Empirical Likelihood-Based Constrained Nonparametric Regression with an Application to Option Price and State Price Density Estimation

Transcription

1 Empirical Likeliood-Based Constrained Nonparametric Regression wit an Application to Option Price and State Price Density Estimation Guangyi Ma y Texas A&M University Tis version: January, Abstract Economic models often imply tat te proposed functional relationsip between economic variables must satisfy certain sape restrictions. Tis paper develops a constrained nonparametric regression metod to estimate a function and its derivatives subject to suc restrictions. e construct a set of constrained local quadratic (CLQ) estimators based on empirical likeliood. Under standard regularity conditions, te proposed CLQ estimators are sown to be weakly consistent and ave te same rst order asymptotic distribution as te conventional unconstrained estimators. Te CLQ estimators are guaranteed to be witin te inequality constraints imposed by economic teory, and display similar smootness as te unconstrained estimators. At a location were te unconstrained estimator for a curve (e.g., te second derivative) violates a restriction, te corresponding CLQ estimator is adjusted towards to te true function. Interestingly, suc bias reduction can also be acieved wen te binding e ect is from a restriction on anoter curve (e.g., te rst derivative). Tis nite sample gain is acieved troug joint estimation of te functions, using te same empirical likeliood weigts. e apply tis procedure to estimate te day-to-day option pricing function and te corresponding state-price density function wit respect to di erent strike prices. Key ords: Constrained Nonparametric Regression, Empirical Likeliood, Derivative Estimation, Option Pricing, State-Price Density. JEL Classi cation Numbers: C3, C4, C58. Address: Department of Economics, Texas A&M University, 48 TAMU, College Station, TX 77843, USA, telepone: , gma@econmail.tamu.edu. y I am indebted to Qi Li and Ke-Li Xu for teir guidance. I am also grateful to te participants of te Econometrics Seminar at te Department of Economics, Texas A&M University. All errors are my own responsibility.

2 Introduction Nonparametric regression metods are known to be robust to functional form misspeci cation, ence tey are useful wen te researcer does not ave a teory specifying te exact relationsip between economic variables. However, in many cases, economic teory indicates tat te functional relationsip between two variables X and Y, say, Y = m (X), sould be under certain sape restrictions suc as monotonicity, convexity, omogeneity, etc. Because te estimation results from nonparametric regression are not guaranteed to satisfy tese ex-ante model restrictions, it is desirable to develop a metodology to accommodate suc conventional restrictions in nonparametric estimation. In previous literature, various approaces to nonparametric regression wic satisfy monotonic restriction ave been developed. See Matzkin (994) for a compreensive survey. A popular approac in te existing researc literature is te isotonic regression metod (e.g., Hansen, Pledger, and rigt, 973; Dykstra, 983; Goldman and Rudd, 99; Rudd, 995; etc.). A less desirable feature of te isotonic regression tecnique is tat te estimated function migt not be smoot. To produce monotonic yet still smoot estimation results, one can add a kernel-based smooting step wit te isotonic regression (e.g., Mukerjee, 988; Mammen, 99). Recently, Aït-Saalia and Duarte (3) proposed a similar two-step procedure to estimate option price function nonparametrically. In te rst step, tey adopt Dykstra s (983) constrained least square algoritm to trim te data so tat te estimates from te succeeding kernel smooting step are guaranteed to be monotonic and convex. In tis constrained least square metod, te numerical searc is performed iteratively in a subset of te n-dimensional Euclidean space, were n is te sample size. Tis algoritm is potentially computation-intensive wen te sample size is large. Consequently, it is practically useful to combine teory-imposed restrictions wit a procedure (e.g., kernel smooting metod) tat will yield smoot estimated functions, and reduce computational burden. Anoter vast line in te literature is "constrained smooting splines" (e.g., Yatcew and Bos (997) etc.). See a recent survey by Henderson and Parmeter (9) for tis and oter metods.

3 In tis paper, we explore te possibilities of imposing sape restrictions in te local polynomial regression framework. e construct constrained local quadratic (CLQ) estimators speci cally for te functions m (X), m (X), and m (X). Te proposed estimators can be viewed as reweigted versions of te standard local quadratic (LQ) estimators and te weigts are determined via empirical likeliood (EL) maximization. Empirical likeliood (Owen, 988, 99, 99) is a nonparametric likeliood metod, in contrast to te widely known parametric likeliood metod. See Kitamura (6) for a compreensive survey of EL in econometrics. EL can be applied in bot parametric and nonparametric models. In parametric estimation, a generalized empirical likeliood estimator is sown to ave advantages, in terms of iger order asymptotic properties, over te GMM estimator (Newey and Smit 4). Te idea of parametric estimation via EL is to maximize a nonparametric likeliood ratio Q n i= (np i) between a probability measure fp i : i = ; ; ng given on te sample points and te empirical distribution f=n; ; =ng, subject to moment conditions (Qin and Lawless, 994; Kitamura, Tripati, and An, 4). EL can also be used in combination wit nonparametric models. For a given nonparametric estimator, con dence intervals via EL as demonstrated advantages over asymptotic normality-based approaces. Refer to Hall and Owen (993), Cen (996) for density function estimation; Cen and Qin (), Qin and Tsao (5) for local linear estimators of conditional mean function; Cai () for conditional distribution and regression quantiles; Xu (9) for local linear estimators in continuous-time di usion models. Te common approac in tis literature is to maximize te nonparametric likeliood ratio Q n i= (np i) subject to te EL-weigted estimating equations wic can be viewed as counterparts of te moment conditions in parametric settings. Following tis line, we consider te empirical likeliood pro le fp i : i = ; ; ng embedded on a set of local quadratic estimators and we maximize Q n i= (np i) under te desired sape restrictions. If te restrictions are true for te underlying data generating process, ten te empirical likeliood pro le asymptotically converges to f=n; ; =ng as n goes to in nity. Hence our CLQ estimators ave te same rst order asymptotic distribution as te 3

4 standard local quadratic estimators. Our procedure o ers estimation results tat are smoot functions, and reduces te dimensions of numerical optimization from sample size n to te number of restrictions. Moreover, te procedure estimates te function Y = m (X) and its rst and second derivative simultaneously, so it is particularly useful wen one is interested in estimating te derivatives. en multiple nonparametric functions are jointly estimated, it is common for only some of te restrictions to be violated by te unconstrained estimator. By adjusting tose violations to meet te constraints, our EL approac can meanwile tune oter functional estimates at te same location towards te corresponding true value. In addition, te procedure can be applied in more general situations wen constraints on te function and its derivatives vary by location, unlike te constant constraints considered in existing metods. Hall and Huang () proposed an EL-based nonparametric regression approac to estimate a function, subject to monotonicity constraints. Under certain assumptions of te weigt functions of te original estimator (kernel or local linear weigts, etc.), tey sow te existence of a set of "location independent" EL weigts wic guarantee te reweigted estimator to be monotonic. Racine, Parmeter, and Du (9) extend Hall and Huang s () approac to multivariate and multi-constraint cases. Our study is di erent from tese two papers in several aspects. First, we jointly estimate te regression function and its derivatives and investigate te asymptotic distribution of te EL weigted CLQ estimators, wereas te above two papers focus on te estimation of te regression function itself. Second, our EL weigts are "location dependent" so tat we can accommodate constraints varying upon te domain of X. Tird, we allow constraints on te regression function as well as its derivatives, wile Racine, Parmeter, and Du s (9) teoretical results are not directly applicable to suc a case were tere are multiple constraints on te rst and second derivatives wit respect to te same explanatory variable X. As an application, we use te EL-based CLQ estimator to investigate te nonparametric estimation of daily call option prices C as a function of strike prices X. As implied 4

