Closed-form likelihood approximation and estimation of jump-diffusions with an application to the...

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Journal of Econometrics 141 (2007) 1245–1280

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Closed-form likelihood approximation andestimation of jump-diffusions with an application

to the realignment risk of the Chinese Yuan

Jialin Yu�

Department of Finance and Economics, Columbia University, 421 Uris Hall, 3022 Broadway,

New York, NY 10027, USA

Available online 27 March 2007

Abstract

This paper provides closed-form likelihood approximations for multivariate jump-diffusion

processes widely used in finance. For a fixed order of approximation, the maximum-likelihood

estimator (MLE) computed from this approximate likelihood achieves the asymptotic efficiency of

the true yet uncomputable MLE as the sampling interval shrinks. This method is used to uncover the

realignment probability of the Chinese Yuan. Since February 2002, the market-implied realignment

intensity has increased fivefold. The term structure of the forward realignment rate, which completely

characterizes future realignment probabilities, is hump-shaped and peaks at mid-2004. The

realignment probability responds quickly to economic news releases and government interventions.

r 2007 Elsevier B.V. All rights reserved.

JEL classification: C13; C22; C32; G15

Keywords: Maximum likelihood estimation; Jump diffusion; Discrete sampling; Chinese Yuan; Currency

realignment

1. Introduction

Jump-diffusions are very useful for modeling various economic phenomena such ascurrency crises, financial market crashes, defaults etc. There are now substantial evidence

see front matter r 2007 Elsevier B.V. All rights reserved.

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ARTICLE IN PRESSJ. Yu / Journal of Econometrics 141 (2007) 1245–12801246

of jumps in financial markets. See for example Andersen et al. (2006), Barndorff-Nielsenand Shephard (2004) and Huang and Tauchen (2005) on jumps in modeling andforecasting return volatility, Andersen et al. (2002) and Eraker et al. (2003) on jumps instock market, Piazzesi (2005) and Johannes (2004) on bond market, Bates (1996) oncurrency market, Duffie and Singleton (2003) on credit risk. From a statistical point ofview, jump-diffusion nests diffusion as a special case. Being a more general model, jump-diffusion can approximate the true data-generating process better as measured by, forexample, Kullback–Leibler information criterion (KLIC) (see White, 1982). As illustratedby Merton (1992), jumps model ‘‘Rare Events’’ that can have a big impact over a shortperiod of time. These rare events can have very different implications from diffusive events.For example, it is known in the derivative pricing theory that, with jumps, the arbitragepricing argument leading to the Black–Scholes option pricing formula breaks down. Liuet al. (2003) have shown that the risks brought by jumps and stochastic volatilitydramatically change an investor’s optimal portfolio choice.To estimate jump-diffusions, likelihood-based methods such as maximum-likelihood

estimation are preferred. This optimality is well documented in the statistics literature.However, maximum-likelihood estimation is difficult to implement because the likelihoodfunction is available in closed-form for only a handful of processes. This difficulty can beseen from Sundaresan (2000): ‘‘The challenge to the econometricians is to present aframework for estimating such multivariate diffusion processes, which are becoming moreand more common in financial economics in recent times. . . . The development ofestimation procedures for multivariate AJD processes is certainly a very important steptoward realizing this hope.’’ Note the method proposed in this paper will apply to bothaffine jump-diffusion (AJD in the quote) and non-AJD. The difficulty of obtaining closed-form likelihood function has led to likelihood approximation using simulation (Pedersen,1995, Brandt and Santa-Clara, 2002) or Fourier inversion of the characteristic function(Singleton, 2001; Aıt-Sahalia and Yu, 2006). Generalized method of moments estimation(Hansen and Scheinkman, 1995; Kessler and Sørensen, 1999; Duffie and Glynn,2004; Carrasco et al., 2007) has also been proposed to sidestep likelihood evaluation.Non-parametric and semiparametric estimations have also been proposed for diffusions(Aıt-Sahalia, 1996; Bandi and Phillips, 2003; Kristensen, 2004) and for jump-diffusions(Bandi and Nguyen, 2003).The first step towards a closed-form likelihood approximation is taken by Aıt-Sahalia

(2002) who provides a likelihood expansion for univariate diffusions. Such closed-formapproximations are shown to be extremely accurate and are fast to compute byMonte-Carlo studies (see Jensen and Poulsen, 2002). The method was subsequently refinedby Bakshi and Ju (2005) and Bakshi et al. (2006) under the same setup of univariatediffusions and was extended to multivariate diffusions by Aıt-Sahalia (2006) andunivariate Levy-driven processes by Schaumburg (2001).Building on this closed-form approximation approach, this paper provides a closed-

form approximation of the likelihood function of multivariate jump-diffusion processes. Itextends Aıt-Sahalia (2002) and Aıt-Sahalia (2006) by constructing an alternative form ofleading term that captures the jump behavior. The approximate likelihood function is thensolved from Kolmogorov equations. It extends Schaumburg (2001) by relaxing the i.i.d.property inherent in Levy-driven randomness and by addressing multivariate processes.The maximum-likelihood estimators (MLEs) using the approximate likelihood functionprovided in this paper are shown to achieve the asymptotic efficiency of the true yet

