Simulation-Based Estimation of Contingent-Claims Prices

Transcription

1 Simulatio-Based Estimatio of Cotiget-Claims Prices Peter C. B. Phillips Yale Uiversity, Uiversity of Aucklad, Uiversity of York, ad Sigapore Maagemet Uiversity Ju Yu Sigapore Maagemet Uiversity A ew methodology is proposed to estimate theoretical prices of fiacial cotiget claims whose values are depedet o some other uderlyig fiacial assets. I the literature, the preferred choice of estimator is usually maximum likelihood (ML). ML has strog asymptotic justificatio but is ot ecessarily the best method i fiite samples. This paper proposes a simulatio-based method. Whe it is used i coectio with ML, it ca improve the fiite-sample performace of the ML estimator while maitaiig its good asymptotic properties. The method is implemeted ad evaluated here i the Black-Scholes optio pricig model ad i the Vasicek bod ad bod optio pricig model. It is especially favored whe the bias i ML is large due to strog persistece i the data or strog oliearity i pricig fuctios. Mote Carlo studies show that the proposed procedures achieve bias reductios over ML estimatio i pricig cotiget claims whe ML is biased. The bias reductios are sometimes accompaied by reductios i variace. Empirical applicatios to U.S. Treasury bills highlight the differeces betwee the bod prices implied by the simulatio-based approach ad those delivered by ML. Some cosequeces for the statistical testig of cotiget-claim pricig models are discussed. (JEL C11, C15, G12) Pricig fiacial cotiget claims, whose values deped o the price of a uderlyig asset, has bee a importat topic i moder fiacial ecoomics. Some well-kow examples iclude Black-Scholes (1973); Merto (1973); Vasicek (1977); Cox, Igersoll, ad Ross (1985); Hesto (1993); Dua (1996); ad Duffie, Pa, ad Sigleto (2000). Ofte the uderlyig asset is assumed We thak Joel Hasbrouck (the editor), David Bates, Ji-Chua Dua, Negjiu Ju, James MacKio, Adria Paga, Mitch Warachka, ad participats at the 2007 Sigapore Ecoometric Study Group meetig, 2007 Iteratioal Symposium o Ecoometric Theory ad Applicatios, 2007 Iteratioal Symposium o Fiacial Egieerig ad Risk, ad 2008 Chia Iteratioal Coferece i Fiace for helpful discussios. The commets from a aoymous referee were especially helpful ad led to sigificat improvemets i the paper. Phillips gratefully ackowledges support from the Natioal Sciece Foudatio uder grat os. SES ad SES Yu gratefully ackowledges support from the Sigapore Miistry of Educatio AcRF Tier 2 fud uder grat o. T206B4301-RS. Sed correspodece to Ju Yu, School of Ecoomics, Sigapore Maagemet Uiversity, 90 Stamford Road, Sigapore yuju@smu.edu.sg. C The Author Published by Oxford Uiversity Press o behalf of The Society for Fiacial Studies. All rights reserved. For Permissios, please jourals.permissios@oxfordjourals.org. doi: /rfs/hhp009 Advace Access publicatio March 2, 2009

2 The Review of Fiacial Studies / v to follow a parametric time-series model, commoly formulated i cotiuous time, ad the price of the cotiget claim, ofte kow as the theoretical price, is derived by usig o-arbitrage argumets. The resultig price of the cotiget claim is a fuctio of the parameters i the time-series model. The fuctioal form of this depedece is almost always complicated ad oliear. Sice the parameters of the uderlyig asset are usually ukow, they are geerally replaced by time-series estimates i the cotiget-claim pricig formulas. Cosequetly, the statistical properties of the theoretical cotigetclaim price estimate critically hige o those of the parameter estimates. For example, the samplig variatio i the estimated cotiget-claim price depeds o the samplig variatio i the estimated parameters. Hece, the choice of method for parameter estimatio is importat ad the topic has received a great deal of attetio i the literature (see, for example, Boyle ad Aathaarayaa 1977; Ball ad Torous 1984; Lo 1986; Cha et al. 1992; Aït-Sahalia 1999). Perhaps the most direct method for parameter estimatio is to use historical time-series data o the uderlyig asset price. It has ofte bee argued that whe the model for the uderlyig asset is correctly specified, the preferred basis for estimatio ad iferece should be maximum likelihood (ML) (see, for example, Ball ad Torous 1984; Lo 1986; Aït-Sahalia 1999, 2002). There are strog reasos for this choice. Primary amog these is the fact that the ML estimator (MLE) has desirable asymptotic properties of cosistecy, ormality, ad efficiecy uder broad coditios (Huber 1967) i statioary time-series settigs. Moreover, whe the MLE is used i pricig formulas, oe aturally expects the good asymptotic properties of the MLE to trasfer over to the correspodig cotiget-claim price. The theoretical price of a cotiget claim is a smooth oliear fuctio of the system parameters beig estimated, so that plug-i estimates of cotiget-claim prices are themselves MLEs i view of the ivariace property of maximum likelihood (e.g., Zeha 1966). I cosequece, these plug-i pricig estimates have all the desirable asymptotic properties of the MLE. Of course, ML is a very geeral tool of estimatio ad iferece so that it has wide applicability i this cotext ad, at least for statioary time series, its good asymptotic properties are well established. The ML approach therefore provides a coveiet framework for estimatio ad iferece i asset pricig models (cf. Lo 1986). Despite its geerally good asymptotic properties, ML is ot ecessarily the best estimatio method for cotiget-claim prices i fiite samples for three reasos. First, sice the price of a cotiget claim is a oliear trasformatio of the system parameters, isertio of eve ubiased estimators ito the pricig formulas will ot assure ubiased estimatio of a cotiget-claim price (Igersoll 1976). The stroger the oliearity, the larger the bias. Secod, although log-spa samples are ow available for may fiacial variables, makig asymptotic properties of ecoometric estimators more relevat, full 3670