5 by nance teory, under te assumption of market completeness and no arbitrage opportunities, te price of a call option C = C (X) must be a decreasing and convex function of te option s strike price X. Tese sape restrictions can be expressed as C t; (X) max ; St e t; Xe rt; ; S t e t;, C t; (X) [ e rt; ; ], and C t; (X) [; ), were t is te current time, is te time-to-expiration, r is te risk free interest rate, and is te dividend yield of te underlying asset wit price S t. e estimate C t; (X), C t; (X), and C t; (X) under tese constraints and compare te results of our CLQ estimation wit te results of standard LQ estimation. In a simulation study, we adopt te same simulation set-up as in Aït-Saalia and Duarte (3), and nd tat our results are comparable wit teirs in tis extremely small sample setting, wereas our procedure exibits potential advantages suc as less intensive computation wen te sample size becomes larger. Last, we apply tis metod to estimate te S&P 5 index options in a typical trading day in May 9. Te remainder of tis paper is organized as follows. In Section we introduce te de nition of EL for te local quadratic estimators, subject to inequality constraints. Ten we sow te equivalence of two saddlepoint problems from te EL formulation to ease te following asymptotic analysis. Next, in Section 3 we study te asymptotic properties of te EL-based CLQ estimator. In Section 4 we apply te constrained estimation procedure to estimate option price function and te state-price density. Section 5 concludes. Te proofs and gures are presented in te Appendix. Empirical Likeliood-Based Constrained Local Quadratic Regression e consider a random sample f(x i ; Y i ) : i = ; ; ng generated from a bivariate distribution. Let us denote te conditional mean function of Y given X by m (x) = E (Y jx = x) and te conditional variance function by (x) = V ar (Y jx = x), ten te nonparamet- 5

6 ric regression model under consideration is Y = m (X) + (X) u, were E (ujx) = and V ar (ujx) =. e also denote te marginal density of X by f (). In tis section we develop te empirical likeliood formulation in te context of local quadratic regression model subject to inequality constraints.. Te Local Quadratic Estimator Because our empirical motivation is to impose teory-motivated constraints on te estimators of te functions m (), m (), and m (), we focus on te local quadratic regression model, wic provides estimators for te tree functions simultaneously. Te local quadratic estimators can be derived from te following minimization problem min nx j (x): j=;; i= Yi (x) (x) (X i x) (x) (X i x) Xi x K : () Denote K i = K ((X i x) =), and ereafter we sall sligtly abuse te notation and write (m () ; m () ; m ()) = m () ; m () () ; m () () =, ten te local quadratic estimator for (m (x) ; m (x) ; m (x)) can be written as b (x) = b (x) ; b (x) ; b (x) ; were for j = ; ;, b j (x) = n ji (x) Y i i= j n i= i (x) 6

7 and " i (x) = s s 4 s Xi x # 3 (s s 4 s s 3 ) s X i x s s 3 K i ; " # i (x) = (s s 3 s s 4 ) s X i x Xi x s s 4 (s s 3 s s ) K i ; " i (x) = s s 3 s Xi x # (s s 3 s s ) s X i x s s K i ; s j = X j n Xi x K i for j = ; ; ; 3; 4: i=. Empirical Likeliood for te Local Quadratic Estimating Equations In tis subsection we construct empirical likeliood for local quadratic regression model. Let fp ; ; p n g be a discrete probability distribution on te sample f(x i ; Y i ) : i = ; ; ng. Tat is, fp ; ; p n g is a set of nonnegative numbers adding to unity. At a location x in te domain of X, te pro le empirical likeliood ratio at a set of candidate values (x) = ( (x) ; (x) ; (x)) of i E b (x) = i i i E b (x) ; E b (x) ; E b (x) is de ned as L () = ny n max np i j p i > ; p i = ; o p iu i () = ; () fp ; ;p ng i= i= i= i were U i () = (U i () ; U i () ; U i ()) and U ji (x; (x)) = ji (x) Y i (X i x) j j (x). Te tree equations i= p iu i () = (3) Hereafter, wen it is clear we sall omit te explicit dependence of a variable on te location x for brevity of notations. 7

8 are labelled as "estimating equations" in te empirical likeliood literature. Heuristically, if we take fp ; ; p n g = f=n; ; =ng, ten te estimating equations become i n ji (x) Y i (X i x) j j (x) = ; i= wic can be viewed as reformulations of te rst order conditions of te weigted least square problem (). From te above equations, we can solve, for j = ; ;, by recognizing tat j (x) = n n i= ji (x) Y i i= ji (x) (X i x) j = j n n i= ji (x) Y i i= i (x) i (x) = X n i= Xi i (x) i= x = X n Xi i (x) i= x : (4) Tat is, te candidate values j (x) coincide wit te local quadratic estimators b j (x). In general, te candidate values j (x) are not xed at b j (x), and te corresponding fp ; ; p n g are di erent from uniform weigts =n. As a digression on notation, we will reserve D n for te common value in (4). Tat is, we denote D n = X n Xi ji (x) i= x j = s s s 4 s 3 s (s s 4 s s 3 ) s s s s 3 : By using te log empirical likeliood ratio, we can modify te EL maximization problem () to be l () = n max log (np i) j p i > ; p i = ; o p iu i () = : (5) (p ; ;p n) i= i= i= Furter, by introducing Lagrange multipliers () = ( () ; () ; ()) for te esti- 8

9 mating equations (3) respectively, we can form te Lagrangian as L = X log (np n i) p i i= i= n () i= p iu i () ; and solve for p i () = n ( + () U i ()) : Now te log empirical likeliood ratio l () can be expressed as l () = min i log ( + i= () U i ()) = max log ( + i= () U i ()) ; (6) and e () = arg max log ( + i= () U i ()) ; (7) were = () R 3 j + () U i () > =n; i = ; ; n : Te domain of () is derived from p i [; ] and it is needed to ensure tat te arguments of te logaritm are strictly positive..3 Empirical Likeliood Formulation under Inequality Constraints In tis section we investigate te log empirical likeliood ratio (6) under inequality constraints. In te regression model Y = m (X)+ (X) u, we consider sape restrictions imposed by economic teory in te form of lower and upper bounds on te function m () and its derivatives. More speci cally, let b (x) = (b (x) ; b (x) ; b (x)) and b (x) = b (x) ; b (x) ; b (x), 9

