variace+ 1. INTRODUCTION The risk eutral valuatio model for pricig derivative securities is based o the priciple of fidig a uique equivalet risk eutral probability measure that reders the uderlyig discouted asset process ~e+g+, stock, bod, idex! a martigale ad valuig cotiget claims as expectatios+ This paper uses quasilikelihood estimatio ad risk eutral martigale theory to develop a ecoometric framework for ~i! estimatig excess returs of the uderlyig security This paper is based o my dissertatio at Corell Uiversity, ad I thak the chairma of my graduate committee, Professor Robert Jarrow+ I also thak Fred Arditti, Re-Raw Che, Rick Durret, Thomas Epps, Ali Fatemi, Robert Grauer, Yogmiao Hog, Dogcheol Kim, Robert Lid, Carl Luft, ad Lore Switzer for useful commets ad discussios+ Special thaks also to semiar participats at Cocordia Uiversity ~Motreal!, Corell Uiversity, DePaul Uiversity, Simo Fraser Uiversity, Uiversity of Virgiia, Rutgers Uiversity, the 1999 Europea Fiace Associatio Meetig ~Paris!, ad the 1999 Souther Fiace Associatio Meetigs ~Key West, Florida!+ Address correspodece to: Gurupdesh S+ Padher, Departmet of Fiace, DePaul Uiversity, 1 East Jackso Blvd+, Chicago, IL 60604, USA; gpadher@mozart+fi+depaul+edu Cambridge Uiversity Press $

3 786 GURUPDESH S. PANDHER from high frequecy derivative prices, ~ii! testig for risk eutral pricig, ad ~iii! measurig premiums outside the o-arbitrage pricig model+ The strog cosistecy ad asymptotic ormality of the estimator are established i the cotext of a ostatioary uderlyig state process+ The asymptotic properties further imply that the proposed estimator is robust ad estimatio holds whe distributioal assumptios o the uderlyig asset process assumed i the risk eutral model ~e+g+, Browia motio! are relaxed+ A diagostic study is udertake to resolve sample desig issues such as impact of the strike level, strike replicatio, ad shorter maturity cycles o estimatio of excess returs+ The estimatio framework exploits the relatioship betwee a arbitrary claim s partial differetial equatio ad probabilistic represetatios ~Feyma Kac theory! ad uses cotiuous risk eutral pricig ad quasi-likelihood theory to idetify the optimal orthogoality coditio for estimatig excess returs from derivative prices sampled at discrete itervals+ This estimate ca be compared with excess returs estimated directly from the uderlyig asset price process+ Sigificat departures from equivalece imply the existece of additioal premiums i derivative prices outside the o-arbitrage pricig model+ Beyod the risk eutral applicatio of the paper, the proposed estimatio framework also has iterestig empirical derivative pricig applicatios that are beig explored+ Market prices of risk ca be readily costructed from derivative excess returs ad volatility+ Cotiget claims ca the be empirically priced with the risk eutral desity derived from Girsaov s chage of measure formula+ Padher ~2000! exteds the framework to estimate the volatility of the security process from high frequecy derivatives prices+ Quasi-likelihood estimators for excess returs ~ad their variace! are obtaied for both the derivative price process ad the uderlyig asset price process ~e+g+, stock, bod, or idex!+ The results o strog cosistecy ad asymptotic ormality of the estimator are distributio free ad derived uder a milder coditioal secod momet assumptio+ This is satisfied by a large class of stochastic processes with fiite coditioal secod momets ~or fiite variatio!+ Therefore, the proposed estimator is robust to the distributioal assumptios of risk eutral martigale theory where the stochastics are drive by Browia motio+ However, whe the uderlyig state process is close to beig a Browia motio, the quasi-likelihood estimator offers optimal ad efficiet estimatio+ Moreover, the fial feasible estimator developed is a discretized versio of the estimator implied by the cotiuous risk eutral pricig framework+ The covergece results show that i additio to possessig the robustess property, the feasible estimator offers cosistet estimatio ad is asymptotically ormal+ There are a umber of further implicatios of this ecoometric framework for derivatives+ First, the methodology is very geeral ad applicable to ay arbitrary traded derivative icludig calls, futures, ad swaps+ Secod, the estimatio procedure iherits the optimality properties of the quasi-likelihood or estimatio fuctio ~EF! framework ~Godambe, 1960; Godambe ad Heyde, 1987; Thavaeswara ad Thompso, 1986!, esurig that the statistical equa-

4 ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 787 tios used to estimate the implied market rate of retur are ~i! ubiased ad ~ii! of miimum variace i the class of all liear estimatig equatios+ Third, much of the fiace related stochastic processes literature has focused o estimatio of parameters ~e+g+, drift, volatility! from the state process X ~e+g+, idex, stock, bod! usig maximum likelihood, momet coditios, ad oparametric methods ~Broze, 1997; Dohal, 1987; Hase ad Scheikma, 1995; Flores-Zmirou, 1993; ad others!+ This paper cosiders the quasi-likelihood estimatio of the excess retur parameter from the derivative process V~X! overlyig the state process ~ad also from X!+ Furthermore, much of the quasilikelihood literature deals with estimatio i a purely discrete or cotiuous cotext+ Here, the settig is mixed where the estimatig equatios follow from the cotiuous risk eutral pricig model for cotiget claims but where samplig of market derivative prices ~ad uderlyig asset prices! occurs at discrete, perhaps radom, times+ Ecoometric issues coected to the use of discrete data for cotiuoustime derivative pricig models i other estimatio frameworks ~e+g+, maximum likelihood have bee cosidered more recetly by Cherov ad Ghysels ~1998!, Duffie ad Gly ~1998!, Pederse ~1995!, ad others+ This paper differs from the directio take i this work both i focus of estimatio ad the estimatio methodology+ The existig literature has ot dealt with estimatio of excess returs from derivative prices V~X!+ To costruct this estimator, the proposed methodology first idetifies a coditioal martigale differece equatio ~CMDE! by costructig a Itô expasio of the discouted derivative process betwee two give samplig itervals uder the risk eutral measure, the applies the Feyma Kac result to reduce terms, ad last itroduces the parameter of iterest ~excess returs! by switchig to the empirical measure+ Oce the CMDE is costructed, the optimal orthogoality restrictio o the CMDE is obtaied from quasi-likelihood theory+ A discrete feasible estimator is ext developed from this procedure i which all quatities are measurable with respect to iformatio available at the begiig of each samplig period+ Fourth, the proposed method for testig the risk eutral hypothesis does ot require estimatio of the risk eutral probability measure from observed prices ~Baz ad Miller, 1978; Breeda ad Litzeberger, 1978!+ The estimatio regimes of Logstaff ~1991! ad Ait-Sahalia ad Lo ~1998! for call optios estimate a oparametric risk eutral probability desity ~histogram! from a sequece of calls with the same maturity but differet strikes+ Maximum likelihood estimatio ad testig are cosidered by Lo ~1988!+ The approach of ivertig market prices of optios to estimate parameters of the risk eutral measure uder parametric desity models is pursued by Sherrick, Irwi, ad Forster ~1990!+ Bekaert, Hodrick, ad Marshall ~1997! discuss biases i tests of the expectatios hypothesis of the term structure of iterest rates+ Fifth, the arbitrage bod pricig models of Vasicek ~1977!, Brea ad Schwartz ~1979!, ad Artzer ad Delbae ~1987! require a iversio of the