3 Simulatio-Based Estimatio of Cotiget Claims Prices data sets are ot always employed i estimatio because of possible structural chages i log-spa data. Whe short-spa samples are used i estimatio, fiite-sample distributios ca be far from the asymptotic theory. Third, i dyamic models that are used for pricig claims that are cotiget o persistet state variables, the MLE of the system parameters may sustai substatial fiite-sample bias eve i very large samples; ad whe biased estimated parameters are iserted ito the pricig formulas, the bias ca be amplified i the resultig estimates of the cotiget-claim price. Phillips ad Yu (2005) reported evidece of sigificat bias i the MLE of short-term iterest rate models. The preset paper shows that bias i the MLE of volatility models ca also be substatial, especially i worst-case scearios where there is persistece ad oliearity. Some past studies i the literature have addressed the fiite-sample properties of estimators of cotiget-claim prices. Boyle ad Aathaarayaa (1977) examied the exact fiite-sample distributio of the estimated Black-Scholes optio price evaluated at a ubiased estimator of the true variace ad showed the resultat estimator to be biased. To remove the bias, Butler ad Schachter (1986) proposed a estimator based o Taylor series expasios. Kight ad Satchell (1997) showed that the estimator of Butler ad Schachter is oly ubiased for at-the-moey optios. Whe ML is used to estimate oe-factor models for short-term iterest rates, Ball ad Torous (1996) ad Chapma ad Pearso (2000) provided evidece of large fiite-sample biases i the mea reversio parameter. Phillips ad Yu (2005) showed that this bias traslates ito bod pricig ad bod optio pricig ad the pricig biases are ecoomically too sigificat to igore. To reduce these biases, Phillips ad Yu (2005) proposed a ew jackkife procedure. While the method proposed by Butler ad Schachter (1986) is fudametally differet from that of Phillips ad Yu (2005), they share a commo limitig property: relative to ML, both these methods trade off the gai that may be achieved i bias reductio with a loss that arises through icreased variace. The preset paper itroduces a ew methodology of estimatig cotigetclaim prices that ca achieve bias reductio as well as variace reductio, thereby offerig overall gais i mea square estimatio error for cotigetclaim pricig. Istead of isertig a bias-corrected ML estimator ito the pricig formulas, the approach ivolves the direct estimatio of cotiget-claim prices that is complete with a i-built correctio for bias. The proposed method is simulatio-based ad ivolves multiple stages. I a prelimiary stage, the bias i the price estimator is calibrated via simulatio ad at the ext stage a procedure that accouts for this bias is implemeted. Simulatio-based methods have bee successfully used i past work to estimate parameters i various fiacial time-series models. For example, they have bee employed i the cotext of cotiuous-time models to address issues of discretizatio bias (e.g., Duffie ad Sigleto 1993; Mofort 1996; Dai ad Sigleto 2000) ad i the cotext of discrete-time stochastic volatility models 3671

4 The Review of Fiacial Studies / v to deal with itractable likelihoods (e.g., Mofardii 1998; Aderse, Chug, ad Lud 1999). The methods have also bee utilized to correct fiite-sample bias i time-series models (e.g., MacKio ad Smith 1998; Gourieroux, Reault, ad Touzi 2000) ad i dyamic pael models (e.g., Gourieroux, Phillips, ad Yu 2007). The preset work is, to the best of our kowledge, the first implemetatio of such methods i cotiget-claim pricig. Simulatio-based methods have several favorable attributes i the estimatio of cotiget-claim prices. The first is that they do ot require explicit aalytic evaluatio of the bias fuctio sice this fuctio is implicitly calculated by simulatio. This advatage is sigificat as most asset pricig models do ot yield aalytic expressios for the bias fuctio. Simulatio-based methods are therefore applicable i a broad rage of model specificatios where aalytic methods fail. Secod, the simulatio approach described here ca be used i coectio with may differet estimatio methods, icludig exact ML whe it is available, ad various approximate ML techiques. I recet years, buildig upo the pioeerig work of Aït-Sahalia (1999, 2002), a extesive literature has emerged that develops ad applies closed-form ML methods to estimate model parameters i various setups. For example, Aït-Sahalia (1999) ad Egorov, Li, ad Xu (2003) estimated short-term iterest rate models. Aït-Sahalia (2007) geeralized the techique to multivariate diffusios. Aït-Sahalia ad Kimmel (2006); Egorov, Li, ad Ng (2008); ad Thompso (2008) estimated term structure models. Aït-Sahalia ad Kimmel (2007) estimated stochastic volatility models. There are two ice features about the closed-form ML method: (1) the estimator ca approximate the exact MLE highly accurately; ad (2) beig based o a closed aalytic form, it is computatioally efficiet. Whe the simulatio-based method is used i coectio with the exact or the closedform MLE, the resultat estimator is asymptotically equivalet, thereby sharig all the asymptotic properties of the iitial MLE, ad stadard tools of statistical iferece are applicable. I this sese, the estimator may be regarded as a extesio of the closed-form MLE. Third, the preset methods ca deal with both the estimatio bias ad the discretizatio bias that arises whe oliear stochastic differetial equatios are estimated. Sice oliear stochastic differetial equatios typically do ot have closed-form likelihood expressios, exact ML estimatio presets may challeges. While it is straightforward to estimate a discretized model, discretizatio bias is ievitably itroduced i practice. Simulatios permit the samplig iterval to be chose arbitrarily small, thereby providig a importat cotrol o the size of the discretizatio bias. Fourth, simulatio-based methods have the advatage of flexibility ad ca be readily applied i ay practical cotiget-claim pricig situatio. Oe drawback of simulatio-based methods is that they are ievitably computatioally itesive. But umerical methods are ow a importat aspect of most empirical procedures i fiace ad ogoig advaces i 3672

5 Simulatio-Based Estimatio of Cotiget Claims Prices Desity of MLE Desity of Simulatio based Est Actual Bod Price 0.15 Desity Bod Price Figure 1 Distributios of simulatio-based ad ML estimates of bod price for highly persistet data To obtai the distributios of simulatio-based ad ML estimates of bod price, we simulate 5000 data sets from the Vasicek model ds(t) = κ(μ S(t))dt + σdb(t), each with 7500 daily observatios (30 years of daily iterest rates), ad the estimate the price of a three-year discout bod. We choose κ = as a highly depedet case. The graphs show the kerel desity of simulatio-based ad ML estimates of bod price. The solid lie is for the MLE; the dashed lie is for the simulatio-based estimates; the dotted lie is the true value. computig techology cotiue to make umerically itesive computatios less burdesome i practical applicatios. Moreover, the computatioal efficiecy of the closed-form MLE makes it a ideal iitial estimator for our simulatio-based methods. Aother characteristic of simulatio-based methods is that they lack exact reproducibility uless commo seeds ad radom umber geerators are used. This is because the umber of simulatio paths is ievitably fiite i practical applicatios. Our fidigs here idicate that simulatio-based methods provide substatial improvemets i pricig cotiget claims over ML i the case where ML has a substatial bias. To illustrate, Figure 1 compares the distributios of estimates of the price of a discout bod obtaied from 30-year daily data by usig the MLE ad a bias-corrected simulatio method, both i the cotext of a highly persistet Vasicek model. The actual bod price i this case is $ As is apparet i the figure, the simulatio-based estimates are much better cetered o the true bod price ad achieve bias reductio. I additio, the bias reductio comes with a reductio i variace. I fact, the gai i the percetage bias achieved by the simulatio-based method is 64.8% ad the gai i stadard error is 14.59%. However, whe the bias i the MLE is ot substatial, ML may well provide the best estimator i fiite samples. I this evet, simulatiobased estimators typically do ot provide ay improvemet over ML. Figure 2 compares the distributios of estimates of the price of a discout bod obtaied from 30-year daily data by usig the MLE ad a bias-corrected simulatio method, both i the cotext of a less persistet Vasicek model. I this case, the 3673