10 ten te restrictions can be expressed as b (x) 6 m (x) 6 b (x) ; b (x) 6 m (x) 6 b (x) ; b (x) 6 m (x) 6 b (x) : For example, in te estimation of option price function and its derivatives, we know tat b (X) = max ; S t e Xe r ; e r ;, and b (X) = St e ; ;. Our goal is to accommodate tese constraints in te nonparametric estimation of m () and its derivatives. Because te log empirical likeliood ratio (6) depends on candidate values (x) = ( (x) ; (x) ; (x)), we can stack te above inequality constraints and impose tem on te candidate values: b (x) 6 (x) 6 b (x) : (8) Ten (6) is modi ed as were min l () = min max G n (; ) ; (9) b66b b66b G n (; ) = i= log ( + () U i ()) : Tis can be viewed as a saddlepoint problem, and let its solution be e; e. Remark Te objective function in (5) corresponds to times te Kullback Leibler distance between te probability distribution fp ; ; p n g and te empirical distribution f=n; ; =ng. Tus maximizing te log empirical likeliood ratio l () can be interpreted as minimizing te Kullback Leibler distance between fp ; ; p n g and f=n; ; =ng. On te oter and, it is easy to verify tat (5) attains its global maximum at f=n; ; =ng, corresponding to te candidate values (x) being equal to te standard LQ estimators b (x). Indeed b (x) are te minimizers of l () witout imposing te inequality constraints (8). Tus e (x), as te minimizers of l () in (9), are designed to minimally adjust te standard LQ estimators suc

11 tat te inequality constraints (8) are satis ed. i Remark Te empirical likeliood formulated so far is for E b (x) = m (x)+bias, rater tan for m (x). Tis point as been observed in previous studies of empirical likeliood-based inference for nonparametric models (Cen and Qin (), Qin and Tsao (5), etc.). To reduce te bias, we use an "undersmooting" bandwidt condition 7!, as recommended in te literature. e will discuss tis approac furter in te asymptotic analysis. To facilitate te asymptotic analysis of te CLQ estimator anoter saddlepoint problem min max ;R 6 + e; e, we need to introduce G n (; ; ) () were G n (; ; ) = G n (; ) + n (b ) + n b and = ( ; ) is a set of Lagrangian multipliers for te inequalities b 6 ; b 6 : Te following lemma states tat te two problems (9) and () ave te same saddlepoints. Lemma e; e is a saddlepoint of G n (; ) and solve (9) if and only if e; e ; e is a saddlepoint of G n (; ; ) tat solves (), were for j = ; ;, 8 e j x; e >< = >: 8 e j x; e >< = >: P n e j( ) e ji (x)(x i n i= + ( e ) e U i( ) e P n e j( ) e ji (x)(x i n i= + ( e ) e U i( ) e x) j if b j e j = ; if b j e j < ; x) j if e j b j = ; if e j b j < : ()

12 Remark 3 In practical implementation, one can program according to te saddlepoint problem (9). Essentially, at eac evaluating location x, searcing for e (x) is performed in b (x) ; b (x). Tis can be viewed as an "outer loop". ile for eac candidate (x), searcing for () is done in te "inner loop" via maximizing G n (; ). Plugging p i in te EL weigted estimating equations P n i= p iu i () =, we ave U i () n i= + () U i () = ; wic can also be viewed as te rst order conditions for () in (6) divided by n. In te "inner loop", given a candidate value of (x), we can equivalently solve for () from tese rst order conditions. Remark 4 Te saddlepoint problem () is useful in te following asymptotic analysis of te CLQ estimators e (x). Since (9) and () are equivalent, we use te same notation l () in te remaining of tis paper. Tat is, we denote l () = max G ;R 6 n (; ; ) + = max ;R 6 + i= log ( + () U i ()) + n (b ) + n b : 3 Asymptotic Analysis of te Constrained Local Quadratic Estimators In tis section we rst sow tat, under proper regularity conditions, (m) = ( (m) ; (m) ; (m)) converges to zero as n!. Tis result is presented in Teorem. Ten we sow in Teorem tat te CLQ estimators e (x) and te standard LQ estimators b (x) are asymptotically equivalent, tat is, e (x) and b (x) ave te same rst-order asymptotic distribution. As a starting point, we need te following assumptions:

13 Assumption Te kernel function K () is a symmetric bounded density function compactly supported on [ ; ]. Assumption f () and () ave continuous derivatives up to te second order in a neigborood of x, and bot f (x) > and (x) >. Also m () as continuous derivatives up to te tird order in a neigborood of x. Assumption 3!,!, and 7! as n!. Lemma Under Assumptions,, and 3, as n!, we ave te asymptotic distribution p U i (m) i= 3 6 m(3) (x) f 3 d (x) B U! N ; (x) f 5 (x) V U ; () were B U = 4 4 C A ; V U = 3!! 5 C A ;! = ;! = ;! 3 = 4 ;! 5 = + 4 : Remark 5 To investigate te asymptotic beavior of te EL-based CLQ estimators, rst we need to nd te asymptotic distribution of te estimating equations (3). Lemma presents te asymptotic distribution of (3) evaluated at te set of true values m (x) = (m (x) ; m (x) ; m (x)). Tis result can be derived from te asymptotic distribution of te LQ estimators b (x) because (3) can be viewed as a reformulation of b (x) m (x). Essen- 3

14 tially, from Lemma we ave U i (m) = O p () = + 3 : i= Remark 6 Notice tat te leading bias terms of U i (m) and U i= i (m) in i= () are actually zeros because of te symmetry of te kernel K (). As suggested by Cen and Qin (), we use an "undersmooting" condition 7! to reduce te bias of i= U i (m). it tis condition, we can still use te optimal bandwidt = O n =9 for te estimation of te regression function and te second derivative. Lemma 3 Under Assumptions,, and 3, we ave U i (m) U i (m) = U + o p () ; i= were U = f 3 (x) (x)! (x)! (x)! (x)! [ (x) + m (x)]! 3 [ (x) + m (x)]! 4 (x)! [ (x) + m (x)]! 4 [ (x) + m (x)]! 5 C A : Teorem Assume tat E jy i j s < for some s > and tat Assumptions,, and 3 old. Ten (m) = O p () = + 3 = o p n 3=7 ; also (m) = X n U X i (m) U i (m) n i= U i (m) + o p () = + 3 : i= Remark 7 From Teorem we ave tat p i (m) = n ( + (m) U i (m)) converges to =n wit increasing sample size and proper selected bandwidt. Hence te EL-based CLQ 4