5 788 GURUPDESH S. PANDHER term structure to remove the market price of risk whe valuig cotiget claims as the iitial step ~the models of Ho ad ad Health, Jarrow, ad take the bod price process ad forward rate process, respectively, as exogeous ad avoid the iversio!+ There are computatioal difficulties i this iversio because bod pricig formulae are highly oliear ad the spot rate ad bod price processes are ot idepedet of the market price of risk+ The ecoometric approach of this paper offers liear estimatio of excess returs, avertig the oliearity problem, ad may offer a advatage i these models over calibratio-based estimatio+ The estimatio methodology ad its empirical properties are tested ad verified usig a extesive Mote-Carlo diagostic study+ The empirical study also eables resolutio of importat sample desig issues+ The S&P500 idex ad call optios defied o it are simulated usig historical tred ad volatility+ Excess returs ad market prices of risk are estimated separately from both the idex ad call optio prices uder various scearios to ivestigate the impact of the strike level, legth of maturity cycle, ad strike replicatio+ Differeces i the estimated excess returs from calls ad the idex quatify extra premiums ot explaied by the risk eutral pricig model+ The results of the diagostic study verify the ability of the ecoometric model ad estimators to estimate the excess returs correctly ad test the hypothesis of risk eutral pricig+ The call data geerated i the empirical study are based o the risk eutral pricig model ~Black Scholes formula for calls!, ad estimates of the market price of risk from both the idex ad its derivative calls are very close for ay give sample size+ Therefore, the empirical study reveals that o premia are foud whe oe should exist+ Estimatio is uaffected by the strike level of the call+ It is also foud that the additio of replicates based o differet strikes does ot improve the stadard errors of the estimator due to depedece amog strike replicates+ I the market settig, the vast majority of traded calls are of maturities less tha 1 year+ Therefore, it is ot feasible to icrease the sample size by extedig the time to maturity+ A alterative samplig desig that cycles over calls of smaller ~ooverlappig! maturities is cosidered as a way to reduce the variace of estimators+ It is foud that samplig from cycles of shorter maturities with the same effective sample size ~umber of cycles times maturity legth! yields similar ad stable estimatio as a sigle maturity sample of larger but equivalet duratio+ This result gives cofidece i the applicability of the estimatio methodology to market derivative prices where larger samples derived from samplig over multiple ~overlappig! maturities ca be used to reduce variace+ The remaider of the paper is orgaized as follows+ Sectio 2 sets out the probability model ad stochastic processes for the arbitrary derivative process ad itroduces the mai features of the quasi-likelihood ~EF! estimatio framework+ The estimator of excess returs from a arbitrary derivative price pro-

6 ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 789 cess ad its variace are derived i Sectio 3+ Sectio 4 establishes the strog cosistecy ad asymptotic ormality of the feasible EF estimator+ The estimatio of excess returs from the uderlyig asset process is cosidered i Sectio 5+ Sectio 6 presets the results from the Mote-Carlo study i which a idex ad calls are simulated usig the historical volatility ad tred of the S&P500 idex to verify the estimatio ad evaluate the impact of sample size, strike level, strike replicatio, ad maturity legth o the estimatio+ Coclusios follow i Sectio PRELIMINARIES: STOCHASTIC PROCESSES FOR DERIVATIVES AND QUASI-LIKELIHOOD ESTIMATION This sectio defies the probability model ad stochastics for a arbitrary derivative process ad itroduces the Feyma Kac result required to develop the estimator of excess returs+ The essetial features of quasi-likelihood estimatio ~or estimatig fuctio theory! are also preseted The Probability Model ad Stochastic Process for Derivative Claims Fix the probability space ~V, F T,~F t! 0 t T,P! where ~F t! 0 t T $F t ; 0 t T % is the filtratio defied o the evet space V satisfyig the usual coditios ~i+e+, filtratio is right cotiuous ad F 0 cotais all ull sets of F T!+ The probability space is assumed large eough to support a R d -valued stochastic processes X $X t, F T ; 0 t T % that is right cotiuous with left limits ~RCLL! whose elemets geerate the s-fields F t s$x s ; 0 s t%+ The process X will represet the state variable of the pricig model ~e+g+, stock, bod, or idex!+ We view X as a diffusio process followig the geeral stochastic differetial equatio P-a+s+: dx t b~t, X t!dt s~t,x t!dw t, 2.1) where b~t, X # R d r R d is the drift vector, s~t,x R d r R d XR d is the dispersio matrix ~of rak d!, ad W t is a d-dimesioal F t - measurable stadard Browia motio with respect to the probability measure P+ Moreover, b~t, X t! ad s~t,x t! are take to satisfy the global Lipschitz ad liear growth coditios ~see Karatzas ad Shreve, 1991, p+ 289!+ This esures that there exists a strog-form solutio to ~2+1! relative to W $W t, F t ; 0 t T % ad the process X is square itegrable #+ Last, defie a~t, X t! s~t,x t!s T ~t,x t! to be the diffusio matrix+ The precedig quatities are defied with respect to the empirical measure P+ Let Q be the uique equivalet risk eutral measure uder which expectatios of the X process discouted at the risk-free spot iterest rate process r $r t ; 0 t T % are Q-martigales where r is the growth process of the moey