ARTICLE IN PRESSJ. Yu / Journal of Econometrics 141 (2007) 1245–1280 1247

uncomputable MLEs as the sampling interval shrinks for a fixed order of expansion. Thisdiffers from Aıt-Sahalia (2002) where the asymptotic results are obtained when the orderof expansion increases holding fixed the sampling interval. Such infill (small samplinginterval) and long span asymptotics have been employed in non-parametric estimations(see for example Bandi and Phillips, 2003; Bandi and Nguyen, 2003, among others). Thissampling scheme allows the extension of likelihood expansion beyond univariate diffusionsand reducible multivariate diffusions to the more general jump diffusions (see Section 3). Italso permits the use of lower order approximations which is further helped by thenumerical evidence that even a low order approximation can be very accurate for typicallysampling intervals, e.g. daily or weekly data (see Section 2.4.4). The accuracy of theapproximation improves rapidly as higher order correction terms are added or as thesampling interval shrinks. One should caution, however, and not take the asymptoticresults literally by estimating a jump-diffusion model aimed at capturing longer-rundynamics using ultra high frequency data. With ultra high frequency data, one encountersmarket microstructure effects and the parametric model becomes misspecified (see forexample Aıt-Sahalia et al., 2005; Bandi and Russell, 2006). Estimation at such frequenciesas daily or weekly can strike a sensible balance between asymptotic requirements andmarket microstructure effects.

The theory on closed-form likelihood approximation maintains the assumption ofobservable state variables. Nonetheless, these methods can still be applied to models withunobservable states if additional data series are available such as the derivative price seriesin the case of stochastic volatility models. In these cases, one may be able to create amapping between the unobservable volatility variable and the derivative price and conductinference using the approximate likelihood of the observable states and the unobservablestates mapped from the derivative prices. Please see Aıt-Sahalia and Kimmel (2007), Pan(2002) and Ledoit et al. (2002) for more details on this approach. Alternatively, one can usemethods such as efficient method of moments (EMM), indirect inference, and Markovchain Monte Carlo (MCMC) to handle processes with unobservable states, see for exampleGallant and Tauchen (1996), Gourieroux et al. (1993), and Elerian et al. (2001). Andersenet al. (2002) provides more details on alternative estimation methods for jump diffusionswith stochastic volatilities.

Applying the proposed estimation method, the second part of the paper studies therealignment probability of the Chinese Yuan. The Yuan has been essentially pegged to theUS Dollar for the past seven years. A recent export boom has led to diplomatic pressure onChina to allow the Yuan to appreciate. The realignment risk is important to both Chinaand other parts of the world. Foreign trade volume accounts for roughly 40% of China’sGDP in 2002.1 Any shift in the terms of trade and any changes in the competitiveadvantage of exporters brought by currency fluctuation can have a significant impact onChinese economy. Between 1992 and 1999, foreign direct investment (FDI) into Chinaaccounted for 8.2% of worldwide FDI and 26.3% of FDI going into developing countries(see Huang, 2003), all of which can be subject to currency risks. Beginning in August 2003,China started to open its domestic financial markets to foreign investors through qualifiedforeign institutional investors and is planning to allow Chinese citizens to invest inforeign financial markets. Currency risk will be important for these investors of financialmarkets, too.

1Data provided by Datastream International.


This paper uncovers the term structure of the forward realignment rate which completelycharacterizes future realignment probabilities.2 The term structure is hump-shaped andpeaks at six months from the end of 2003. This implies the financial market is anticipatingan appreciation of the Yuan in the next year and, conditioning on no realignments in thatperiod, the chance of a realignment is perceived to be small in the further future. SinceFebruary 2002, the upward realignment intensity for the Yuan implicit in the financialmarket has increased fivefold. The realignment probability responds quickly to newsreleases on Sino–US trade surplus, state-owned enterprise reform, Chinese government taxrevenue and, most importantly, both domestic and foreign government interventions.The paper is organized as follows. Section 2 provides the likelihood approximation.

Likelihood estimation using the approximate likelihood is discussed in Section 3. Section 4studies the realignment probability of the Chinese Yuan and Section 5 concludes.Appendix A collects all the technical assumptions. Appendix B contains the proofs.Appendix C extends the approximation method. Appendix D provides the history of theYuan’s exchange rate regime. Important news releases that influence the realignmentprobability are documented in Appendix E.