15 estimators e j (x) = j i= p i ji (x) Y i i= p i i (x) converge to te unconstrained LQ estimators b j (x) = n ji (x) Y i i= j n i= i (x) (j = ; ; ) (j = ; ; ) Lemma 4 Assume tat Assumptions,, and 3 old, furter assume tat 5! as n!. Ten G n (; ; ) attains its saddlepoint at e; ; e e were e = e ; e ; e is suc tat e j (x) m j (x) 6 j ; e = e is given by (7); and e is given by (). Furter, e; e satis es g n e; e =, g n e; e =, were g n (; ) = i= g n (; ) = i= U i () + () U i () ; D i (x) () + () U i () ; n and D i (x) is a 3 3 matrix suc tat D i (x) = diag ji (x) ((X i x) =) jo. Remark 8 Lemma 4 sows te existence of a saddlepoint of G n (; ; ) in te interior of a (asymptotically srinking) neigborood of m (x), (x) : jj (x) m j (x)j 6 j ; j = ; ; : (3) Tis is acieved by establising a lower bound of l () out of (3) and ten we sow tat tis lower bound is of a larger stocastic order tan l (m). Teorem Suppose tat te assumptions of Teorem old. e also assume tat 5! 5

16 as n!. Ten for j = ; ;, te EL-based CLQ estimator e j (x) = b j (x) + o p +j = + 3 j ; were for eac j, b j (x) is te corresponding LQ estimator. As n!, te asymptotic distribution of e (x) is given by p diag +j e (x) m (x) 6 m(3) d (x) B e! N ; (x) f (x) V e ; (4) were B e = 4 = C A ; V e = V U ( 4 ) : Remark 9 Teorem sows tat te EL-based CLQ estimators e (x) and te standard LQ estimators b (x) ave te same asymptotic distribution up to te rst order. Tis result is naturally expected because, as te sample size increases, b (x) converges to te true function values wic are in te bounded region b (x) ; b (x), ence te inequality constraints become unbinding (e! p ) and e (x) and b (x) are asymptotically rst-order equivalent. 4 Application: Option Pricing Function and State-Price Density Estimation under Sape Restrictions 4. Restrictions Imposed by Option Pricing Teory To sow te usefulness of te CLQ estimation procedure proposed in tis paper, we estimate te daily option pricing function and te state-price density function by incorporating various sape restrictions. In summary, given market completeness and no arbitrage assumptions, implication from nancial market teory suggests tat te price of a call option, as a function 6

17 of its strike price, must be decreasing and convex. Let us consider an European call option wit price C t at time t, and expiration time T. Denote by = T t te maturity, and X te strike price. Also denote by r t; te risk free interest rate and t; te dividend yield of te underlying asset wit price S t. Using tese notations, we can give te call option price C t by Z + C (X; S t ; ; r t; ; t; ) = e rt; max (; S T X) f (S T js t ; ; r t; ; t; ) ds T ; were f (S T js t ; ; r t; ; t; ) is te state-price density (SPD), also called te risk-neutral density (denoted as f (S T ) for brevity in wat follows). Asset pricing teory imposes no arbitrage bounds for te price function as max ; S t e t; Xe rt; 6 C t; (X) 6 S t e t; ; (5) were C t; (X) is used to denote C (X; S t ; ; r t; ; t; ) since we focus on te call option price C as a function of X. For te = Z + e rt; X f (S T ) ds T ; te no-arbitrage assumption requires C to be a decreasing function of X, and te above derivative larger tan e rt;. Tus we ave e rt; 6 C t; (X) 6 (6) from te positivity and integrability to one of te SPD. Te second derivative is C t; (X) = e rt; f (X) > (7) since te SPD must be positive. 7

18 Given te data (X i ; C i ) recorded at time t (typically in one trading day) wit te same maturity, our objective is to estimate C t; (X), C t; (X), and C t; (X) under constraints (5), (6), and (7). 4. Monte-Carlo Simulation To compare te performance of our procedure wit existing researc, we adopt te simulation setup as tat in Aït-Saalia and Duarte (3). Speci cally, te true call option price function is assumed to be parametric as in te Black-Scoles/Merton model C BS (X; F t; ; ; r t; ; ) = e rt; [F t; (d ) X (d )] ; were F t; = S t e (rt; t; ) is te forward price of te underlying asset at time t and d = log (F t;=x) p + p ; d = log (F t;=x) p p ; and = (X=F t; ; ) is te volatility parameter. To generate data for simulation, we calibrate parameter values from real observations of S&P 5 index options on May 3, 999. Te parameter values and domain of strike prices are set as S t = 365; r t; = 4:5%; t; = :5%; = 3=5; X i [; 7] ; i = X i =4 + 43=35: In te rst simulation, te strike prices X are equally spaced between and 7 wit a sample size of 5. Tat is, in eac sample tere are 5 distinct strike prices and eac of tem corresponds to one call option price. In te second simulation, we generate call option prices for eac distinct strike price, so te sample size is 5. 3 To generate option prices, in te rst simulation (n = 5), we add uniform noise to te true option price function, 3 Tis is similar wit te simulation setup in Yatcew and Hardle (6). 8

19 wic ranges from 3% of te true price value for deep in te money options (X = ) to 8% for deep out of te money options (X = 7). e double te noise size in te second simulation (n = 5). 4 e use te Epanecnikov kernel in te local qradratic estimation and adopt a rule-of-tumb bandwidt as in Fan and Mancini (9). In eac simulation experiment, we generate and estimate samples and sow te average, 5%, and 95% quantiles as con dence bands in eac grap. For samples wit 5 observations in te rst simulation, te estimation results are sown in Figure. Te sample size in tis simulation is tiny, so te unconstrained local quadratic estimators, especially te estimators for rst and second derivatives, c C (X) and c C (X), perform poorly and violate te constraints frequently. Altoug di cult to distinguis in te grap, te estimator for te option price, b C (X), also violates te lower bound wen te strike price is low for deep in te money options. Tis violation of constraint can be adjusted in our EL-based estimator, wile not in Ait-Saalia and Duarte (3). Turning to te constrained estimation by our EL-based procedure, we can nd tat all tree estimators, e C (X), f C (X), and f C (X), are guaranteed to satisfy te constraints, and te estimators for rst and second derivatives ave smaller con dence bands in bot of te boundary areas of te domain of X. An interesting nding is tat, by correcting te violation of constraints in te rst derivative estimate, te EL-based procedure also adjusts te second derivative estimate towards to its true function in corresponding boundary areas, altoug te unconstrained estimate itself, cc (X), may not violate its nonnegative lower bound. 5 In te second simulation wit 5 observations (Figure ), performance of te unconstrained local quadratic estimators is better tan in te previous small sample design in spite of te doubled noise size. it a sample size as large as 5, te unconstrained estimate ( b C (X)) of te option price function and its true value become undistinguisable. But for te estimation of derivatives, te unconstrained estimators ( c C (X) and c C (X)) still 4 If we use te same noise design in te second simulation wit larger sample size, te unconstrained local quadratic estimates will violate constrains less so te constrained and unconstrained estimation results will be close. 5 Note tat te 5% quantile of f C (X) corresponds to te 95% quantile of f C (X) and vice versa. 9