7 790 GURUPDESH S. PANDHER market discout factor B~t,T! exp~ * T s t r s ds!+ Risk eutral valuatio theory ~Harriso ad Kreps, 1979; Harriso ad Pliska, 1981! asserts that a attaiable cotiget claim ca the be valued as a discouted expectatio uder the measure Q+ The process of makig the discouted asset a martigale requires the trasformatio dw t dwg t g t ~t,x t!dt, 2.2) where the market price of risk g~t,x R d r R d ad WG t is a Browia motio with respect to the risk eutral measure Q+ The relatioship betwee the equivalet measures P ad Q is readily obtaied from Girsaov s chage of measure formula+ Note that the existece of g t follows from the osigularity of s~t,x t!~see Harriso ad Pliska, 1981!+ Substitutig ~2+2! ito ~2+1! leads to the differetial equatio dx t r t X t dt s~t,x t!dwg t + 2.3) Let f ~t, X t!:@0,t # R d r R d be a fuctio i the class C 2 ~@0,T# R d! defied o the state variable process X with secod order differetial operator E Q ~ f ~t s, X t s! f ~t, X t!6f t! lim ~A t f!~x!, sr0 s where ~A t f!~x![ 1 2 _ d d k 1 d a j,k ~t,x!@]f~x!0]x j ]x k # r t X ~x!0]x j #+ We are ow ready to defie the value process for a arbitrary derivative security+ Let V~X! $V~t, X t!, F t ; 0 t T % be the geeric value process of the derivative claim based o the state variable process X where V~t, X # R d r R is i the class C 2 ~@0,T# R d!+ It is kow from the Feyma Kac theorem that whe X is a diffusio, V~X! has a heat equatio represetatio ad a correspodig probabilistic represetatio as a discouted Q-martigale+ This result is stated here for referece+ Feyma Kac Result ~Karatzas ad Shreve, 1991, p+ 366!+ Let V~t, X t! C 2 ~@0,T# R d! ad B~t,T! be as defied earlier ad cosider the cotiuous fuctios g~t, X # R d r R ad r~t, X # R d r R satisfyig certai bouded coditios+ If V~t, X t! satisfies the heat equatio ]V ]t A t V rv g # R d V~T, x! f ~x!, x R d, 2.4)

8 ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 791 the V~t, X t! admits the uique stochastic represetatio V~t, x! E x Q f ~X T!exp t r~s, X s!ds s 0 T s t g~s, X s!exp t s r~u, X u!du ds6f t 2.5) # R d + A aalytical expressio for the quasi-likelihood estimator of excess returs l b r is possible for ay arbitrary diffusio price process whe the volatility s~t,x t! ad drift b~t, X t! are time-state separable : b~t, X t! b t b~x t! ad s~t,x t! s t f~x t!+ We will retai this assumptio for the ecoometrics of this paper The Quasi-Likelihood Estimatio Framework Estimatig fuctio theory ~a+k+a+ quasi-likelihood! provides a geeral framework for parameter estimatio that icludes maximum likelihood estimatio ~MLE! as a special case whe a exact distributio is specified for the data geeratig process ad icorporates least squares ~LS! estimatio for liear models with o distributioal assumptios+ It borrows the stregths of both approaches while elimiatig their weakesses+ For example, LS estimatio becomes biased whe the variace of the depedet process depeds o parameters appearig i the mea ~see Godambe ad Kale, 1991!+ For a further overview of EF theory, see Heyde ~1989! ad Godambe ad Heyde ~1987!+ Assumig a discrete settig, the geeral approach to idetifyig the optimal estimatig equatio for the parameter u R d is to first form estimatig fuctios H ~h j, F j! of the data Y ad the parameter u from a particular class of fuctios ~e+g+, liear! such that E~h j 6F j! 0, j 1,+++,, with F F j + The optimality criterio of Godambe ~1960! ~or its sufficiet versios! ca the be applied to determie the optimal estimatig equatios H * ~h j *, F j!+ I relatio to geeralized method of momets ~GMM! estimatio ~Hase, 1982!, H * may be viewed as the optimal orthogoality system+ The EF framework, therefore, gives a systematic framework for idetifyig the optimal estimatig fuctio startig with a primitive error or martigale differece restrictio E~h j 6F j! 0, j 1,+++,+ The stress o the estimatig equatio, as opposed to the parameter estimator, of this framework is justified by the followig observatios: ~i! Fischer s iformatio ad the Cramer Rao iequality are both a estimatig equatio property rather tha that of the MLE; ~ii! asymptotic properties of a estimator are almost ivariably obtaied, as i the case of the MLE, via asymptotics of the estimatig equatio; ~iii! estimatig equatios have the property of ivariace

9 792 GURUPDESH S. PANDHER uder oe-to-oe trasformatio of the estimator; ad ~iv! separate estimatig fuctios ca be combied more easily tha the estimators implicitly defied by them+ We will be iterested i fidig the optimal estimatig equatio i the class of liear F j [ F t measurable estimatig equatios such as H H: H a j ~u!h j ~u!, 2.6) where a j ~u! is a predictable F t -measurable process ad E~h j ~u!6f t! 0, j 1,+++,+ The optimal choice of a j ~u! is give by a * j E ]h ' j ]u F t ~Eh j h ' j 6F t! 1, j 1,+++,+ 2.7) which was show by Godambe ~1960! to miimize the ~coditioal! variace of the stadardized estimatig equatio H s E ]H 1 H 2.8) ]u with Var ~ H s! E ]H 1E~HH ]u '! E ]H ) ]u The criterio of miimizig Var~H s! is justified by the dual objective of ~i! miimizig E~HH '! ad ~ii! maximizig the sesitivity of the estimatig fuctio to departures from the true parameter value ~]H0]u!+ 3. ESTIMATION OF EXCESS RETURNS FROM DERIVATIVE PRICES We begi by discussig the structure of the derivative market data to be used i the EF estimatio of excess risk returs from derivative prices+ Let the observed prices for the derivative security be sampled at the poits i the sequece $t 0, t 1,+++,t with t 0 0 idexig the start of the samplig period ad t T represetig the time to maturity+ The, D j t, j 1,+++, is the legth of the period betwee poits i the term structure+ At each samplig poit, a cross sectio of replicate prices may exist idexed by k 1,+++,m ~e+g+, calls of differet strikes K k!+ Further, prices for ooverlappig cycles of maturity times are available give by the sequece $T 1,+++,T g,+++,t p %+ The price data cosist of a sequece of market prices o the derivative claim give by $V k ~,T g![v~,x tj ;K k,t g!,g 1,+++,p, j 1,+++,, k 1,+++,m%+ The exact structure of the price sequece V k ~,T g! will deped o the sample desig ~sigle0multiple maturities, strike replicates, ad legth of maturity cycles!+