6 The Review of Fiacial Studies / v Desity of MLE Desity of Simulatio Est Actual Bod Price Desity Bod Price Figure 2 Distributios of simulatio-based ad ML estimates of bod price for less persistet data To obtai the distributios of simulatio-based ad ML estimates of bod price, we simulate 5000 data sets from the Vasicek model ds(t) = κ(μ S(t))dt + σdb(t), each with 7500 daily observatios (30 years of daily iterest rates), ad the estimate the price of a three-year discout bod. We choose κ = 5 as a less depedet case. The graphs show the kerel desity of simulatio-based ad ML estimates of bod price. The solid lie is for the MLE; the dashed lie is for the simulatio-based estimates; the dotted lie is the true value. bias i ML is egligible ad the two desities are almost idetical. More details of this implemetatio ad compariso are provided i Sectio 2. While simulatio methods ca offer improvemets over ML whe the latter suffers fiite-sample problems, ML cotiues to play a importat role for several reasos. First, i may empirically relevat situatios, ML does have good fiite-sample properties. Secod, eve i cases where ML may have iferior fiite-sample performace, it ca still provide a useful first-stage method o which to base the simulatio-based methods, as it does here. Third, the good asymptotic behavior of ML will be iherited by suitably desiged simulatiobased methods that rely o ML. The paper is orgaized as follows. Sectio 1 reviews some existig methods ad motivates ad itroduces our simulatio-based methods. Usig simulated data, Sectio 2 explais how the simulatio-based methods ca be implemeted i relatio to ML estimatio of call optios prices i the cotext of the Black-Scholes model ad of bod prices i the cotext of the Vasicek model. The performace of these simulatio-based estimates is compared with that of ML. Sectio 3 shows how the simulatio-based methods ca be used to address simultaeously the estimatio bias i pricig ad the discretizatio bias. Sectio 4 examies the practical effects of simulatio-based methods i a empirical applicatio with mothly zero-coupo bod data. Sectio 5 cocludes ad outlies some further applicatios ad implicatios of the approach. 3674

7 Simulatio-Based Estimatio of Cotiget Claims Prices 1. Estimatio Methods for Cotiget-Claim Prices 1.1 Maximum likelihood ad idirect iferece Let S(t) deote the price of a uderlyig asset whose dyamics are captured by the followig stochastic differetial equatio: ds(t) = μ(s(t), t; θ)dt + σ(s(t), t; θ)db(t), (1) where B(t) is a stadard Browia motio, σ(s(t), t; θ) is some specified diffusio fuctio, μ(s(t), t; θ) is a give drift fuctio, ad θ is a ukow parameter or a vector of ukow parameters. This class of parametric model has bee widely used to characterize the temporal dyamics of fiacial variables, icludig stock prices, iterest rates, ad exchage rates. Although we use a cotiuous-time model here for S(t), the proposed simulatio-based methods will apply more geerally to other time-seriesgeeratig models for S(t). Uless specified otherwise, the market price of risk is assumed to be zero i this paper. Cosequetly, the physical measure is idetical to the risk-eutral measure. Suppose a sequece of time-series observatios S = (S h, S 2h,...,S h ) take with a samplig iterval h is available over a time period [0, T (= h)] ad we wish to price a fiacial asset whose payoff is cotiget upo the value of S(t). Whe there is o cofusio, we write these observatios as {S t } t=1.usig the o-arbitrage argumet, oe ca derive the price of the cotiget claim. Deote by P(θ) the price of this cotiget claim. I geeral, P may also deped o other parameters that occur i the settig ad such depedecies ca be accouted for i our approach. But for coveiece ad expositio, we write P as a fuctio solely of θ. A commo strategy for estimatig P(θ) is to first estimate the parameter vector from the uderlyig model (such as Equatio (1)) based o the data S, leadig to the estimate ˆθ, ad the proceed to isert ˆθ i the pricig fuctio P, givig ˆP = P(ˆθ). It has bee argued that oe should use ML to estimate θ wheever ML is feasible (see Aït-Sahalia 2002 ad Durham ad Gallat 2002). Sice the model (1) has the Markov property, oe ca write the log-likelihood fuctio as l(θ) = l f (S t S t 1 ; θ), (2) t=2 where f (S t S t 1 ) deotes the coditioal desity fuctio of S t give S t 1. Maximizig the log-likelihood fuctio with respect to θ leads to the MLE ˆθ ML, which is cosistet, asymptotically ormal, ad asymptotically efficiet uder usual regularity coditios for statioary dyamic models. I such 3675