20 violate te constraints wen te strike price is very low or very ig. In comparison, te EL-based constrained estimators ( C f (X), and C f (X)) are strictly witin te constraints and ave muc narrower con dence bands, specially, in te left boundary area. Last, we compare te integrated mean squared errors (IMSE) from constrained and unconstrained estimation in Figure 3. e focus on te rst simulation design wit sample size 5. Te plots sow tat te IMSE s are muc lower for te constrained estimators in all tree functional estimations. Also we nd a U-saped IMSE curve in all tree cases, sowing tat tere exists an optimal bandwidt minimizing te IMSE. 4.3 Empirical Analysis To investigate te empirical performance of our EL-based CLQ estimators, we estimate te option price function and te state price density (a scaled second derivative of te option price function). e consider closing prices of European call options on te S&P 5 index (symbol SPX). Te SPX index option is one of te most actively traded options and as been studied extensively in empirical option pricing literature. Te data are downloaded from OptionMetrics. e collect options on May 8, 9 for a maturity of 6 days corresponding to te expiration on July 8, 9. Following Aït-Saalia and Lo (998), Fan and Mancini (9), we use te bid-ask average of closing price as te option price, and we delete less liquid options wit implied volatility larger tan 7%, or price less tan or equal to.5. Finally we reac a sample of 8 call option prices wit strike prices ranging from 75 to 5. Te closing spot price of te S&P 5 index on tat day was 99.7, and te risk free interest rate for te -mont maturity was.5%. Te dividend yield is retrieved from te put-call parity. Figure 4 presents te estimation results of tis daily cross-sectional option prices data set. From te results we can nd tat te unconstrained estimate of te rst derivative signi cantly violates te constraints at bot te in-te-money and te out-ofte-money areas. In contrast, te constrained estimate of tis function is bounded in bot areas.

21 5 Conclusions e propose an empirical likeliood-based constrained local quadratic regression procedure to accomodate general sape restrictions imposed by economic teory. Te resulted estimates can satisfy te constraints on te function and its rst and second derivatives. Compared wit te traditional "isotonic regression and smooting" two-step metod, te EL-based approac can be less computationally intensive, and can accommodate more general constraints, ence it sows potential to be useful in a wide range of applications. e study te empirical performance of te constrained estimation metod in bot simulations and real data applications. A part of te follow-up work is to analyze more extensively te EL-based CLQ estimators, bot asymptotically and in nite sample, suc as te comparison of mean squared errors between te constrained and unconstrained estimators. Anoter direction wic migt enric te scope of tis paper is to develop tests on sape restrictions suc as monotonicity and convexity, based on te asymptotic ci-square distribution of te log EL ratio statistic.

22 References Aït-Saalia, Y., Duarte, J., 3. Nonparametric option pricing under sape restrictions. Journal of Econometrics 6, Aït-Saalia, Y., Lo, A., 998. Nonparametric estimation of state-price densities implicit in nancial asset prices. Journal of Finance 53, Cai, Z.,. eigted Nadaraya-atson regression estimation. Statistics and Probability Letters 5, Cai, Z.,. Regression quantiles for time series. Econometric Teory 8, Cen, S.X., 996. Empirical likeliood con dence intervals for nonparametric density estimation. Biometrika 83, Cen, S.X., Qin, Y.-S.,. Empirical likeliood con dence interval for a local linear smooter. Biometrika 87, Dykstra, R.L., 983. An algoritm for restricted least squares. Journal of te American Statistical Association 78, Fan, J., Gijbels, I., 996. Local Polynomial Modelling and its Applications. Capman & Hall, London. Fan, J., Mancini, L., 9. Option pricing wit model-guided nonparametric metods. Journal of te American Statistical Association 4, Hall, P., Huang, H.,. Nonparametric kernel regression subject to monotonicity constraints. Te Annals of Statistics 9, Hall, P., Owen, A.B., 993. Empirical likeliood con dence bands in density estimation. Journal of Computational and Grapical Statistics, Hall, P., Presnell, B Intentionally biased bootstrap metods. Journal of te Royal Statistical Society, Series B, 6, Henderson, D., Parmeter C., 9. Imposing economic constraints on nonparametric regression: Survey, implementation and extensions. In: Li, Q., Racine, J. S. (Eds.), Advances in Econometrics: Nonparametric Metods. Elsevier Science, Vol. 5,

23 Matzkin, R.L., 994. Restrictions of economic teory in nonparametric metods. In: Engle, R.F., McFadden, D.L. (Eds.), Handbook of Econometrics, Vol. 4, Nort Holland, Amsterdam. Moon, H.R., Scorfeide, F., 9. Estimation wit overidentifying inequality moment conditions. Journal of Econometrics 53, Kitamura, Y., 6. Empirical likeliood metods in econometrics: Teory and practice. In: Blundell, R., Torsten, P., Newey,.K. (Eds.), Advances in Economics and Econometrics, Teory and Applications, Nint orld Congress. Cambridge University Press, Cambridge. Kitamura, Y., Tripati G., An H., 4. Empirical likeliood based inference in conditional moment restriction models, Econometrica 7, Li, Q., Racine, J., 7. Nonparametric Econometrics: Teory and Practice. Princeton, NJ: Princeton University Press. Newey,.K., Smit R.J., 4. Higer order properties of GMM and generalized empirical likeliood estimators, Econometrica 7, Owen, A.B., 988. Empirical likeliood ratio con dence intervals for a single functional. Biometrika 75, Owen, A.B., 99. Empirical likeliood con dence regions. Annals of Statistics 8, 9. Owen, A.B., 99. Empirical likeliood for linear models. Annals of Statistics 9, Owen, A.B.,. Empirical Likeliood. Capman and Hall, New York. Pagan, A., Ulla, A., 999. Nonparametric Econometrics. New York: Cambridge University Press. Qin, G., Tsao, M., 5. Empirical likeliood based inference for te derivative of te nonparametric regression function. Bernoulli, Qin, J., Lawless, J., 994. Empirical likeliood and general estimating equations. Annals of Statistics,

24 Racine, J.S., Parmeter, C.F., Du, P., 9. Constrained nonparametric kernel regression: Estimation and inference. Virginia Tec AAEC orking Paper. Xu, K.-L., 9. Empirical likeliood based inference for recurrent nonparametric di usions. Journal of Econometrics 53, Xu, K.-L., 9, Re-weigted functional estimation of di usion models. Econometric Teory 6, Yatcew, A.J., Bos, L., 997. Nonparametric regression and testing in economic models. Journal of Quantitative Economics 3, 8 3. Yatcew, A., Hardle,., 6. Nonparametric state price density estimation using constrained least squares and te bootstrap. Journal of Econometrics 33,