2. Likelihood approximation

2.1. Multivariate jump-diffusion process

We consider a multivariate jump-diffusion process X defined on a probability spaceðO;F;PÞ with filtration fFtg satisfying the usual conditions (see, for example, Protter, 1990).

dX t ¼ mðX t; yÞdtþ sðX t; yÞdW t þ Jt dNt, (1)

where X t is an n-dimensional state vector and W t is a standard d-dimensional Brownianmotion. y 2 Rp is a finite dimensional parameter. mð:; yÞ : Rn ! Rn is the drift and sð:; yÞ :Rn ! Rn�d is the diffusion. The pure jump process N has stochastic intensity lðX t; yÞ andjump size 1.3 The jump size Jt is independent of Ft� and has probability density nð:; yÞ :Rn ! R with support C 2 Rn. In this section, we consider the case where C has a non-empty interior in Rn.4 V ðx; yÞ � sðx; yÞsðx; yÞT is the variance matrix, sðx; yÞT denotes thematrix transposition of sðx; yÞ.Equivalently, X is a Markov process with infinitesimal generator AB, defined on

bounded function f : Rn ! R with bounded and continuous first and second derivatives,given by

ABf ðxÞ ¼Xn

i¼1

miðx; yÞqqxi

f ðxÞ þ1

2

Xn

i¼1

Xn

j¼1

vijðx; yÞq2

qxiqxj

f ðxÞ

þ lðx; yÞZ

C

½f ðxþ cÞ � f ðxÞ�nðc; yÞdc, ð2Þ

where miðx; yÞ, vijðx; yÞ and xi are, respectively, elements of mðx; yÞ, V ðx; yÞ and x.5

2The forward realignment rate is defined in Section 4.2.1.3See page 27–28 of Bremaud (1981) for definition of stochastic intensity.4This implies, when N jumps, all the state variables can jump. Cases where some state variables do not jump,

together with some other extensions, are considered in Appendix C.5See, for example, Revuz and Yor (1999) for details on infinitesimal generators.


Definition. The transition probability density pðD; yjx; yÞ, when it exists, is the conditionaldensity of X tþD ¼ y 2 Rn given X t ¼ x 2 Rn.

To save notation, the dependence on y of the functions mð:; yÞ, sð:; yÞ, V ð:; yÞ, lð:; yÞ; nð:; yÞand pðD; yjx; yÞ will not be made explicit when there is no confusion.

Proposition 1. Under Assumptions 1–2, the transition density satisfies the backward and

forward Kolmogorov equations given by

qqD

pðD; yjxÞ ¼ ABpðD; yjxÞ, (3)

qqD

pðD; yjxÞ ¼ AF pðD; yjxÞ. (4)

(3) and (4) are, respectively, the backward and forward equations. The infinitesimalgenerators AB and AF are defined as

ABpðD; yjxÞ ¼ LBpðD; yjxÞ þ lðxÞZ

C

½pðD; yjxþ cÞ � pðD; yjxÞ�nðcÞdc,

AF pðD; yjxÞ ¼ LF pðD; yjxÞ þZ

C

½lðy� cÞpðD; y� cjxÞ � lðyÞpðD; yjxÞ�nðcÞdc.

The operators LB and LF are given by

LBpðD; yjxÞ ¼Xn

i¼1

miðxÞqqxi

pðD; yjxÞ þ1

2

Xn

i¼1

Xn

j¼1

vijðxÞq2

qxiqxj

pðD; yjxÞ,

LF pðD; yjxÞ ¼ �Xn

i¼1

qqyi

½miðyÞpðD; yjxÞ� þ1

2

Xn

i¼1

Xn

j¼1

q2

qyiqyj

½vijðyÞpðD; yjxÞ�. ð5Þ

2.2. Method of transition density approximation

In this paper, we will find a closed-form approximation to pðD; yjxÞ using the backwardand forward equations. Specifically, we conjecture

pðD; yjxÞ ¼ D�n=2 exp �Cð�1Þðx; yÞ

D

� �X1k¼0

CðkÞðx; yÞDk þX1k¼1

DðkÞðx; yÞDk (6)

for some function CðkÞðx; yÞ and DðkÞðx; yÞ to be determined. We then plug (6) into thebackward and forward equations, match the terms with the same orders of D orD exp½�Cð�1Þðx; yÞ=D�, set their coefficients to 0 and solve for CðkÞðx; yÞ and DðkÞðx; yÞ. Anapproximation of order m40 is obtained by ignoring terms of higher orders.

pðmÞðD; yjxÞ ¼ D�n=2 exp �Cð�1Þðx; yÞ

D

� �Xm

k¼0

CðkÞðx; yÞDk þXm

k¼1

DðkÞðx; yÞDk. (7)

The transition density further satisfies the following two conditions.

Condition 1. Cð�1Þðx; yÞ ¼ 0 if and only if x ¼ y.

Condition 2. Cð0Þðx;xÞ ¼ ð2pÞ�n=2½detV ðxÞ��1=2.