8 The Review of Fiacial Studies / v circumstaces, the limit distributio of ˆθ ML is give by (ˆθ ML T θ ) d N(0, I 1 (θ)), (3) where I (θ) is the limitig iformatio matrix, ad the MLE is cosidered optimal i the Hajék-LeCam sese, achievig the Cramér-Rao boud ad havig the highest possible estimatio precisio i the limit whe. By virtue of the priciple of ivariace, the MLE of P(θ) is obtaied simply by replacig θ i P(θ) with ˆθ ML, leadig to ˆP ML = P(ˆθ ML ).1 By stadard delta method argumets, the followig asymptotic behavior for ˆP ML holds: ( ˆP ML P ) d N(0, VP ), (4) where V P = P θ I 1 (θ) P θ. (5) Sice the estimator ˆP ML is the MLE, it has the highest possible precisio whe, ad i cosequece, this plug-i estimator has bee argued to be the preferred approach (see, for example, Lo 1986). There are at least two problems with this use of the exact ML approach. First, to calculate the exact MLE, oe eeds a closed-form expressio for l f (S t S t 1 ; θ), which is available oly i rare cases. Recet years have witessed a growig iterest i approximatig l f (S t S t 1 ; θ) with closedform approximatios. Importat cotributios iclude Aït-Sahalia (1999, 2002, 2007) ad Aït-Sahalia ad Yu (2006). Sectio 3 describes how to deal with this difficulty i the preset cotext. Secod, while ˆP ML has the highest possible precisio asymptotically, it does ot ecessarily perform the best i fiite samples, where it may suffer substatial bias. For example, whe the time-series behavior i S t is highly persistet ad μ(s t ; θ) is a affie fuctio i S t, (e.g., μ(s t ; θ) = κ(μ S t )), the exact MLE of κ is substatially upward biased eve i large samples (Phillips ad Yu 2005). This upward bias i κ ML traslates to bias i ˆP ML ad may be large eough to be of ecoomic sigificace i practice. To appreciate the circumstaces where ˆP ML suffers bias, expad P(ˆθ ML ) aroud θ (assumig P(θ) to be twice differetiable ad θ to be scalar) ad, takig expectatios, we have the approximate expressio E ( ˆP ML ) P(θ) + E (ˆθ ML θ ) P(θ) + 1 ML) 2 P(θ) Var(ˆθ. (6) θ 2 θ 2 Equatio (6) idicates three situatios where ˆP ML will icur substatial bias: first, whe ˆθ ML is itself strogly biased; secod, whe P(θ) is highly oliear 1 While we assume i this paper that P is ot a fuctio of the uderlyig asset price, this assumptio may be relaxed ad, coditioal o the uderlyig asset price, the priciple of ivariace still applies. 3676

9 Simulatio-Based Estimatio of Cotiget Claims Prices ad P/ θ is large; ad third, whe Var(ˆθ ML ) is large, which is typically the case i small-sample situatios. To illustrate the possible fiite-sample problems that ca arise with MLE, we cosider three examples here. These ivolve worst-case scearios to illustrate the difficulties. A wider set of examples is give i Sectio 2 for some of which MLE performs very well. I the first example, we estimate the price of a very deep out-of-the-moey optio i the cotext of the Black-Scholes model. I the secod example, we estimate a bod price ad a bod optio price i the cotext of the Vasicek model. The third example looks at the optio price i the cotext of the stochastic volatility model of Hull ad White (1987). Some further details of the examples are give i Sectio 2. I the first example, let S(t) be the price of a uderlyig stock at time t, which is assumed to follow the geometric Browia motio process (Black ad Scholes 1973): ds(t) = μs(t)dt + σs(t)db(t), (7) ad let {S t } t=0 be a sample of equispaced time-series observatios o S(t) with samplig iterval h ad T = h. I the Black-Scholes optio pricig formula, the oly ukow quatity is σ 2. Sice ˆσ 2,ML 1 1 T t=0 (l S t+1 S t 1 1 t=0 l S t+1 S t ) 2 is the MLE of σ 2, ˆP ML = P(ˆσ 2,ML )isthemleofp, a estimator advocated i Lo (1986). Moreover, Lo showed that ( ˆP ML T P ) d N (0, τ ) 2 S2 σ 2 φ 2 (d 1 ), (8) where τ is the time to maturity ad φ is the desity of the stadard ormal distributio. We use 250 (simulated) daily stock returs (i.e., h = 1/250) to obtai the ML estimates of σ 2 ad the price of a deep out-of-the-moey Europea call optio that matures i oe week. The experimet is replicated 5000 times to obtai the mea, the percetage bias, ad the root mea square error (RMSE). Table 1 reports o the results. It ca be see that while the MLE of σ 2 has little bias ( 0.48%), the bias i ˆP ML sigificat. Obviously, i this case the bias i ˆP ML the oliearity i the fuctio P. is substatial (19.6%) ad ecoomically comes almost etirely from To uderstad why the bias is so severe i ˆP ML whe there is little bias i,weplotifigure3 l P(σ 2 )/ σ as a fuctio of σ 2 for optios with ˆσ 2,ML differet degree of moeyess. It ca be see that the deep out-of-the-moey optio is highly oliear while the other two optios are early liear. As a result, the secod ad third terms o the right-had side of Equatio (6) are egligible whe pricig i-the-moey ad ear-moey optios but oegligible whe pricig a deep-out-of-the-moey optio. I Sectio 2, we will provide examples where the bias i ˆP ML i-the-moey or ear-the-moey. is egligible whe the moeyess is 3677

10 The Review of Fiacial Studies / v Table 1 Fiite-sample properties of MLE of σ 2 ad P i the Black-Scholes model σ 2,ML True Mea Bias (i %) RMSE The table reports o the true value, the mea, the bias (i percetage), ad the RMSE of MLE of σ 2 ad P i the Black-Scholes model obtaied from simulatios. We simulate 5000 data sets from the Black-Scholes model each with 250 daily observatios. ds(t) = μs(t)dt + σs(t)db(t), ˆP ML Deep Out of the moey At the moey I the moey l(p)/ σ σ 2 Figure 3 Derivative of the Black-Scholes price with respect to volatility as a fuctio volatility The graphs plot l P(σ 2 )/ σ as a fuctio of σ 2 for optios with differet degree of moeyess. P(σ 2 )is the Black-Scholes price defied by S (d 1 ) Xe rτ (d 2 ), where d 1 = (l(s/ X) + (r + 0.5σ 2 )τ)/σ τ ad d 2 = d 1 σ τ. I the secod example, the short-term iterest rate S(t) is assumed to follow the Orstei-Uhlebeck process (Vasicek 1977): ds(t) = κ(μ S(t))dt + σdb(t), (9) ad {S t } t=1 is a sample of equispaced time-series observatios o S(t) over [0, T (= h)] with samplig iterval h. I the Vasicek bod pricig formula, the ukow quatities are κ, μ, ad σ 2. It is kow that μ ad σ 2 ca be estimated with little bias by exact ML (Tag ad Che 2007), so we fix these two parameters ad let κ be the oly ukow parameter i the simulatio. We use 7500 (simulated) daily observatios (i.e., h = 1/250) to obtai the ML estimates of κ, the price of a three-year discout bod (Bod Price or BP 3678