25 Appendix A: Proofs Proof of Lemma Proof. (i) Let e; ; e e be a saddlepoint of G n (; ; ) solving (). First we look at te upper bounds b. Suppose tere is j f; ; g suc tat e j > b j, ten tere must exist j > e j > suc tat ej j b j > e ej j b j, so G e; e n ; e; e j ; j > G e; e n ; e, wic contradicts wit te de nition of e; ; e e. Terefore e j 6 b j for all j = ; ;. Tis implies tat e e b 6 since e >. Furter, if e j < b j, ten e j =. Togeter we ave e e b =. Similarly we can sow tat e b e =. So G n e; ; e e = G n e; e > G n (; ) for any b; b and. Tat is, e; e is a saddlepoint of G n (; ) tat solves (9). (ii) Let e; e be a saddlepoint of G n (; ) solving (9) and e as de ned in te lemma. e want to sow tat e; ; e e is a saddlepoint of (). First, since e e b = and e b e = by te de nition of e, also since e b; b, we ave e e b > e b and e b e > b e for any = ( ; ) R 6 +, ence G n e; ; e e > G n e; e ; : (8) Second, let e = ej ; e j suc tat e j = b j (or e j = b j ) and e j b j ; b j. Ten for any b; b, we make te same partition = ( j ; j ) and ave G n ; ; e e = G n j ; j ; ; e e = G n j ; e j ; ; e e ; were te second equality olds because by de nition te part in e corresponding to j are 5

26 zeros. Furter, we ave G n j ; e j ; ; e e > G ej n ; e j ; ; e e since G n (; ; ) is globally convex in j, and by de nition of n (; ; j e; e ;e = : Togeter we ave G n ; ; e e > G e; e n ; e : (9) Finally, (8) and (9) imply tat e; ; e e is a saddlepoint of (). Proof of Lemma For te general local polynomial estimators, te asymptotic conditional bias and variance terms are discussed in Fan and Gijbels (996), Teorem 3.. Following teir notations, we denote, in te case of local quadratic estimator, S = 3 C A ; S = 3 C A ; c = 3 4 C A ; ec = 4 5 C A ; were j = R u j K (u) du, j = R u j K (u) du. Note tat =, and for a symmetric kernel, = 3 = 5 = = 3 =. Ten te asymptotic bias is given by Bias bj (x) jx = e j+ S m (3) (x) c 3 j + o p 3 j 6 6

27 for j =, and Bias bj (x) jx = e j+ S ec m (4) (x) + 4m (3) (x) f () (x) 4 j + o p 4 j 4 f (x) for j = ;. Te asymptotic variances are given by V ar bj (x) jx = e j+ S S S (x) e j+ f (x) + o +j p +j for j = ; ;. It is known tat te leading term in te asymptotic bias is of a smaller order for j being even tan in te case for j being odd. Explicitly, we ave Bias b (x) jx = m (4) (x) + 4m (3) (x) f () (x) + o 4 4 p 4 ; f (x) Bias b (x) jx = 4 m (3) (x) + o p ; 6 Bias b (x) jx = 6 4 m (4) (x) + 4m (3) (x) f () (x) + o 4 4 p ; f (x) V ar b (x) jx = (x) ( 4 ) f (x) + o p ; V ar b (x) jx = (x) 3 f (x) + o p ; 3 V ar b (x) jx = + 4 (x) 5 ( 4 ) f (x) + o p : 5 To derive te asymptotic distribution for te estimating equations, we need to introduce more notations. Let S = T=D, were t t t T = B t t 3 t 4 A = B A ; t t 4 t D = det (S) = 4 3 ( 4 3 ) 3 ; 7

28 ten S S S = D T S T: Note tat we ave already denoted D n = i (x) i= = s s s 4 s 3 s (s s 4 s s 3 ) s s s s 3 ; tus we ave D n p! f 3 (x) D because s j p! f (x) j for j = ; ; ; 3; 4. Te tree estimating equations evaluated at te true values (m (x) ; m (x) ; m (x)) are U i (m) = b (x) m (x) D n ; i= U i (m) = b (x) m (x) D n ; i= U i (m) = b (x) m (x) D n ; i= so we can derive tat, by assuming!,!, 7! as n!, p U i (m) i= 3 6 m(3) (x) f 3 d (x) T c! N ; (x) f 5 (x) T S T ; were T c = 3 ( 4 3) 4 ( 4 3 ) 5 ( 3 ) 3 ( 3 4 ) 4 ( 4 ) 5 ( 3 ) 3 ( 3 ) 4 ( 3 ) 5 ( ) C A ; T S T = 3! 4!! 4! 5 C A ; 8

29 and! = t + t t + t t + t + t t 3 + t 4 ;! = t t + t t 3 + t + (t t 4 + t t + t t 3 ) + (t t 4 + t t 3 ) 3 + t t 4 4 ;! = t t + (t t 4 + t t ) + t t 5 + t t 4 + t + (t t 4 + t t 5 ) 3 + t t 5 4 ;! 3 = t + t t 3 + t t 4 + t3 + t 3 t t 4 4 ;! 4 = t t + (t t 4 + t t 3 ) + (t t 5 + t t 4 + t 3 t 4 ) + t 3 t 5 + t4 3 + t 4 t 5 4 ;! 5 = t + t t 4 + t t 5 + t4 + t 4 t t 5 4 : For a symmetric kernel K (), remind tat = 3 = 5 = = 3 =, so t = 4, t =, t 3 = 4, t 5 =, t = t 4 =, and T c = B 4 4 A ; ( 4 + ) 4 T S T = B ( 4 = 4 + ( 4 + ) C A : Proof of Lemma 3 Lemma 3 states te stocastic order for te squared sums of U i (m). Te proof is similar as tat of Lemma in Qin and Tsao (5). Using te same notation as in Section., we let 9

30 K i = K ((X i x) =), and i (x) = " s s 4 s 3 Xi (s s 4 s s 3 ) x " # Xi x Xi x = T + T + T K i ; " i (x) = (s s 3 s s 4 ) s X i x s s 4 " # Xi x Xi x = T + T 3 + T 4 K i ; " i (x) = s s 3 s Xi x (s s 3 s s ) " # Xi x Xi x = T + T 4 + T 5 K i ; # s X i x s s 3 K i Xi (s s 3 s s ) # x # s X i x s s K i K i ten for j = ; ;, U ji (m) = ji (x) Y i m j (x) (X i x) ji : i= i= Te conclusion in Lemma 3 can be veri ed as follows. Lemma A U i (m ) = (x) f 3 (x)! + o p (). i= Proof. rite X U i (m ) = X i (x) [Y i m (x)] = X i (x) [Y i m (X i )] + X i (x) [m (X i ) m (x)] + = J + J + J 3 : X i (x) [Y i m (X i )] [m (X i ) m (x)] 3