When D! 0, Condition 1 guarantees the transition density peaks at x and Condition 2comes from the requirement that the density integrates to one with respect to y as Dbecomes small (see Appendix B for proof).The approximation is a small-D approximation in that it does not need more correction

terms to deliver better approximation for fixed D. Instead, for a fixed number of correctionterms, the approximation gets better as D shrinks, much like Taylor expansion aroundD ¼ 0.Before calculating CðkÞðx; yÞ and DðkÞðx; yÞ, we discuss why (6) is the right form of

approximation to consider. Intuitively, the first term in (6) captures the behavior ofpðD; yjxÞ at y near x where the diffusion term dominates and the second term captures thetail behavior corresponding to jumps. Let At;D denote the event of no jumps between time t

and tþ D. Act;D denotes set complementation.

pðD; yjxÞ ¼ PrðAt;DjX t ¼ xÞpdf ðX tþD ¼ yjX t ¼ x,At;DÞ

þ PrðAct;DjX t ¼ xÞpdf ðX tþD ¼ yjX t ¼ x; Ac

t;DÞ. ð8Þ

The Poisson arrival rate implies the second term is OðDÞ as D! 0. This gives the secondterm in (6).Conditioning on no jumps, pdf ðX tþD ¼ yjX t ¼ x,At;DÞ is the transition density for a

diffusion. As shown in Varadhan (1967) (Theorem 2.2, see also Section 5.1 in Aıt-Sahalia,2006),

limD!0½�2D log pdf ðX tþD ¼ yjX t ¼ x;At;DÞ� ¼ d2ðx; yÞ

for some function d2ðx; yÞ.6 Let Cð�1Þðx; yÞ ¼ 12d2ðx; yÞ, we obtain the leading term

exp½�Cð�1Þðx; yÞ=D� in (6). It is pre-multiplied by D�n=2 because, as D! 0, the density aty ¼ x goes to infinity at the same speed as a standard n-dimensional normal density withvariance of order OðDÞ in light of the driving Brownian motion.

Pmk¼0C

ðkÞðx; yÞDk correctsfor the fact that D is not 0. When there is no jump, the expansion coincides with that inAıt-Sahalia (2006).

2.3. Closed-form expression of the approximate transition density

Theorems 1 and 2 in this section give a set of restrictions on CðkÞðx; yÞ and DðkÞðx; yÞimposed by the forward and backward equations which, together with conditions 1 and 2,can be used to solve for the approximate transition density (7). Corollaries 1 and 2 giveexplicit expressions of CðkÞðx; yÞ and DðkÞðx; yÞ in the univariate case.

Theorem 1. The backward equation imposes the following restrictions,

0 ¼ Cð�1Þðx; yÞ �1

2

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cð�1Þðx; yÞ

� �,

0 ¼ Cð0Þ LBCð�1Þ �n

2

h iþ

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cð0Þðx; yÞ

� �,

6d2ðx; yÞ is the square of the shortest distance from x to y measured by the Riemannian metric defined locally as

ds2 ¼ dxT � V�1ðxÞ � dx where V�1ðxÞ is the matrix inverse to V ðxÞ, dx is the vector ðdx1; dx2; . . . ;dxnÞT.


0 ¼ Cðkþ1Þ LBCð�1Þ þ ðk þ 1Þ �n

2

h iþ

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cðkþ1Þðx; yÞ

� �þ ½lðxÞ � LB�CðkÞ for nonnegative k,

0 ¼ Dð1Þ � lðxÞnðy� xÞ,

0 ¼ Dðkþ1Þ �1

1þ kABDðkÞ þ ð2pÞn=2lðxÞ

Xk

r¼0

1

ð2rÞ!

Xs2Sn

2r

Mns

qs

qwsgk�rðx; y;wÞ

��w¼0

24 35 for k40,

where gkðx; y;wÞ � CðkÞðw�1B ðwÞ; yÞnðw�1B ðwÞ � xÞ=j detðq=quTÞwBðu; yÞju¼w�1

BðwÞj, wBðx; yÞ �

½V1=2ðxÞ�Tðq=qxÞCð�1Þðx; yÞ: Fixing y, wBð:; yÞ is invertible in a neighborhood of x ¼ y and

w�1B ð:Þ is its inverse function in this neighborhood. (For the ease of notation, the dependence of

w�1B ð:Þ on y is not made explicit henceforward.)

Snr in the theorem denotes the set of n-tuple non-negative even integers ðs1; s2; . . . ; snÞ

with the property that jðs1; s2; . . . ; snÞj � s1 þ s2 þ � � � þ sn ¼ r. For s 2 Snr and x 2 Rn, xs �

xs11 x

s22 . . . x

snn and differentiation of a function qð:Þ : Rn ! R is denoted ðqs=qxsÞ

qðxÞ � ðqjsj=qxs11 . . . qxsn

n ÞqðxÞ. Mns denotes the sth moment of n-dimensional standard

normal distribution given by Mns � ð2pÞ

�n=2 RRn exp½�wTw=2�ws dw. detð:Þ is the determi-

nant of a square matrix. For ease of notation, we sometimes use CðkÞ and DðkÞ for CðkÞðx; yÞand DðkÞðx; yÞ. LBCð�1Þðx; yÞ is defined the same way as in (5) with Cð�1Þðx; yÞ replacingpðD; yjxÞ. Similar use extends to LF , AB and AF .