11 Simulatio-Based Estimatio of Cotiget Claims Prices Table 2 Fiite-sample properties of MLE of κ, BP, adop i the Vasicek model κ ML BP ML ÔP ML True Mea Bias (i %) RMSE The table reports o the true value, the mea, the bias (i percetage), ad the RMSE of MLE of κ, BP, adop i the Vasicek model obtaied from simulatios. We simulate 5000 data sets from the Vasicek model ds(t) = κ(μ S(t))dt + σdb(t), each with 7500 daily observatios, ad price a three-year discout bod ad a two-year Europea call optio writte o the discout bod. BP is defied i Equatio (21) while OP is defied i Equatio (22). hereafter), ad the price of a two-year Europea call optio o the discout bod (Optio Price or OP hereafter). The experimet is replicated 5000 times to obtai the mea, the percetage bias, ad the RMSE. Table 2 reports o the results. It ca be see that the bias i κ is substatial ad sice the true value of κ = 0.018, this may be iterpreted as a maifestatio of the ear uit root problem. The bias aturally traslates ito BP ML ad ÔPML, which is of ecoomic sigificace (see, for example, Hull 2000). I the last example, S(t) is a stock price, which is assumed to follow the stochastic volatility (SV) model (Hull ad White 1987): ds(t) = σ S S(t)σ(t)dB 1 (t), d l σ 2 (t) = κl σ 2 (t)dt + γdb 2 (t), ad {S t } t=1 is agai a sample of equispaced time-series observatios o S(t) with samplig iterval h. Uder certai assumptios, Hull ad White (1987) showed that the value of a Europea call optio is the Black-Scholes price itegrated over the distributio of the mea volatility. Ufortuately, the optio price does ot have a closed-form solutio. A flexible way for calculatig optio prices is via Mote Carlo simulatios. For example, Hull ad White (1987) desiged a efficiet procedure of carryig out the Mote Carlo simulatio to calculate a Europea call optio. I geeral, the price depeds o κ, σ S, ad γ. For the SV model, it is well kow that the likelihood fuctio has o closed-form expressio (Durham ad Gallat 2002; Kim, Shephard, ad Chib 1998). 2 Several simulatio-based ML methods have bee proposed i recet years. I this paper, a discretized versio of the SV model is estimated by the ML method of Skaug ad Yu (2007). We use 500 (simulated) daily 2 Whe oly price data are used to estimate the SV model, the latet volatility process has to be itegrated out from the joit desity of prices ad volatility, makig the evaluatio of likelihood umerically demadig. However, whe the latet volatility is obtaied from optio prices, Aït-Sahalia ad Kimmel (2007) showed that a approximate ML is feasible. I this case, if volatility is highly persistet, the same fiite-sample problem i ML ca be expected to occur as for the persistet Vasicek model. 3679

12 The Review of Fiacial Studies / v Table 3 Fiite-sample properties of MLE of σ S, κ, γ,adp i the logormal SV model σ ML S, κ ML γ ML ˆP ML True Mea Bias (i %) RMSE The table reports o the true value, the mea, the bias (i percetage), ad the RMSE of MLE of σ S, κ, γ, ad P i the logormal stochastic volatility model obtaied from simulatios. We simulate 500 data sets from the logormal stochastic volatility model of Hull ad White (1987), each with 500 daily observatios. ds(t) = σ S S(t)σ(t)dB 1 (t), d l σ 2 (t) = κl σ 2 (t)dt + γdb 2 (t), observatios (h = 1/250) to obtai the ML estimates of σ S, κ, γ, ad the price of a out-of-the-moey Europea call optio. 3 The optio prices are calculated based o 1000 simulated paths. The experimet is replicated 500 times to obtai the mea, the percetage bias, ad the RMSE. Table 3 reports o the results. As i the Vasicek model, the bias i κ is substatial ad is agai a maifestatio of the ear-uit root problem. 4 The bias aturally traslates ito ˆP ML ad is ecoomically very sigificat. To the best of our kowledge, this bias i estimatig κ ad i pricig seems ot to have bee oticed i the cotext of SV models. All three examples clearly poit to a eed to correct the bias i ML estimatio of cotiget-claim prices. The first example suggests that isertio of a bias-corrected parameter estimate ito the cotiget-claim price does ot ecessarily work well. The preset paper seeks to address this problem by the use of the idirect iferece procedure applied directly to the cotiget-claim price. Idirect iferece is a simulatio-based method developed by Smith (1993) for estimatig models where the likelihood is difficult to costruct aalytically but where the model may be readily simulated. 5 It is closely related to the simulated GMM method of Duffie ad Sigleto (1993) ad the efficiet method of momets (EMM) techique of Gallat ad Tauche (1996). This method also has the property that it ca successfully correct for estimatio bias i time-series parameter estimatio. The applicatio of idirect iferece here proceeds as follows. Let ˆθ ML T deote the MLE that is obtaied from the actual data ad ivolves some fiite-sample estimatio bias. For ay give parameter choice θ, let S k (θ) ={ S k 1, S k 2,..., S k } be data simulated from the time-series 3 The spot price is $10, the time to maturity for the optio cotract is 0.5 years, the iterest rate is 10%, the strike price is $11.56, ad the iitial value of σ 2 is Sice the samplig iterval is very small at the daily frequecy, the discretizatio bias is egligible (Phillips ad Yu 2007). 5 The ame idirect iferece was coied by Gourieroux, Mofort, ad Reault (1993), who further developed the methodology. 3680

13 Simulatio-Based Estimatio of Cotiget Claims Prices model (1), where k = 1,...,K ad K is the umber of simulated paths. The umber of observatios i S k (θ) is chose to be the same as the umber of actual observatios i S so that the exact fiite-sample properties of ˆθ ML,icludig its fiite-sample bias, may be calibrated. Let φ ML,k (θ) deote the MLE of θ obtaied i this way from the kth simulated path. By costructio, this simulatio-based estimate aturally carries ay fiite-sample estimatio bias of the MLE i the give model ad for this sample size. The idea behid the procedure that leads to bias correctio is to choose θ so that the average behavior of φ ML,k (θ) is matched agaist the umerical estimate ˆθ ML obtaied with the observed data. I particular, the idirect iferece estimator is defied by ˆθ,K II = argmi θ ˆθ ML 1 K φ ML,k (θ) K, (10) where is some fiite-dimesioal distace metric ad the regio of extremum estimatio is a compact set. I the case where K teds to ifiity, the law of large umbers, K 1 K ML,k k=1 φ (θ) p E( φ ML,k (θ)), applies by virtue of the ature of the simulatio ad the the idirect iferece estimator becomes k=1 ˆθ II = argmi θ ˆθ ML b (θ), (11) where b (θ) = E( φ ML,k (θ)) is called the bidig fuctio. Whe b is ivertible, the idirect iferece estimator may be writte directly as ˆθ II = b 1 ( θ ML T ). The procedure essetially builds i a fiite-sample bias correctio to ˆθ ML, with the bias beig computed directly by simulatio. Ay bias that occurs i ˆθ ML will also be preset i the bidig fuctio b (θ). Hece, with the bias correctio that is built ito the iversio fuctioal ˆθ II = b 1 ML (ˆθ ), the estimator ˆθ II becomes exactly b -mea-ubiased for θ. That is, E(b (ˆθ II)) = b (θ). Moreover, i typical cases where lim E(ˆθ ML ) = θ ad ˆθ ML is asymptotically ubiased, we have ˆθ II ˆθ ML i the limit as. The, the idirect iferece estimator is asymptotically equivalet to the MLE so that ˆθ II shares the same good asymptotic properties of ˆθ ML, while havig improved fiite-sample performace. 1.2 Direct simulatio-based methods of pricig While the idirect iferece estimator of θ, ˆθ II, may have better fiite-sample properties tha ˆθ ML, isertig ˆθ II ito P(θ) does ot ecessarily lead to a better estimator tha ˆP ML due to the oliearity i the pricig fuctio. This pheomeo was explicitly addressed i Phillips ad Yu (2005), where it was foud that jackkifig the quatity of iterest clearly performs better tha 3681