31 First, J = T X K i (X i ) u i + T T + T T + T X X i X X i + T T x x X X i Ki (X i ) u i 3 Ki (X i ) u i + T x K i (X i ) u i X X i x 4 K i (X i ) u i ; since for j = ; ; ; 3; 4, X X i " j X # j x Ki (X i ) u x i = E K (X ) u + o p () Z = (x) f (x) u j K (u) du + o p () = (x) f (x) j + o p () ; and for j = ; ; ; 3; 4, T j = f (x) t j + o p () ; so J = (x) f 3 (x) t + t t + t t + t + t t 3 + t 4 + op () = (x) f 3 (x)! + o p () : 3

32 Second, J = o p () since for j = ; ; ; 3; 4, X j X i x Ki (X i ) u i (m (X i ) m (x)) " X # j x = E K (X ) u (m (X ) m (x)) + o p () " X # j x = E K (X ) E (u jx ) (m (X ) m (x)) + o p () = o p () : Tird, J 3 = o p () since for j = ; ; ; 3; 4, X X i = X X i = m() (x) X X i n + o p X X i x x j Ki [m (X i ) m (x)] j x Ki m () (x) (X i x) + o p () x j+ Ki + o p () m() (x) X X i n j K i x j+ K i = m () (x) [ j+ f (x) + o p ()] + o p () m () (x) [ j+ f (x) + o p ()] + o p [ j f (x) + o p ()] = O p + o p = o p () : were te rst equality is because te kernel function is bounded in [ ; ]. Lemma A U i (m ) = [ (x) + m (x)] f 3 (x)! 3 + o p (). i= 3

33 Proof. rite X U i (m ) = X i (x) Y i m () (x) (X i x) = X i (x) [Y i m (X i )] + X i (x) [Y i m (X i )] m (X i ) m () (x) (X i x) X i (x) m (X i ) m () (x) (X i x) + = J + J + J 3 ; were J = (x) f 3 (x)! 3 + o p (), J = o p () because of similar proof for corresponding parts in Lemma A. Next, J 3 = m (x) X m (x) X i (x) + i (x) m (X i ) m (x) m () (x) (X i x) + X i (x) m (X i ) m (x) m () (x) (X i x) = m (x) J 4 + m (x) J 5 + J 6 ; were m (x) J 4 = m (x) f 3 (x)! 3 + o p (), since for j = ; ; ; 3; 4, X X i x j K i = E " X x # j K + o p () = f (x) j + o p () ; and for j = ; ; ; 3; 4, T j = f (x) t j + o p () : 33

34 Also, J 5 = X i (x) m() (x) (X i x) + o p = m() (x) X i (x) (X i x) + o p J 4 = O p + o p = o p () ; J 6 = X i (x) m() (x) (X i x) + o p = O p 4 + o p 4 = o p () : Lemma A 3 U i (m ) = [ (x) + m (x)] f 3 (x)! 5 + o p (). i= Proof. rite X U i (m ) = X i (x) Y i m() (x) (X i x) = X i (x) [Y i m (X i )] + + X i (x) [Y i m (X i )] m (X i ) X i (x) m (X i ) = J + J + J 3 ; m() (x) (X i x) m() (x) (X i x) were J = (x) f 3 (x)! 5 + o p (), J = o p () because of similar proof for corresponding parts in Lemma A. Next, let A = m (X i ) m (x) m () (x) (X i x) A = m (x) + m () (x) (X i x) ; m() (x) (X i x) = o p ; 34

35 ten J 3 = X i (x) A + X i (x) A A + X i (x) A = J 4 + J 5 + J 6 ; were J 4 = X i (x) o p = op 4 ; X i (x) o p + m() (x) J 5 = m (x) J 6 = m (x) X i (x) + m (x) m() (x) = J 7 + O p () + O p ; X i (x) o p (X i x) = o p + o p 3 ; X m () (x) X i (x) (X i x) + i (x) (X i x) and J 7 = m (x) f 3 (x)! 5 + o p () as J 4 in Lemma A. So J 3 = m (x) f 3 (x)! 5 + o p (). Lemma A 4 U i (m ) U i (m ) = (x) f 3 (x)! + o p (). i= Proof. rite X Ui (m ) U i (m ) = X i (x) i (x) [Y i m (x)] Y i m () (x) (X i x) = X i (x) i (x) [Y i m (X i )] X i (x) i (x) [Y i m (X i )] m (X i ) m () (x) (X i x) X i (x) i (x) [Y i m (X i )] [m (X i ) m (x)] X i (x) i (x) [m (X i ) m (x)] m (X i ) m () (x) (X i x) = J + J + J 3 + J 4 ; 35

36 were J = (x) f 3 (x)! + o p (), J = J 3 = o p () because of similar proof for corresponding parts in Lemma A, and J 4 = X i (x) i (x) [m (X i ) m (x)] + X i (x) i (x) [m (X i ) m (x)] m (x) m () (x) (X i x) = J 4 + J 4 ; were J 4 = o p () as J 3 in Lemma A, and J 4 = X i (x) i (x) m () (x) (X i x) + o p () m (x) m () (x) (X i x) = m (x) X i (x) i (x) m () (x) (X i x) + o p () m () (x) X i (x) i (x) m () (x) (X i x) + o p () (X i x) = O p () + o p () + O p + o p = o p () : Lemma A 5 U i (m ) U i (m ) = (x) f 3 (x)! + o p (). i= 36

37 Proof. rite X Ui (m ) U i (m ) = X i (x) i (x) [Y i = X i (x) i (x) [Y i m (X i )] + X i (x) i (x) [Y i m (X i )] m (X i ) m (x)] Y i m() (x) (X i x) + X i (x) i (x) [Y i m (X i )] [m (X i ) m (x)] + X i (x) i (x) [m (X i ) m (x)] m (X i ) = J + J + J 3 + J 4 ; m() (x) (X i x) m() (x) (X i x) were J = (x) f 3 (x)! + o p (), J = J 3 = o p () because of similar proof for corresponding parts in Lemma A, and J 4 = X i (x) i (x) [m (X i ) m (x)] + X i (x) i (x) [m (X i ) m (x)] m (x) m() (x) (X i x) = J 4 + J 4 ; were J 4 = o p () as J 3 in Lemma A, and J 4 = X i (x) i (x) m () (x) (X i x) + o p () m (x) m() (x) (X i x) = m (x) X i (x) i (x) m () (x) (X i x) + o p () m () (x) X i (x) i (x) m () (x) (X i x) + o p () (X i x) = O p () + o p () + O p 3 + o p 3 = o p () : 37