Theorem 2. The forward equation imposes the following restrictions,

0 ¼ Cð�1Þðx; yÞ �1

2

qqy

Cð�1Þðx; yÞ

� �TV ðyÞ

qqy

Cð�1Þðx; yÞ

� �,

0 ¼ Cð0Þ �Xn

i¼1

miðyÞqqyi

Cð�1Þ þ1

2

Xn

i¼1

Xn

j¼1

Hijðx; yÞ �n

2

" #

þqqy

Cð�1Þðx; yÞ

� �TV ðyÞ

qqy

Cð0Þðx; yÞ

� �,

0 ¼ Cðkþ1Þ �Xn

i¼1

miðyÞqqyi

Cð�1Þ þ1

2

Xn

i¼1

Xn

j¼1

Hijðx; yÞ �n

2þ ðk þ 1Þ

" #

þqqy

Cð�1Þðx; yÞ

� �TV ðyÞ

qqy

Cðkþ1Þðx; yÞ

� �þ lðyÞ � LF� �

CðkÞ for non�negative k,

0 ¼ Dð1Þ � lðxÞnðy� xÞ,

0 ¼ Dðkþ1Þ �1

1þ kAF DðkÞ þ ð2pÞn=2

Xk

r¼0

1

ð2rÞ!

Xs2Sn

2r

Mns

qs

qws½lðw�1F ðwÞÞhk�rðx; y;wÞ�

��w¼0

24 35for k positive,


where Hijðx; yÞ�½ðq=qyiÞvijðyÞ�½ðq=qyjÞCð�1Þðx; yÞ� þ ½ðq=qyjÞvijðyÞ�½ðq=qyiÞC

ð�1Þðx; yÞ� þ vijðyÞ

ðq2=qyiqyjÞCð�1Þðx;yÞ, hkðx;y;wÞ�CðkÞðx;w�1F ðwÞÞnðy�w�1F ðwÞÞ=j detðq=quTÞ wF ðx; uÞju¼w�1

FðwÞj,

wF ðx; yÞ � ½V1=2ðyÞ�Tðq=qyÞCð�1Þðx; yÞ. Fixing x, wF ðx; :Þ is invertible in a neighborhood of

y ¼ x and w�1F ð:Þ is its inverse function in this neighborhood. (For the ease of notation, the

dependence of w�1F ð:Þ on x is not made explicit henceforward.)

The first equation in either theorem characterizes Cð�1Þðx; yÞ. Knowing Cð�1Þðx; yÞ, thesecond equation can be solved for Cð0Þðx; yÞ. Cðkþ1Þðx; yÞ is then solved recursively throughthe third equation. The last two equations give DðkÞðx; yÞ which do not require solvingdifferential equations.

Implementation. To compute the approximate transition density, we need to use Conditions 1and 2, together with either Theorem 1 or 2. Regarding the inverse functions w�1B ðwÞ and

w�1F ðwÞ in Theorems 1 and 2, only the values of these inverse functions at w ¼ 0 are required

to evaluate the transition density. At w ¼ 0, w�1B ðw ¼ 0Þ ¼ y and w�1F ðw ¼ 0Þ ¼ x. The

derivatives involving the inverse function can be calculated using the implicit function theorem

(see for example Rudin, 1976), ðq=qwTÞw�1B ðwÞ ¼ ððq=qxTÞwBðx; yÞjx¼w�1BðwÞÞ�1 and

ðq=qwTÞw�1F ðwÞ ¼ ððq=qyTÞwF ðx; yÞjy¼w�1FðwÞÞ�1.

In the univariate case, n ¼ 1, the functions CðkÞðx; yÞ and DðkÞðx; yÞ can be solvedexplicitly.

Corollary 1. Univariate case. From the backward equation,

Cð�1Þðx; yÞ ¼1

2

Z y

x

sðsÞ�1 ds

� �2,

Cð0Þðx; yÞ ¼1ffiffiffiffiffiffi

2pp

sðyÞexp

Z y

x

mðsÞs2ðsÞ

�s0ðsÞ2sðsÞ

ds

� �,

Cðkþ1Þðx; yÞ ¼ �

Z y

x

sðsÞ�1 ds

� ��ðkþ1Þ Z y

x

expR x

ss0ðuÞ2sðuÞ �

mðuÞs2ðuÞ

duh i

sðsÞ�1R y

ssðuÞ�1 du

� �k½lðsÞ � LB�CðkÞðs; yÞ

8><>:9>=>;ds

for kX0,

Dð1Þðx; yÞ ¼ lðxÞnðy� xÞ,

Dðkþ1Þðx; yÞ ¼1

1þ kABDðkÞðx; yÞ þ

ffiffiffiffiffiffi2pp

lðxÞXk

r¼0

M12r

ð2rÞ!

q2r

qw2rgk�rðx; y;wÞ

��w¼0

" #for k40,

where gkðx; y;wÞ � CðkÞðw�1B ðwÞ; yÞnðw�1B ðwÞ � xÞsðw�1B ðwÞÞ, M1

2r � 1=ffiffiffiffiffiffi2pp R

Rexpð�s2=2Þs2r

ds and wBðx; yÞ ¼R x

ysðsÞ�1 ds.