10 I the cotext of obtaiig the estimatig fuctio for the excess retur parameter l from derivative prices, we will cosider the class of liear estimatig fuctios give by H H: H p m g 1 k 1 a jkg ~l!h jkg ~l! + 3.1) We described the mai features of the estimatig fuctio theory i Sectio 2+2+ The key remaiig issues are the specificatio of the martigale differece fuctios h jkg ~l! ad the weightig factors a jkg ~l!, g 1,+++,p, j 1,+++,, k 1,+++,m i ~3+1!+ The optimal estimatig fuctio ad implied quasi-likelihood estimator for l ca the be idetified by choosig a * jkh ~l! optimally ad will deped o the sample desig used+ Three cases are cosidered: ~i! sigle strike ad maturity ~T!, ~ii! multiple strike replicates o a sigle maturity, ad ~iii! replicates o multiple ooverlappig maturity cycles+ The key results relatig to the EF estimator of l ad its variace are derived i Propositios 1 8+ The strog asymptotic cosistecy ad ormality of the estimator are established i Sectio 4+ Without loss of geerality, we start by obtaiig the estimatig fuctio for l uder the first case ~ p 1, m 1!+ PROPOSITION 1 ~The Estimatig Fuctio h j ~l! for Excess Returs Strike ad Maturity#+ Let g~t, X # R d r R ad r~t, X # R d r R be a cotiuous fuctio where the value process V~t, X t! C 2 ~@0,T# R d! satisfies the partial differetial equatio ]V ]t A t V rv g # R d V~T, X T! f ~X T!, X T R d + 3.2) The, give the market derivative prices $V~,T! [ V~,X tj ;K,T!, j 1,+++,%, the estimatig fuctios h j ~l!, j 1,+++, i the liear class H H: H are give by ~d 1! a j ~l!h j ~l! 2.6) h j ~l! Y j lze j V~,T!B~t,! V~t,T! g~u,x u!b~t,u!du u t ]V~u,T! X u B~t,u!du, 3.3) ]X l t ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 793 where l b r is the excess risk retur. s

11 794 GURUPDESH S. PANDHER Proof+ See Appedix A+ The fiacial iterpretatio of ~3+3! is as follows+ The first three terms of h j ~l! represet the depedet Y observatio i the regressio sese, whereas the last term represets the correspodig idepedet X variable+ The first two terms give the chage i the discouted value of the cotiget claim observed over the samplig iterval+ The third term adds back the discouted divideds paid out over this period+ I the case of Europea call optios, this term is zero+ The fourth itegral term ivolvig the delta of the derivative claim represets the cumulative discouted value of the uderlyig asset held to replicate the chage i the claim s value over the iterval ~plus divideds!+ Therefore, et chage i the value of the claim mius its hedge replicatio should be approximately zero+ Some of the basic properties of h j ~l! required i determiig the estimator are summarized i Propositio 2+ PROPOSITION 2 ~Properties of the Estimatig Fuctio h j ~l!!+ The first ad secod momets of h j ~l!, j 1,+++, determied i Propositio 1 satisfy i) E~h j ~l!6f t! 0. d ii) E~h 2 j ~l!6f t! u t E ]V~u, T! 2 ]X j ~X u s! 2 F t B u 2 du+ iii) E~h j ~l!h k ~l!6f t! 0, k j. iv) ]h t j~l! j ]V~u,T! X u B u du ~d 1 case!+ ]l u t ]X Proof+ Property ~i! is immediate from the defiitio of h j ~l!, which is a stochastic itegral w+r+t+ Browia motio with probability measure P+ The secod property follows from the isometry property of the squared stochastic itegral ~Karatzas ad Shreve, 1991, p+ 137! which i this case delivers d E~h 2 j ~l!6f t! u t E ]V~u, T! 2 ]X j ~X u s! 2 F t B u 2 d@w j # u, 3.4) where j # u du is the quadratic variatio of the Browia motio W j u + The third result follows from the disjoitess of the stochastic itegrals, ad the fourth property is immediate from ~3+3!+ PROPOSITION 3 ~Optimal Estimatig Equatio, Estimator for l ad Variace: Sigle Strike ad Maturity!+ Let $V~u,T![V~u,X u ;T!,u,,+++,% defie the sequece of derivative prices with maturity T. The the followig hold: i) The optimal estimatig fuctio for l i the liear class of estimatig equatios H H: H a j ~l!h j ~l! 2.6)

12 is give by H * ~l! ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 795 EZE j ~Y j l E EWG j Z j! E ]V~u,T! ]X X u u t F t B~t,u!du u t E ]V~u, T! 2 ~X u s! 2 ]X F t B 2 ~t,u!du t V~ j,t!b~t,! V~t,T! g~u, X u!b~t,u!du u t yieldig ad Zl a * l t Z Var ~ Zl * a! E ]V~u,T! X u B~t ]X,u!du, 3.5) EZE j ~EW! G 1 j Y j EZE j ~EW! G 1 j ZE j 1 EZE j ~EW G! 1 j ZE j, where the coditioig is doe over the stochastic regressors ZE ~ ZE 1,+++, ZE!. * The coditioal fiite sample distributio of Zl a6ze is give by * Zl a6ze ; N l, 1 EZE j ~EW! G 1 j ZE j + t ii) Defie Y j V~,T!B~t,! V~t,T! * j u t g~u, X u!b~t,u!du, Z j t * j t u t ~]V~t,T!0]X!X t B~t,u!du, ad W j * j u t ~]V~t,T!0]X! 2 ~X t s! 2 B 2 ~t,u!du. The, the feasible EF estimator for * Zl a ad its variace V~ Zl * a! implied by the optimal estimatig fuctio are Zl a Z j W j 1 Y j Z j W j 1 Z j 3.6)

13 [ 796 GURUPDESH S. PANDHER ad Z Var ~ Zl a! 1 Z j W j 1 Z j + 3.7) Some remarks are i order before discussig the proof of Propositio 3+ ~i! The direct EF estimator Zl * a is ot computable because it requires iformatio i the # that is ot available betwee samplig poits+ ~ii! Also, ZE j is a radom variable with respect to the iformatio set F t + The feasible EF estimator is developed by replacig ukow quatities Zl * a with their F t -measurable surrogates defied i Propositio 3 to obtai Zl a ~3+6!+ ~iii! The volatility parameter s is assumed costat oly over the samplig period ~e+g+, day! ad does ot ifluece the feasible EF estimator Zl a as it cacels out i the umerator ad deomiator; however, it is required i the computatio of the variace estimator ~3+7! ad is embedded i the weights W j + It ca be estimated cosistetly from the state price process X usig stadard methods ~Campbell, Lo, ad McKilay, 1997, p+ 36!: s 2 1 N where a[ 10N ~l~x k! l~x k 1! ad [ j! 2, The proof of Propositio 3 follows+ ~l~x k! l~x k 1!! ad N D j + Proof+ For the liear class of orthogoal estimatig fuctios defied by H i ~2+6!, the optimal estimatig equatio ~3+5! follows from choosig a j ~l! accordig to ~2+7! ad makig use of the terms defied i Propositio 2: a j * E ]h j ' ]l F t E u t E u t ~Eh j h ' j 6F t! 1 ]V~u,T! X u ]X j F t B~t,u!du ]V~u, T! 2 ]X j ~X u s! 2 6F t B2 ~t,u!du EZE j, EWG j j 1,+++,+ 3.8) Next ote that a * j must be F t -measurable ad the coditioal expectatios i ~3+8! is ot kow iside the samplig iterval+ Therefore, to obtai the feasible estimator, we replace the coditioal expectatios i a * j by their best available F t -measurable surrogate to obtai a[ * j Z j, W j j 1,+++,+