38 Lemma A 6 U i (m ) U i (m ) = [ (x) + m (x)] f 3 (x)! 4 + o p (). i= Proof. rite X Ui (m ) U i (m ) = X i (x) i (x) Y i m () (x) (X i x) Y i m() (x) (X i x) = X i (x) i (x) [Y i m (X i )] + X i (x) i (x) [Y i m (X i )] m (X i ) m() (x) (X i x) + X i (x) i (x) [Y i m (X i )] m (X i ) m () (x) (X i x) + X i (x) i (x) m (X i ) m () (x) (X i x) m (X i ) = J + J + J 3 + J 4 ; m() (x) (X i x) were J = (x) f 3 (x)! 4 + o p (), J = J 3 = o p () because of similar proof for corresponding parts in Lemma A, and J 4 = X i (x) i (x) m (X i ) m (x) m () (x) (X i x) + m (x) m (X i ) m (x) m () (x) (X i x) m() (x) (X i x) + m (x) + m () (x) (X i x) = X i (x) i (x) [o p () + m (x)] o p + m (x) + m () (x) (X i x) = m (x) X i (x) i (x) + m (x) m() (x) = J 5 + O p () + o p () ; X i (x) i (x) (X i x) + o p () were J 5 = m (x) f 3 (x)! 4 + o p () as J 4 in Lemma A. 38

39 Proof of Teorem Proof. rite (m) = were > and kk =. Also denote U = P P n j= i= U ji (m j ) for j = ; ;, and U = P n i= U i (m) U i (m). Note tat p i = n ( + U i (m)) [; ] ; from wic we ave + U i (m) >. From te tree EL weigted estimating equations, P n i= p iu i (m) =, we ave = U i (m) i= + U i (m) > U i (m) i= + U i (m) = U i (m) i= > U i (m) U i (m) i= + U i (m) > U U ; + Z n i= U i (m) [ U i (m)] + U i (m) U were in te rigt and side of te last inequality, Z n = max 6i6n ku i (m)k so Z n > U i (m) for eac i. Terefore implies + Z n U 6 U U Z n U 6 U : Since (i) by Lemma, U = Op () = + 3, (ii) by Lemma 3, U = U + o p (), (iii) Z n = o p n =s from te assumption of E jy i j s < for s >, we ave k (m)k = = O p () = + 3 : 39

40 Moreover, by a Taylor expansion of te EL weigted estimating equations at =, we ave ence = U i (m) i= X n U i (m) U i (m) (m) + o (k (m)k) ; i= (m) = X n U X i (m) U i (m) n i= U i (m) + o p () = + 3 : i= Proof of Lemma 4 Proof. itout losing generality, for te saddlepoint e; e ; e of G n (; ; ), we only consider te case e =. Tat is, te inequality constraints b 6 6 b are not binding in te large sample context. Terefore te "inner" optimization problem max log ( + ;R 6 i= () U i ()) + n (b ) + n b + is simpli ed as l () = max log ( + i= () U i ()) : e point out tat te following proof also olds witout tis simpli cation. Denote = ; ; ;, and for j = ; ;, j = m j j u j, were u j R is suc tat u = (u ; u ; u ), kuk =. First, following te argument in te proof of Lemma in Qin and Lawless(994), we establis a lower bound for l () at. To do tis, notice tat: 4

41 (i) by Lemma, " U ji j = u j i= i= ji (x) Xi # j x = u j f 3 (x) D + o p + O p () = + 3 = u j f 3 (x) D + o p ; + U ji (m j ) i= since D n = P n i= ji (x) ((X i x) =) j = f 3 (x) D + o p (); (ii) by Lemma 3, U i U i = i= U i (m) U i (m) + o p () = U + o p () ; i= were U i = i= U i = i= U i = i= U i Ui = i= U i Ui = i= U i Ui = i= i= U i (m ) + O p 3 ; i= U i (m ) + O p ; i= U i (m ) + O p ; i= U i (m ) U i (m ) + O p ; i= U i (m ) U i (m ) + O p ; i= U i (m ) U i (m ) + O p : As in te proof of Teorem, from (i) and (ii), we ave = U i U i i= U i + o p () i= = O p : 4

42 Terefore by a Taylor expansion at = and by (), l = = = U i i= U i i= uf 3 (x) D + o p U > 5 (c ) ; U i U i + o p i= U i U i i= U i + o p 5 i= uf 3 (x) D + o p + o p 5 were c > and c is te smallest eigenvalue of f 6 (x) D u U u. Similarly, l (m) = X n U i (m) i= + o p () = + 3 U i (m) U i (m) i= = O p () = + 3 U O p () = o p 7 = O p 7 : U i (m) i= Since l () is continuous in te interior of (x) : jj (x) m j (x)j 6 j ; j = ; ; ; () l () attains minimum value e in (). Moreover, we = (@ () =@) = e + = i= i= (@U i () =@) () + () U i () U i () + () U i () = e () = e 4

43 Note tat we already ave g n e; e = i= U i () + () = U i () = e as discussed in Remark. Terefore by (), i= (@U i () =@) () + () U i () = ; = e n i () =@ = diag ji (x) (X i x) jo. Denote H 3 = diag f j g, ten D i (x) = (@U i () =@) H 3 and g n e; e = i= D i (x) () + () = : U i () = e Proof of Teorem Proof. Taking derivatives of g n (; ) and g n (; ) and evaluating at (m; ), we n (m; n (m; n (m; ) n (m; ) i= (@U i () =@) = i= U i (m) U i (m) ; i= D i (x) : X n D i (x) H 3 ; i= 43

44 P Note tat n i= D P i (x) = D n I 3 since D n = n i= ji (x) ((X i x) =) j for j = ; ;. By Taylor expanding g n e; e and g n e; e at (m; ), we ave = g n e; e = g n (m; ) n (m; ) e = U i (m) D n H 3 e m i= = g n e; e n (m; e + o p () e U i (m) U i (m) i= = g n (m; ) n (m; ) e m n (m; ) e p () = + e m D n I 3 e + o p () ; + o p () ; were = H 3 e m + e. Hence we ave H3 e e m = g P n i= U i (m) + o p () o p () ; were g = D ni 3 P n i= U i (m) U i (m) D n I 3 C A p! U f 3 (x) DI 3 f 3 (x) DI 3 C A : P By tis and n i= U i (m) = O p () = +, 3 we know tat = O p () = + 3. For te limit distribution of e, we ave H 3 e m X n = Dn U i (m) + o p () = + 3 ; i= 44

45 tat is, for j = ; ;, e j (x) m j (x) = j ji (x) Y i m j (x) (X i x) ji i= = b j (x) m j (x) + o p ji (x) X i x i= j +j = + 3 j : + j o p () = + 3 Tus p +j ej (x) m j (x) = p +j bj (x) m j (x) + p +j o p + o p + p 7 = p +j bj (x) m j (x) +j = + 3 : j 45

46 spd spd first derivative first derivative option price option price Appendix B: Figures Price Function strike Price Function strike. First Strike Derivative. First Strike Derivative strike strike 5 SPD 5 SPD log return log return Figure : Simulation results for n = 5 Left column from top to bottom: Unconstrained estimates C b (X), C c (X), and e rt; C c (X). Rigt column from top to bottom: Constrained estimates C e (X), C f (X), and e rt; C f (X). Legend: Solid black line: True function; Solid blue line: Average estimate; Dot blue line: 95% con dence band; Dot red line: Constraints. 46