Corollary 2. Univariate case. From forward equation,


2

Z y

x

sðsÞ�1 ds

� �2,

Cð0Þðx; yÞ ¼1ffiffiffiffiffiffi

2pp

sðxÞexp

Z y

x

mðsÞs2ðsÞ

�3s0ðsÞ2sðsÞ

ds

� �,


Cðkþ1Þðx; yÞ ¼ �

Z y

x

sðsÞ�1 ds

� ��ðkþ1Þ Z y

x

expR s

y3s0ðuÞ2sðuÞ �

mðuÞs2ðuÞ

duh i

sðsÞ�1R s

xsðuÞ�1 du

� �k½lðsÞ � LF �CðkÞðx; sÞ

8><>:9>=>;ds

for kX0,

Dð1Þðx; yÞ ¼ lðxÞnðy� xÞ,


1þ kAF DðkÞðx; yÞ þ

ffiffiffiffiffiffi2pp Xk

r¼0

M12r

ð2rÞ!

q2r

qw2rlðw�1F ðwÞÞhk�rðx; y;wÞ� ��

w¼0

" #for k40,

where hkðx; y;wÞ � CðkÞðx;w�1F ðwÞÞnðy� w�1F ðwÞÞsðw�1F ðwÞÞ, M1

2r � 1=ffiffiffiffiffiffi2pp R

Rexpð�s2=2Þs2r

ds and wF ðx; yÞ ¼R y

xsðsÞ�1 ds.

The choice of Theorem 1 or 2 in practice largely depends on computational ease. Forexample, it is easier to use the results from the forward equation to verify that, inunivariate case without jumps, the approximate transition density obtained here coincideswith that in Aıt-Sahalia (2002). Aıt-Sahalia (2006) gives approximation for the log-likelihood of multivariate diffusion. Without jumps, the approximate log-likelihoodobtained in this paper takes the form

log pðmÞðD; yjxÞ ¼ �n

2logD�

Cð�1Þðx; yÞ

Dþ log

Xm

k¼0

CðkÞðx; yÞDk

which coincides with the approximation in Aıt-Sahalia (2006) when the last term is Taylorexpanded around D ¼ 0.

2.4. Examples

2.4.1. Brownian motion

Let dX t ¼ sdW t. The true transition density is Nð0; s2DÞ. In this case, Cð�1Þðx; yÞ ¼ðy� xÞ2=ð2s2Þ, Cð0Þðx; yÞ ¼ 1=

ffiffiffiffiffiffiffiffiffiffi2ps2p

, CðkÞðx; yÞ ¼ DðkÞðx; yÞ ¼ 0 for kX1. The approximatedensity is exact.

pðmÞðD; yjxÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2ps2Dp exp �

ðy� xÞ2

2s2D

� �.

2.4.2. Brownian motion with drift

For the process dX t ¼ mdtþ sdW t, the true transition density is NðmD; s2DÞ, orequivalently

pðD; yjxÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2ps2Dp exp �

ðy� xÞ2

2s2Dþ

ms2ðy� xÞ

� �exp �

m2

2s2D

� .

We can calculate that Cð�1Þðx; yÞ ¼ ðy� xÞ2=ð2s2Þ, Cð0Þðx; yÞ ¼ ð1=ffiffiffiffiffiffiffiffiffiffi2ps2p

Þ exp½m=s2

ðy� xÞ�, Cð1Þðx; yÞ ¼ �Cð0Þðx; yÞm2=ð2s2Þ, DðkÞðx; yÞ ¼ 0 for all k. The approximate densitypð1ÞðD; yjxÞ is !

pð1ÞðD; yjxÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2ps2Dp exp �

ðy� xÞ2

2s2Dþ

ms2ðy� xÞ

� �1�

m2

2s2D

�


which approximates pðD; yjxÞ by replacing expð�ðm2=ð2s2ÞÞDÞ with its first-order Taylorexpansion.

2.4.3. Jump-diffusion

Consider the univariate jump-diffusion dX t ¼ mdtþ sdW t þ St dNt, where Nt is aPoisson process with arrival rate l. The jump size St is i.i.d. NðmS;s

2SÞ. Conditioning on j

jumps, the increment of X is Nðxþ mDþ jmS;s2Dþ js2SÞ. The true and approximate

transition densities are

pðD; yjxÞ ¼X1j¼0

e�lDðlDÞj

j!

1ffiffiffiffiffiffi2pp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2Dþ js2Sq exp �

ðy� x� mD� jmSÞ2

2ðs2Dþ js2SÞ

� �

¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2ps2Dp exp �

ðy� xÞ2

2s2Dþ

ms2ðy� xÞ

� �exp �

m2

2s2þ l

� D

� �þ

e�lDlffiffiffiffiffiffi2pp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2Dþ s2Sq exp �

ðy� x� mD� mSÞ2

2ðs2Dþ s2SÞ

� �DþOðD2Þ,

pð1ÞðD; yjxÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2ps2Dp exp �

ðy� xÞ2

2s2Dþ

ms2ðy� xÞ

� �1�

m2

2s2þ l

� D

� �þ

lffiffiffiffiffiffiffiffiffiffiffi2ps2S

q exp �ðy� x� mSÞ

2

2s2S

� �D.