14 Z ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 797 The estimator Zl * a follows from solvig H * ~l! 0, ad V~ Zl * a! follows from applyig the variace operator i two steps, first coditioig o the stochastic regressors ZE ~ ZE 1,+++, ZE!~g 0w+l+g+!: V~Zl * a! E Z * a 6Z!# E V Zl * a 6 Z!# E 1 E V EZE j ~EW G! 1 j EZE j * Coditioal o Z ~ ZE 1,+++, ZE!, the fiite sample ormality of Zl a6ze follows directly from the error-side represetatio of h j give by the stochastic t itegral h j * j u t V X ~u,t!x u sb~t,u!dw u, where from Propositio 2, E~h 2 t j 6F t! * j u t E~V 2 X ~u,t!x 2 u 6F t!s 2 B 2 u du [ EWG j ad E~h j 6F t! 0+ * Usig this i Zl a6ze directly alog with the fact that a stochastic itegral w+r+t+ Browia motio ~with predictable itegrads! is Gaussia yields the ormality of the coditioal EF estimator Zl a6ze + * The feasible estimators Zl a ad V~ Zl a! are fially obtaied by replacig the * * itegrals ad expectatios i Zl a6ze ad V~ Zl a6ze! with their best available F t -measurable surrogates yieldig ~3+6! ad ~3+7!+ It will be show i Sectio 4 ~Propositios 6 9! that the feasible EF estimator is strogly cosistet ad asymptotically ormal+ The ext propositio gives the estimators for excess returs i the case of usig derivative replicates ~e+g+, calls of differet strikes with the same maturity!+ The itroductio of replicates o the same uderlyig asset process itroduces depedece amog the martigale differece fuctios h jk ~l!+ Aside from this complicatio the developmet of the estimator follows Propositio 3, ad the details are omitted for brevity+ The fial result is stated i Propositio 4+ PROPOSITION 4 ~Optimal Estimatig Equatio, Estimator for l ad Variace: Strike Replicates of Sigle Maturity!+ Let $V k ~u,t![v~u,x u ;K k,t!, u,,+++,, k 1,+++,m% defie the sequece of derivative prices with replicates e.g., strikes) k 1,+++,m. Also, let V X k ~u,t! [ ]V~u,X u ;K k,t!0]x ad defie Y jk V k ~,T!B~t,! V k ~t,t! t * j t u t g~u, X u!b~t,u!du, Z jk * j u t V X k ~t,t!x t B~t,u!du, W jk t * j u t ~V X k ~t,t! 2 ~X t s! 2 B 2 t ~t,u!du, Cov~Y jk,y jl! * j u t E~V X k ~u,t! V X l ~u,t!~x u s! 2 6F t!b 2 t ~t,u!du, ad Cov~Y jk,y jl! * j u t V X k ~t,t! V X l ~t,t!~x t s! 2 B 2 ~t,u!du. The, the feasible EF estimator for l ad its variace implied by the optimal estimatig equatio are Zl b m k 1 m k 1 Z jk W jk 1 Y jk Z jk W jk 1 Z jk 3.9)

15 Z 798 GURUPDESH S. PANDHER ad Z Var ~ Zl b! m k 1 m l 1 1 Z jk W jk m k 1 Cov~Y Z jk,y jl!w 1 jk Z jk Z jk W jk 1 Z jk ) Propositio 5, which follows, gives the optimal estimatig fuctio, estimator, ad its variace for excess risk returs if replicates of derivative prices ~e+g+, calls of differet strikes! are used over multiple cycles of ooverlappig maturities+ The proof follows easily from Propositios 3 ad 4 upo otig that the sequece of maturity cycles is ooverlappig+ Hece, summatios over g 1,+++,p are aalogous to summatios over ooverlappig itervals idexed by j 1,+++,, ad so the maturity cycles do ot affect the correlatio structure iduced by the uderlyig Browia motio i the stochastic itegrals+ PROPOSITION 5 ~Optimal Estimatig Equatio, Estimator for l ad Variace: Strike Replicates with Multiple Nooverlappig Maturities!+ Let $V k ~u,t g![v~u,x u ;K k,t g!,u,g 1,+++,p, j 1,+++,, k 1,+++,m% defie the sequece of derivative market prices with ooverlappig sequece of maturities $T 1,+++,T p % ad replicates e.g., strikes) k 1,+++,m. Also, let V X k ~u,t g![]v~u,x u ;K k,t g!0]x ad defie Y jkg V k ~,T g!b~t,! t V k ~t,t g! * j t u t g~u, X u!b~t,u!du, Z jkg * j u t V X k ~t,t g! t X t B~t,u!du, W jkg * j u t ~V X k ~t,t g!! 2 ~X t s! 2 B 2 ~t,u!du, ad Cov~Y jkg,y jlg! * u t V X k ~t,t g!v X l ~t,t g!~x t s! 2 B 2 ~t,u!du. The, the feasible EF estimator for l ad its variace implied by the optimal estimatig fuctio are ad Zl c Z Var ~ Zl c! p m g 1 k 1 p m g 1 k 1 p g 1 m k 1 Z jkg W 1 jkg Y jkg 3.11) Z jkg W 1 jkg Z jkg m l 1 1 Z jkg W jkg p m g 1 k 1 Cov~Y Z jkg,y jlg!w 1 jlg Z jlg Z jkg W jk 1 Z jkg ) 4. ASYMPTOTIC CONSISTENCY AND NORMALITY OF FEASIBLE EF EXCESS RETURNS ESTIMATOR This sectio establishes the strog cosistecy ad asymptotic ormality of the feasible EF estimator of excess returs+ I the diffusio cotext, the radom-