14 The Review of Fiacial Studies / v the method that iserts the jackkife (hece bias-reduced) estimator ad the media ubiased (hece bias-corrected) estimator ito the pricig formulas. To improve the fiite-sample properties of ˆP ML, we propose to apply simulatiobased methods directly i the estimatio of cotiget-claim prices. We first focus o the case where θ is a scalar. As above, we deote by ˆθ ML the MLE of θ that is obtaied from the actual data, ad write ˆP ML = P(ˆθ ML ). ˆP ML ivolves fiite-sample estimatio bias due to the oliearity of the pricig fuctio P i θ, or the use of the biased estimate ˆθ ML, or both these effects. The simulatio approach ivolves the followig steps. (1) Give a value for the cotiget-claim price p, compute P 1 (p) (call it θ(p)), where P 1 ( ) is the iverse of the pricig fuctio P(θ). (2) Let S k (p) ={ S k 1, S k 2,..., S k T } be data simulated from the time-series model (1) give θ(p), where k = 1,...,K with K beig the umber of simulated paths. As argued above, we choose the umber of observatios i S k (p) to be the same as the umber of actual observatios i S for the express purpose of fiite-sample bias calibratio. (3) Obtai φ ML,k (p), the MLE of θ, from the kth simulated path, ad calculate P ML,k (p) = P( φ ML,k (p)). (4) Choose p so that the average behavior of P ML,k (p) is matched with ˆP ML to produce a ew bias-corrected estimate. Wheever bias occurs i ˆP ML ad from whatever source, this bias will also be preset i P ML,k (p) for the same reasos. Hece, the procedure builds i a fiite-sample bias correctio directly to correct ˆP ML. The resultat estimator is differet from simply isertig a simulatio-based estimator of θ ito the pricig fuctioal P, because this approach cosiders the quatity of iterest directly. We propose usig two quatities to represet the average behavior of P( φ ML,k (p)) as the bidig fuctio. The first oe is the mea, which correspods to the idirect iferece estimatio approach of Smith (1993) ad Gourieroux, Mofort, ad Reault (1993), while the secod is the media, correspodig to the media ubiased estimatio approach of Adrews (1993). Of course, the media is more robust to outliers tha the mea. Hece, whe the is highly skewed, it may be preferable to use the media i this approach. I geeral, however, the bidig fuctio caot be computed aalytically i either case ad simulatios are eeded to calculate the bidig fuctios. If the mea is chose to be the bidig fuctio, the simulatio-based estimator is defied as distributio of ˆP ML ˆP SM,1,K = argmi p ˆP ML 1 K K P ( φ ML,k (p) ). (12) k=1 3682

15 Simulatio-Based Estimatio of Cotiget Claims Prices I the case where K teds to ifiity, this simulatio-based estimator becomes ˆP SM,1 = argmi p ˆP ML b,1 (p), (13) where the bidig fuctio b,1 (p) ise(p( φ ML,k (p))). If b,1 (p) is ivertible, we the have ˆP SM,2,K ˆP SM,1 = b 1,1 ( ˆP ML ). (14) If the media is chose to be the bidig fuctio, the simulatio estimator is defied as = argmi p ˆP ML ˆρ 0.5 P ( φ ML,k (p) ), (15) where ˆρ τ is the τth sample quatile obtaied from {P( φ ML,1 (p)),..., P( φ ML,K (p))}. 6 I the case where K teds to ifiity, this simulatio-based estimator becomes ˆP SM,2 = argmi p ˆP ML b,2 (p), (16) where the bidig fuctio b,2 (p)isρ 0.5 (P( φ ML,k (p))). If b,2 (p) is ivertible, we have ˆP SM,2 = b 1,2 ( ˆP ML ). (17) Equatio (14) implies that ˆP SM,1 is exactly b -mea-ubiased for θ i the sese that E(b,1 ( ˆP SM,1 )) = b,1 (P). Similarly, from Equatio (17) it ca be show that ˆP SM,2 is exactly b -media-ubiased for θ i the sese that ρ 0.5 (b,2 ( ˆP SM,2 )) = b,2 (P). If b,1 (P) is liear i P, exact b -meaubiasedess implies exact mea ubiasedess, i.e., E( ˆP SM,1 ) = P.Ifb,2 (P) is strictly mootoic i P, exact b -media-ubiasedess implies exact media ubiasedess, i.e., ρ 0.5 ( ˆP SM,1 ) = P. Thus, the sufficiet coditio for esurig exact mea ubiasedess of ˆP SM,1 is stroger tha the sufficiet coditio for esurig exact media ubiasedess of ˆP SM,2. Whe lim E( ˆP ML ) = lim ρ 0.5 ( ˆP ML ) = P ad the slopes of the fuctios b,1 (P) ad b,2 (P) are uity as, the two simulatio-based estimators ˆP SM,1 ad ˆP SM,2 are asymptotically equivalet to the MLE ˆP ML. Applyig the delta method to Equatios (14) ad (17), we obtai for i = 1, 2 ad ˆP SM,i Var ( ˆP SM,i = b 1,i ( ˆP ML ) ( b,i (P) P ) = b 1,i ) 2 Var ( ˆP ML ( b,i (P) + ˆP ML b,i (P) ), ( ) ) b,i (P) 2 V P P, (18) 6 A umber m is the τth quatile of a radom variable X if Prob(X m) = 1 τ ad Prob(X < m) = τ. The sample quatile is the sample couterpart of the quatile. 3683