Here Cð�1Þðx; yÞ ¼ ðy� xÞ2=ð2s2Þ, Cð0Þðx; yÞ ¼ 1=ffiffiffiffiffiffiffiffiffiffi2ps2p

exp½ðm=s2Þðy� xÞ�, Cð1Þðx; yÞ ¼

�Cð0Þðx; yÞðm2=ð2s2Þ þ lÞ, Dð1Þðx; yÞ ¼ ðl=ffiffiffiffiffiffiffiffiffiffiffi2ps2S

qÞ exp½�ðy� x� mSÞ

2=ð2s2SÞ�. pðD; yjxÞ is

approximated by expanding exp½�ðm2=ð2s2Þ þ lÞD� to its first-order around D ¼ 0, by

approximating ðe�lDl=ffiffiffiffiffiffi2pp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

s2Dþ s2Sq

Þ exp½�ðy� x� mD� mSÞ2=ð2ðs2Dþ s2SÞÞ� with its

limit at D ¼ 0 and by ignoring the terms corresponding to at least two jumps which are of

order D2.

2.4.4. A numerical example

Consider the Ornstein–Uhlenbeck process with jump dX t ¼ �kX t dtþ sdW t þ St dNt

where k;s40, N is a standard Poisson process with constant intensity l. St is i.i.d. and hasdouble exponential distribution with mean 0 and standard deviation sS, which has a fattertail than normal distribution.

X is an affine process whose characteristic function is known according to Duffie et al.(2000).7 An approximate transition density, pFFTðD;XDjX 0Þ, can be obtained via fastFourier transform. Treating log pFFTðD;XDjX 0Þ as the ‘‘true’’ log-likelihood, we nowinvestigate numerically the accuracy of the closed-form log-likelihood approximation andhow the accuracy varies with the sampling interval D and the order of the approximation.8

The first and the second graphs in Fig. 1 show the effect of adding correction terms. Thefirst graph uses first order approximation while the second uses second order

7E½eiuXD jX 0 ¼ x� ¼ exp½i uxe�kD þ ðe�2kD � 1Þu2s2=ð4kÞ�ðð1þ e�2kDu2s2SÞ=ð1þ u2s2SÞÞl=2k.

8The parameters are set to k ¼ 0:5, s ¼ 0:2, l ¼ 13, mS ¼ 0, sS ¼ 0:2, X 0 ¼ 0.

ARTICLE IN PRESS

-0.5 -0.25 0 0.25 0.5

0

5

10

15

x 10-3

1st order, weekly

-0.5 -0.25 0 0.25 0.5

-20

-10

0

x 10-5

2nd order, weekly

-0.5 -0.25 0 0.25 0.5

-10

-5

0

x 10-6

2nd order, daily

Fig. 1. Approximation error of log-likelihood.

J. Yu / Journal of Econometrics 141 (2007) 1245–1280 1255

approximation. Both graph correspond to weekly sampling. The accuracy of theapproximation increases rapidly with additional terms. For weekly sampling, the standarddeviation of XDjX 0 ¼ 0 is 0.03. The first two graphs in fact show the log-likelihoodapproximation is good from negative 20 to positive 20 standard deviations. That theapproximation is good in the large deviation area is useful in case a rare event occurs in theobservations.

The second and the third graphs in Fig. 1 show the effect of shrinking sampling interval.The third graph uses daily instead of weekly sampling in the second graph. Both graphs usesecond order approximation. The approximation improves quickly as sampling intervalshrinks.

2.5. Calculate Cð�1Þðx; yÞ

The equations characterizing Cð�1Þðx; yÞ in Section 2.3 are, in multivariate cases, PDEswhich do not always have explicit solutions. In this section, we will discuss the conditionsunder which they can be solved explicitly. When they cannot be explicitly solved, we willdiscuss how to find an approximate solution with a second expansion in the state space.

2.5.1. The reducible case

Consider the equation characterizing Cð�1Þðx; yÞ in Theorem 1.


2

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cð�1Þðx; yÞ

� �.

Remember wBðx; yÞ is defined as wBðx; yÞ � ½V1=2ðxÞ�Tðq=qxÞCð�1Þðx; yÞ in Theorem 1, we

have


2wT

Bðx; yÞwBðx; yÞ,

qqx

Cð�1Þðx; yÞ ¼qdx

wTBðx; yÞ

� �wBðx; yÞ.

Therefore,

wBðx; yÞ ¼ ½V1=2ðxÞ�T

qdx

wTBðx; yÞ

� �wBðx; yÞ. (9)


Let In be the n-dimensional identity matrix. It is easy to see ½V1=2ðxÞ�T½ðq=qxÞwTBðx; yÞ� ¼

In characterizes a solution for wBðx; yÞ and hence a solution for Cð�1Þðx; yÞ. Let V�1=2ðxÞ,whose element in the ith row and jth column is denoted v

�1=2ij ðxÞ, be the inverse matrix of

V1=2ðxÞ. The condition for the existence of a vector function wBðx; yÞ whose first derivativeðq=qxTÞwBðx; yÞ ¼ V�1=2ðxÞ is, intuitively, that the second derivative matrix of each elementof wBðx; yÞ is symmetric,

qqxk

v�1=2ij ðxÞ ¼

qqxj

v�1=2ik ðxÞ for all i; j; k ¼ 1; . . . ; n. (10)

This is the reducibility condition given by Proposition 1 in Aıt-Sahalia (2006). All theunivariate cases are reducible. Therefore, it is not surprising that Cð�1Þðx; yÞ can be solvedexplicitly for the univariate case in Corollaries 1 and 2.