16 ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 799 ess is drive by stochastic itegrals with respect to Browia motio+ Although this leads to a exact fiite sample Gaussia distributio for the exact * coditioal EF estimator Zl a6ze, the asymptotic results are importat to situatios where the uderlyig X process is oly approximately a Browia diffusio ~e+g+, other stochastic processes such as Poisso jumps are mixed with the uderlyig diffusio!+ Moreover, the feasible EF estimator of Sectio 3 is a discretized approximatio to the exact estimator implied by the cotiuous risk eutral pricig framework+ It is importat to establish its cosistecy ad asymptotic ormality+ Results o strog cosistecy ad asymptotic ormality imply that the feasible EF estimators developed i Sectio 3 are robust ad cotiue to hold whe the exact distributio of the uderlyig X process is ot completely kow+ The fiite sample distributioal assumptio of a Browia motio drivig the diffusio process is replaced by a milder coditioal secod momet assumptio+ Without loss of geerality, the results o strog cosistecy ad asymptotic distributio are obtaied i the case of a sigle strike with multiple maturity cycles ~see Propositios 1 ad 3! ad exted easily to the case of strike replicates+ With a fixed time to maturity, the sample size ca oly be icreased by cyclig over ooverlappig maturity cycles+ Therefore, the effective sample size is p, ad the estimators ivolve a double idex over j 1,+++, ad g 1,+++,p+ To keep the otatio simple, the asymptotics that follow will view as the effective sample size without ay loss of geerality+ We keep i mid that the effective sample size becomes large oly whe the maturity cycles p are icreased while the umber of sample poits i each maturity cycle remais fixed Strog Cosistecy of lz The first result gives a boud i L 2 orm ~E~6 Zl 6 2! 102! for the sequece $ Zl % that will be useful to establish cosistecy+ PROPOSITION 6 ~A Boud 1 for the Sequece $ Zl % i L 2 Norm!+ The sequece of EF estimators for l obtaied i Propositio 3 give by Zl Z j W j 1 Y j Z j W j 1 Z j is bouded i L 2 orm by E E~X tj 2 6F t! ) X t Proof+ See Appedix B+ The ext propositio establishes a sufficiet coditio for the strog cosistecy of the EF estimator Zl + Propositio 8 shows this coditio is met, thereby establishig its strog cosistecy+

17 800 GURUPDESH S. PANDHER PROPOSITION 7 ~Sufficiet Coditio for Strog Cosistecy of EF Estimator of l!+ The EF estimator of Propositio 3 is strogly coverget: Zl Z j W j 1 Y j Z j W j 1 Z j r l, a+s+, o the set E~X tj 2 6F t! r ) 2 2 X t Proof+ Defie S Zl l ad E $v:6s 6 e%, e 0+ We also have the followig set relatioships: lim sup r` E flim m $sup 6 e#% $sup 6 e#% because $sup 6 e#% is a decreasig sequece of sets i m+ Therefore, we have 6S 6 e P~lim sup r` 6S 6 e! P sup 1 lim r` P max 1 j 1 lim r` e E~6S 6 2! 2 6S 6 e 1 lim r` e ~le~a2 2! E~B 2!! ~l2 k 1 k 2! 2 e 2 E~X E 2 tj 6F t! X t 2 r 0 4.3) as r ` by the hypothesis of Propositio 7 ~k 1 ad k 2 are positive costats!+ The third iequality of ~4+3! follows from Kolgomorov s iequality ~Hall ad Heyde, 1980!, the fourth iequality is obtaied i the proof of Propositio 6 ~see Appedix B, equatio ~B+7!!, ad the fifth iequality follows from Propositio 7+ This establishes the result Zl r l, a+s+, o the set E~X 2 tj 6F t!0 2 2 X t r 0%+ $ It ow remais to show that the sufficiet coditio for the strog cosistecy of the EF estimator Zl derived i Propositios 6 ad 7 holds for stochastic differetial equatios satisfyig coditios for strog solutios+ This is verified i Propositio 8, thereby establishig the strog cosistecy of the EF estimator+ PROPOSITION 8 ~The Sufficiet Cosistecy Coditio of Propositio 7 Is Satisfied!+ For the diffusio process X defied i 2.1) satisfyig the Lipschitz ad growth coditios E~X 2 tj 6F t! r ) 2 2 X t

18 [ ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 801 Proof+ We assume the state variable X follows the stochastic differetial equatio dx t b~t, X t!dt s~t,x t!dw t, 2.1) where b~t, X # R d r R d is the drift vector, s~t,x R d r R d R d is the dispersio matrix ~of rak d!, ad dw t is a d-dimesioal F t -measurable stadard Browia motio with respect to the probability measure P+ If b~t, X t! ad s~t,x t! satisfy the global Lipschitz ad liear growth coditios ~see Karatzas ad Shreve, 1991, p+ 289!, the a L 2 boud o X give i Duffie ~1992, p+ 292! ca be writte as E~6X tj 6 2 6F t! Ce Ct ~1 6X t 6 2! O~6X t 6 2! 4.5) 2 for some costat P+ Because X t is F t measurable, this leads to E~6X tj 6 2 6F t! 2 6X t O 1 r 0 as r ` Asymptotic Normality of Zl By Propositio 3, the fiite sample distributio of the coditioal EF estimator is Gaussia because the uderlyig state process X is drive by a Browia motio ad the terms i the estimator ivolve stochastic itegrals w+r+t+ this Browia motio+ The asymptotic distributio is relevat whe this distributioal assumptio is relaxed ad replaced by a weaker coditioal secod momet restrictio+ The asymptotic cosistecy ad large sample ormality allow iferece for the feasible EF estimator eve whe the exact distributio of the uderlyig X process is ot completely kow+ The asymptotic ormality of Zl is established i Propositio 9, which follows+ PROPOSITION 9 ~Asymptotic Normality!+ If 6F E~6X tj 6 2 6F t! the Zl r l, p, ad E~6h j 6 2 Z j W j 1 Z 02~ Zl l! N~0,1!+ 4.6) Proof+ Defie the coditioal variace I ~l! [ Z j W 1 j E~h 2 j 6F t!w 1 j Z j ad let H^ ~ Zl![]H ~l *!0]l Z j W 1 j Z j +The the first-order Taylor expasio of the feasible optimal estimatio fuctio H ~ Zl! a * j h j Z j W 1 j h j yields 0 H ~ Zl! H ~l! ]H ~ Zl *! ~ Zl l! ]l H ~l! H^ ~l!~ Zl H^ ~l! H^ ~ Zl *!# ~ Zl l!, 4.7)