16 The Review of Fiacial Studies / v where V P is give i Equatio (5). The asymptotic approximatio (18) suggests that the simulatio-based estimators should iherit some of the efficiecy properties of the ML estimator. I fact, the chage i the variace depeds largely o b,i (P)/ P, the slope of the bidig fuctio, as see above. has a smaller variace tha the MLE, ad for b,i (P)/ P < 1, ˆP SM,i has a larger variace tha the MLE. We ow cosider the case where θ is a M θ -dimesioal vector. Deote by ˆθ ML the MLE of θ, obtaied from actual data. A importat first step i the simulatio-based method is to back out θ from cotiget-claim prices. To achieve idetificatio, we have to estimate M p M θ cotiget-claim prices p to esure the existece ad uiqueess of the iverse mappig P 1 (p). These cotiget claims may differ i maturities, strike prices, or other features. If the umber of cotiget claims M p exceeds M θ, the iverse P 1 (p) will ot geerally exist uless the equatios p = P(θ) are fully cosistet, although we may compute the least squares solutio: For b,i (P)/ P > 1, ˆP SM,i θ mi = argmi P(θ) p, P (θ) p = (P (θ) p) (P(θ) p). θ If the dimesio M θ of θ outumbers the cotiget claims M p, the there is geerally isufficiet iformatio to recover θ from p = P (θ) ad θ is ot idetified. We will therefore assume i what follows that M θ = M p ad that P is ivertible. After the iversio, the same steps are used to obtai the simulatio-based estimator of P. Sice P is ow multidimesioal, Equatio (18) becomes Var ( ˆP SM,i ) ( b,i (P) P ( b,i (P) P ) 1 Var ( ˆP ML ) 1 V P ) 1 ) ( b,i (P) P ( ) b,i (P) 1. (19) To reduce the computatio cost, oe ca choose a fie grid of discrete poits, P, from a exteded Euclidea space ad obtai the bidig fuctio o the grid via simulatios. The stadard iterpolatio ad extrapolatio methods ca be used to approximate the bidig fuctios at ay poit. I this paper, a liear iterpolatio ad extrapolatio method is used. 1.3 Simulatio-based methods for cross-sectioal data Ulike the time-series case where the gold stadard method of estimatio is ML, i the cross-sectio case o sigle estimatio method is regarded as a gold stadard. To apply the simulatio-based methods, the models have to be assumed for the theoretical cotiget-claim prices ad for the relatio betwee the theoretical prices ad observed prices. Suppose τ i ad X i are, respectively, the time-to-maturity ad the strike price of a optio, where i = 1,...,.Let ˆP i (τ i, X i ) be its observed price ad P 3684

17 Simulatio-Based Estimatio of Cotiget Claims Prices P i (θ; τ i, X i ) its model price as determied by the optio formula correspodig to model (1). Assume that ˆP i = P i (θ; τ i, X i ) + ɛ i, ɛ i N ( 0, σ 2 e). (20) Uder this specificatio, we ca use OLS to estimate θ from cross-sectioal data o optio prices { ˆP i } i=1,givig A estimate of σe 2 is obtaied by ˆθ = argmi θ ( ˆP i P i (θ; τ i, X i )) 2. ˆσ 2 e = 1 1 ( ˆP i P i (ˆθ; τ i, X i )) 2. i=1 To estimate the price of a optio with a ew time-to-maturity ad a ew strike price (say, τ +1 ad X +1, respectively), a commoly used estimator i the literature (see, for example, Bakshi, Cao, ad Che 1997) is ˆP +1 = P +1 (ˆθ; τ +1, X +1 ). To implemet the simulatio-based method, we suggest the followig steps: (1) Give a value for the optio price p, compute the implied parameter at the ew time-to-maturity ad a ew strike price. Defie the implied parameter by P 1 +1 (p; τ +1, X +1 ). (2) Let P k (p) = ( P k 1 (p),..., P k (p)) be data simulated from (20), give P 1 +1 (p; τ +1, X +1 ), {τ i, X i } i=1, ˆσ2 e, where k = 1,...,K with K beig the umber of simulated paths. (3) From the kth simulated path, obtai the OLS estimate of θ (call it θ k ) ad the estimate of P +1 (call it P k +1 ). (4) Choose p so that the average behavior of P k +1 (p) is matched with ˆP +1 to produce a simulatio-based estimate. 2. Illustratios ad Mote Carlo Evidece This sectio illustrates the bias problem i the estimatio of cotiget-claim prices i the cotext of both the Black-Scholes optio pricig model ad the Vasicek bod ad optio pricig model. The reaso for cosiderig these two specific models i the Mote Carlo study is that they both have closed-form expressios for the coditioal desities ad we ca therefore perform exact ML estimatio of P, providig a useful bechmark of compariso. Moreover, the cotiget-claim prices have closed-form expressios i these two models. We also discuss situatios whereby ML does ot suffer from bias problems ad simulatio-based methods do ot offer ay improvemet over ML. 3685

18 The Review of Fiacial Studies / v Black-Scholes optio pricig As show i Sectio 1, i the Black-Scholes model, for deep out-of-the-moey optios with a short time-to-maturity, ˆP ML ca be substatially biased. It is therefore of particular iterest to see how bias reductio strategies work i this case. First, we defie the followig otatio: X = Strike price, τ = Time to maturity, r = Iterest rate, ˆσ 2,ML = MLE of σ 2 defied by 1 1 T t=0 (l S t+1 S t 1 1 t=0 l S t+1 S t ) 2, s 2 = Bias-corrected MLE of σ 2 defied by 1 ˆσ2,ML d 1 = 1 σ τ (l(s/ X) + (r + 0.5σ2 )τ), d 2 = d 1 σ τ, T, = Cumulative distributio fuctio of stadard ormal distributio, P = Price of a Europea call optio obtaied from S (d 1 ) Xe rτ (d 2 ). I fiite samples, however, ˆσ 2,ML is slightly biased, while s 2 is ubiased. Isertig s 2 ito P(σ2 ) is a alterative estimator of P that has received a great deal of attetio (see, for example, Boyle ad Aathaarayaa 1977 ad Butler ad Schachter 1986). I particular, Boyle ad Aathaarayaa (1977) obtaied exact fiite-sample momets of P(s 2) ad showed that P(s2 )isa biased estimator of P. They further provided evidece of the small magitude of the bias for ear- ad i-the-moey optios. However, whe the optio is deep out-of-the-moey, the size of the bias becomes large. Based o a Taylor series expasio of the cumulative distributio fuctio of the stadard ormal distributio ad the distributio of the miimum variace ubiased estimator of σ 2, Butler ad Schachter (1986) derived a ubiased estimator of P. Kight ad Satchell (1997) showed that a uiformly miimum variace ubiased estimator of P exists if ad oly if the optio is at-the-moey. We ow compare the performace of some existig methods with the proposed simulatio-based methods usig simulated data. I particular, a simple simulatio study is coducted to compare the performace of ˆP ML P(s 2), ˆP SM,1 values are used:, ad ˆP SM,2,. Throughout the simulatios, the followig parameter S = $100 τ = 5/250 r = 5% = 250 h = 1/250. That is, we use 250 daily stock returs to estimate the price of a Europea call optio that matures i oe week ad obtai the estimates ˆP ML, P(s2 ˆP ), SM,1, ad ˆP SM,2. The experimet is replicated 5000 times to obtai the 3686