19 ^ ^ ^ 802 GURUPDESH S. PANDHER where Zl * g Zl ~1 g!l+ Suppose a is a icreasig sequece a r ` 1 chose such that a H ~l! N~0,1! ad rewrite ~4+7! as a 1 H ~l!~ Zl l! a 1 H ~l! a H^ ~l! H^ ~ Zl *!# ~ Zl * l!, 4.8) where i the case at had H^ ~l! H^ ~ Zl *! 0 by the liearity of the estimatig fuctio+ The left had side coverges i distributio to a stadard ormal variate if a 1 H ~l! N~0,1!+ It remais ow to fid this sequece a ad prove the covergece+ Cosider the ormalized coditioal variace Var ~a 1 H ~l!! a 2 2 I ~l! a Z j W 1 j E~h 2 j 6F t!w 1 j Z j + 4.9) It is easy to check that if E~6h j 6 2 6F E~6X tj 6 2 6F t!, the this coditioal variace is bouded by the F t -measurable term 6X t 6 2 : E~6h j 6 2 6F E~6X tj 6 2 6F t! Ce Ct ~1 6X t 6 2! O~6X t 6 2!, 4.10) where the boud follows from ~4+5! ad derives from the global Lipschitz ad liear growth coditios o the drift ad volatility parameters of the diffusio X+ It is easy to verify that ~4+10! allows the ormalized coditioal variace ~4+9! to be writte as Var ~a 1 H ~l!! a 2 O which implies choosig a ~ Z j W j 1 Z j, 4.11) Z j W 1 j Z j! 102 so that Var~a 1 H ~l!! O~1!+ Because the coditioal variace I ~l! is bouded uder the assumptio of Propositio 10 ~see 4+10!, the result a 1 H ~l! Z j W 1 j Z j ~ 1!02H ~l! N~0,1! 4.12) follows from applicatio of the cetral limit theorem for martigales with bouded coditioal secod momets E~6h j 6 2 6F t! ~see Billigsley, 1986, Theorem 35+9, p+ 498!+ This shows that the right had side of ~4+8! coverges i distributio to a ormal stadard variate ad gives the desired result for the left had side: a 1 H ~l!~ Zl l! Z j W j 1 Z 02~ Zl l! N~0,1!+ 4.13)

20 5. ESTIMATING EXCESS RETURNS FROM THE UNDERLYING ASSET PROCESS Estimatio of excess returs from observed market derivative prices usig EF theory was developed i Sectio 4+ This sectio discusses the estimatio of l from the uderlyig asset process ~stock, bod, or idex! o which the derivative securities are defied+ The paper s methodology for estimatig premiums ad testig for risk eutral pricig rests o comparig these two estimates for equality+ The relevat class of liear estimatig fuctios for the excess returs parameter l is H H: H a j ~l!h j ~l! + 5.1) The compoet fuctios h j ~l! for the stock process, the optimal weights a j * ~l!, ad the estimators for l ad its variace are derived i the propositios that follow+ Note that the issues of sample desig pertaiig to strikebased replicates ad maturity cycles do ot arise i this situatio+ PROPOSITION 10 ~The Estimatig Fuctio for l Based o the Stock Process X!+ Let X be a diffusio process followig the stochastic differetial equatio dx t b~t, X t!dt s~t,x t!dw t, P-a+s+, 5.2) where b~t, X # R d r R d is the drift vector, s~t,x R d r R d R d is the dispersio matrix of rak d), ad dw t is a d-dimesioal F t - measurable stadard Browia motio with respect to the empirical probability measure P. The, uder the further assumptio of time-state separability see Sectio 2.1), the coditioal martigale differece fuctios h j ~l!, j 1,+++,, i the liear class H H: H are give by ~d 1! ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 803 a j ~l!h j ~l! 5.3) h j ~l! X~!B~t,! X~t! l t X~u!B~t,u!du, 5.4) where l b r is the excess retur. Proof+ The proof is idetical to the developmet of Propositio 1 with the discouted price process f ~X s, B s! X s B~t, s! formig the basis of the Itô expasio+ The optimal estimatig equatio ad the EF estimator for ad its variace estimator are stated i Propositio 11+ The proof parallels Propositio 3+

21 804 GURUPDESH S. PANDHER PROPOSITION 11 ~Optimal Estimatig Equatio, Estimator for l ad Variace from Stock Process!+ Let $X~!, j 1,+++,% defie the sequece of market stock idex) prices. The optimal estimatig fuctio for l is give by H * ~l! u t u t E~X u 6F!B~t t,u!du E~~X u s! 2 6F!B2 t ~t,u!du Defie X~!B~t,! X~t! l t X u B~t,u!du + 5.5) Y j X~!B~t,! X~t!B~t!, Z j u t X t B~t,u!du ad W j u t ~X t s!2 B 2 ~t,u!du+ The, the feasible estimator for l ad its variace implied by the optimal estimatig fuctio are Zl d Z j W j 1 Y j Z j W j 1 Z j 5.6) ad Z Var ~ Zl d! 1 Z j W j 1 Z j + 5.7) 6. DIAGNOSTIC STUDY The empirical properties of the estimatio framework developed i the previous sectios are tested ad evaluated usig a extesive Mote-Carlo study+ The empirical study also eables resolutio of importat sample desig issues for derivative prices ~impact of strike level ad maturity legth!, verifies certai theoretical implicatios ~e+g+, strike replicates do ot reduce variace

22 ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 805 compare ~3+7! ad ~3+10!!, ad gives cofidece i implemetatio to market derivative data+ I the study, the S&P500 idex ad call optios defied o it are simulated usig historical tred ad volatility, ad excess returs are estimated from both the uderlyig asset ad call optio prices uder various scearios with differig strike levels ad strike replicatio ad shorter maturity legths+ Differeces i the estimated excess returs quatify extra premiums ot explaied by the risk eutral pricig model+ Major coclusios of the study are summarized first before detailig the aalysis ad results Major Fidigs The results obtaied from the diagostic study verify the ability of the feasible EF estimator to correctly estimate excess returs from derivative prices ad test the hypothesis of risk eutral pricig+ The simulatios are based o assumptios satisfyig the risk eutral ull hypothesis ~Black Scholes valuatio of calls!, ad estimates of excess returs ~ad equivaletly market price of risk! from both the idex ad its derivative calls are very close for ay give sample size+ Therefore, the empirical study reveals that o pricig errors ad premiums are foud whe oe should exist+ Stadard errors ted to be large but go dow with larger sample sizes+ Secod, the estimatio is foud to be ivariat to the strike level ~ moeyess! of the calls+ Third, it is foud that additio of replicates based o differet strikes does ot improve the stadard errors of the estimator because of high depedece amog replicates+ I the market settig, it is rare to fid calls of large maturity+ Therefore, it is ot feasible to icrease the sample size by extedig the time to maturity+ A alterative samplig desig that cycles over calls of smaller ~ooverlappig! maturities is cosidered as a way to improve precisio+ It is foud that estimatio remais stable whe samplig from cycles of shorter maturities with the same effective sample size ~cycles times maturity legth!+ This result of the study gives cofidece i the applicability of the estimatio methodology to derivative market prices where large samples based o cyclig over shorter maturities ~3 moths best! may be used to reduce variace Geeratio of S&P500 Idex ad Call Optios The S&P500 idex used i the diagostic study is geerated usig tred ad historical volatility estimates obtaied from the Bloomberg olie system+ Estimates of the aualized historical volatility over a 260 day ~1 tradig year! tradig period over the last 6 moths of 1998 fluctuated i the rage 17 23%+ For the empirical study, the aualized volatility was set at s 20%+ From a similar examiatio of the S&P500 idex, a aualized price appreciatio of b 9% was chose, ad the startig value of the idex was set at X 0 1,000 ~the estimatio holds uiformly for data geerated uder other parameter val-

23 806 GURUPDESH S. PANDHER ues for drift ad volatility!+ The S&P idex process was simulated as a discrete geometric Browia motio usig the recursive formula X j [ X tj X exp b s 2 D j s!d j Z, j 1,+++,, 6.1) 2 where D j ad Z ; N~0,1! is a stadard ormal radom variate ~geerated by a radom umber geerator!+ Followig a suggestio of a aoymous referee, the iitial 1,000 recursios of ~6+1! were discarded to remove startup problems i the series, ad the simulatio size was expaded to 2,000 ~from 100!+ This improved the performace of the excess retur estimator eve at the lowest sample size of 100 ~see Table 1! ad whe call optios are sampled from shorter maturity cycles ~Table 3!+ The empirical study was carried out i the Gauss programmig laguage+ Call optios $V k ~,T g![v~,x tj ;K k,t g!,g 1,+++,p, j 1,+++, g, k 1,+++,m g % o the S&P500 idex were geerated by usig ~6+1! i the Black Scholes formula: V k ~,T g![v k ~,X tj ;K k,t g! X tj N~d 1! Kexp~ r!n~d 2!, l X r K 1 2 s 2 d 1 s!, d 2 d 1 s!, 6.2) where g i D j, j 1,+++, g is the time to maturity for each ooverlappig maturity cycle with legths $T 1,+++,T g,+++,t p %+ A radom draw of the idex ad calls ~at the moey! over 500 tradig days is plotted i Figure 1+ I the plot, the idex value is added to the call prices Performace of EF Estimator ad Effect of Sample Size The performace of the proposed estimatio framework i correctly estimatig the excess retur l b r 9%, ad equivaletly the market price of risk g l0s +45, as a fuctio of sample size is first examied+ It is importat to ote that these are theoretical true values uder the assumptio of a cotiuous geometric Browia motio+ As a result of the discretizatio ivolved i geeratig the idex recursively with formula ~6+2!, some distortio is itroduced, ad the actual true values for the excess retur ad market price of risk will differ from 9% ad +45, respectively+ This should be kept i mid whe comparig the performace of the estimator with the true value at differet sample sizes+ Sample sizes ~tradig days; with 260 tradig days per year! varyig from 100 to 10,000 with sigle maturity ad strike ~at the moey! were used i the

24 ESTIMATION AND TESTING FOR RISK NEUTRAL PRICING 807 Figure 1. Radom draw of S&P500 idex ad calls+ first experimet+ The results averaged over 2,000 simulatios are reported i Table 1+ EF estimates of g~ +45! ad l~ 9%! from both the call ad S&P idex are very close at each sample size+ Eve at a small sample size of 100, the mea of estimates is withi 1+9% of the theoretical true value+ [ [ [ [ [ [ Table 1. Estimatig fuctio estimates of l ad g by sample size Sample Size ~Tradig Days! Estimator Average ,000 2,000 5,000 10,000 EF-call g c, EF Var ~ g c, EF! SE~ g c, EF! Zl c, EF 9+17% 9+28% 9+17% 9+16% 9+03% 8+97% EF-idex g x, EF Var ~ g x, EF! SE~ g x, EF! Zl x, EF 9+19% 9+27% 9+17% 9+15% 9+03% 8+97%

25 808 GURUPDESH S. PANDHER 6.4. Testig for Risk Neutral Pricig ad Additioal Premiums The hypothesis that the market call prices are derived from risk eutral pricig ~ad o additioal premiums! ca be tested by comparig the excess returs from call prices with the same from the S&P500 idex+ More formally, our ull hypothesis is H 0 : l c, EF l x, EF 0+ From Sectio 4, we kow that uder H 0, Zl c, EF Zl asy x, EF ; N~0,Var ~ Zl c, EF Zl x, EF!!, implyig the test statistic Zl c, EF Zl x, EF SE~ Zl c, EF Zl x, EF! + PROPOSITION 12 ~Variace of Zl c, EF Zl x, EF!+ Let Y c, j ad Y c, j be defied as i Propositios 3 ad 11, respectively. Similarly for Z c, j, W c, j, Z x, j, t ad W x, j. Further, defie Cov~Y c, j,y x, j! * j u t ~V X ~t, T!!~X t s! 2 B 2 ~t,u!du. The, the variace of Zl c, EF Zl x, EF is give by Var ~ Zl c, EF Zl x, EF! 2 1 Z c, j W 1 c, j Z c, j 1 Z x, j W 1 x, j Z x, j Z v, j W 1 c, j Cov~Y c, j,y x, j!w 1 x, j Y x, j Z c, j W c, j 1 Z c, j Z x, j W 1 x, j Z x, j + Proof+ The expressio follows directly by cosiderig the variace ad covariace terms of Zl c, EF Zl x, EF ad replacig expectatios by the appropriate * * F t -measurable surrogates as show i the proof of Propositio 3+ The ull hypothesis of risk eutral pricig was tested over differet sample sizes as give i Table 2+ Because the call prices are geerated uder assumptios satisfyig the ull hypothesis ~Black Scholes valuatio!, the ull should ot be rejected by the data+ Ideed, as Table 2 shows, differeces i the excess returs obtaied from the S&P500 calls ad the idex, g[ c, EF g[ x, EF, are isigificat, ad the stadard errors will ot reject the ull hypothesis+ The variace of the differece is dramatically reduced by the covariace term as the sample size icreases+ These results show that EF estimatio reveals o pricig errors ad premiums i call prices whe oe should exist Effect of Strike Level or Moeyess The results so far were obtaied usig strikes at the moey+ The impact o the estimatio of chagig the strike level ~ moeyess! was also ivesti-