19 Simulatio-Based Estimatio of Cotiget Claims Prices Table 4 Fiite-sample properties of ˆP ML, P(s 2 ), ˆP SM,1,ad ˆP SM,2 i the Black-Scholes model for a at-the-moey optio ad a i-the-moey optio At-the-moey optio price I-the-moey optio price True value P = True value P = Estimators ˆP ML P(s 2) ˆP SM,1 ˆP SM,2 ˆP ML P(s 2) ˆP SM,1 ˆP SM,2 Mea Bias (i %) Std err RMSE Media The table reports o the mea, the bias (i percetage), the stadard error, the RMSE, ad the media of ˆP ML, P(s 2), ˆP SM,1,ad ˆP SM,2 i the Black-Scholes model obtaied from simulatios. We simulate 5000 data sets from the Black-Scholes model ds(t) = μs(t)dt + σs(t)db(t), each with 250 daily observatios. The strike price X is set to be 0.95 S exp(rτ) (i-the-moey) ad S exp(rτ) (at-the-moey), respectively, ad σ 2 = 0.4. meas, stadard errors, RMSEs, ad medias of all four estimates. For the two simulatio-based estimates, we choose the umber of simulated paths to be K =5000. It is well kow that σ 2 ca be accurately estimated from daily data. Hece, we expect little fiite-sample bias i ˆσ 2,ML. Table 4 shows the results whe σ 2 = 0.4 ad X = 0.95 S exp(rτ) (i-themoey), X = S exp(r τ) (at-the-moey), respectively. I this case, the actual optio prices are $ ad $ Several coclusios ca be draw from the results reported i the table. First, cosistet with what has bee documeted, we foud that ˆP ML has very small percetage bias ( % ad %). Moreover, it has the smallest variace amog the four estimators. This is ot surprisig sice σ 2 ca be accurately estimated ad there is o strog oliearity i the optios. Secod, compared with ˆP ML, the use of a ubiased plug-i estimator, P(s 2 ), reduces the percetage bias to % ad %, respectively, but slightly icreases the variace. Third, the bias is further reduced by the simulatio estimators ˆP SM,1 (0%) ad ˆP SM,2 (0.0249% ad %). Note that ˆP SM,1 is exactly mea-ubiased ad ˆP SM,2 is exactly media-ubiased. While both simulatio methods offer bias reductio over ˆP ML, they also margially icrease the variace. Fially, all four estimators perform similarly i terms of RMSE. It is ot surprisig that the two simulatio estimators do ot improve over ML because there is little bias i ˆσ 2,ML ad there is little oliearity i P(σ 2 ). Hece, by desig ML is expected to have good fiite-sample properties i this case ad the simulatio estimators have very similar performace. To help uderstad the performace of the two simulatio-based methods, we plot the bidig fuctios i Figure 4 for the at-the-moey optio. Several features are apparet i the figure. First, both bidig fuctios are very close to the 45 lie, suggestig that oly a small amout of bias correctio is eeded i 3687

20 The Review of Fiacial Studies / v degree lie Bidig fuctio 1 Bidig fuctio Bidig Fuctios Optio Price Figure 4 Bidig fuctios of the two simulatio-based methods for the at-the-moey optio The graphs plot b,i (P) as a fuctio of p for i = 1, 2, respectively, obtaied from the Black-Scholes model for a at-the-moey optio, with 5000 data sets simulated, each with 250 daily observatios. The dashed lie is for b,1 (P); the dotted lie for b,2 (P). The 45 lie is plotted for compariso. the two simulatio-based methods i this case. Secod, both bidig fuctios are virtually liear, implyig that ˆP SM,1 should be exactly mea ubiased ad ˆP SM,2 should be exactly media ubiased, a result cosistet with the Mote Carlo fidigs i Table 4. Third, the slopes of the two bidig fuctios are close to but slightly less tha 1, suggestig that the variaces of the two simulatio-based estimators are close to, but slightly larger tha, that of ˆP ML. This fidig also corroborates the results foud i Table 4. I light of the fidig i the literature that stadard estimatio methods ted to geerate large percetage biases for deep out-of-the-moey optios, we desiged a experimet to compare the performace of the four methods whe X = 1.4S exp(rτ) (i.e., a deep out-of-the-moey optio) ad σ 2 = 0.4. This is of course a worst-case sceario but may be practically relevat for some stocks. Table 5 reports o the meas, stadard errors, RMSEs, ad medias, each multiplied by 1000, of all four estimates across 5000 replicatios. The actual call optio price (multiplied by 10,000) is $2.12. Several fidigs emerge from Table 5. First, cosistet with fidigs i the literature, ˆP ML has a large percetage bias (19.60%) eve though the bias i ˆσ 2,ML is very small. Moreover, this estimator o loger has the smallest variace. Secod, compared with ˆP ML, istead of reducig the bias, P(s2 ) icreases the percetage bias to 23.51%, so the effect of pluggig i a ubiased estimator icreases bias. This is a typical example where pluggig the bias-corrected ML estimator ito the cotiget-claim price does ot ecessarily lead to desirable fiite-sample properties. Third ad most importatly, the bias is reduced i ˆP SM,1 (1.13%) i terms of the mea ad i ˆP SM,2 (0%) i terms of the media. The performace of ˆP SM,1 is particularly ecouragig. This estimate ot oly reduces the bias i terms of the mea, but also decreases variace, 3688