2.5.2. The irreducible case

When the reducibility condition (10) does not hold, we can no longer solve (9) forCð�1Þðx; yÞ explicitly in general. In this case, we will approximate Cð�1Þðx; yÞ instead. Toapproximate Cð�1Þðx; yÞ, we notice that the derivatives of Cð�1Þðx; yÞ are needed to computeCð0Þðx; yÞ whose derivatives will in turn be used to compute Cð1Þðx; yÞ and so on. So we haveto approximate not only Cð�1Þðx; yÞ but also its derivatives of various orders. A naturalmethod of approximation is therefore Taylor expansion of Cð�1Þðx; yÞ in the state spaceproposed by Aıt-Sahalia (2006). We will Taylor expand Cð�1Þðx; yÞ around y ¼ x withcoefficients to be determined, plug the expansion into the differential equation in Theorem1 or 2, match and set to 0 the coefficients of terms with the same orders of y� x, and solvefor the Taylor expansion of Cð�1Þðx; yÞ. This expansion is discussed in detail in Aıt-Sahalia(2006) (see especially Theorem 2 and the remarks thereafter).We will first give an example to illustrate this second expansion in the state space and

then discuss how to determine the order of the Taylor expansion. We will use Cð�1;JÞðx; yÞto denote the expansion of Cð�1Þðx; yÞ to order J and use Cðk;JÞðx; yÞ for kX0 to denoteother coefficients computed from using Cð�1;JÞðx; yÞ. To make the error of pðmÞðD; yjxÞ havethe correct order, J will depend on m. But for notational convenience, this dependence isnot written out explicitly.As an example, we take the equation characterizing Cð�1Þðx; yÞ in Theorem 2 and seek a

second order expansion Cð�1;2Þðx; yÞ. Taylor expand this equation around y ¼ x and noticeCondition 1,

qqy

Cð�1Þðx; yÞ

��y¼x

� ðy� xÞ þ1

2ðy� xÞT �

q2

qy2Cð�1Þðx; yÞ

��y¼x

� ðy� xÞ þ R1

¼1

2

qqy

Cð�1Þðx; yÞ

��y¼x

þq2


��y¼x


" #T� V ðxÞ þ R3½ �

�qqy

Cð�1Þðx; yÞ

��y¼x

þq2


��y¼x


" #,

where R1, R2 and R3 are remainder terms that will not affect the result.


Match the terms with the same orders of ðy� xÞ and set their coefficients to 0. We get

qqy

Cð�1Þðx; yÞ

��y¼x

¼ 0;q2


��y¼x

¼ V�1ðxÞ.

The second order expansion of Cð�1Þðx; yÞ around y ¼ x is therefore

Cð�1;2Þðx; yÞ ¼ 12ðy� xÞT � V�1ðxÞ � ðy� xÞ. (11)

Now we discuss the choice of J. We can achieve an error of opðDmÞ if each termCðk;JÞðx; yÞDk differs from CðkÞðx; yÞDk by opðDmÞ. For jump-diffusions, y� x�Opð

ffiffiffiffiDpÞ. It is

clear then to make Cðk;JÞðx; yÞ � CðkÞðx; yÞ ¼ opðDm�kÞ, we need JX2ðm� kÞ for all kX� 1as shown in Eq. (36) in Aıt-Sahalia (2006). Therefore,

J ¼ 2ðmþ 1Þ.

This choice of J also ensures the relative error incurred byPm

k¼1DðkÞðx; yÞDk is op Dmð Þ,

which can be verified from Theorems 1 and 2.For kX0, Cðk;JÞðx; yÞ are exact solutions to the linear PDEs characterizing them.9

However, they only approximate CðkÞðx; yÞ because these PDEs involve Cð�1Þðx; yÞ forwhich only an approximation Cð�1;JÞðx; yÞ is available. In the case that these linear PDEsare too cumbersome to solve, we can find an approximate solution Cðk;JðkÞÞðx; yÞ in the sameway as the expansion for Cð�1Þ ðx; yÞ with the order of expansion being JðkÞ ¼ 2ðm� kÞ.See also Aıt-Sahalia (2006).

3. Likelihood estimation

To estimate the unknown parameter y 2 Rp in the jump diffusion model, we collectobservations in a time span T with sampling interval D. The sample size is nT ;D ¼ T=D. LetfX 0 ;XD;X 2D; . . . ;X T g denote the observations. Let

lT ;DðyÞ ¼XnT ;D

i¼1

log pðD;X iDjX ði�1ÞD; yÞ

be the log-likelihood function conditioning on X 0. The loss of information contained in thefirst observation X 0 is asymptotically negligible. Let byT ;D � arg maxy2YlT ;DðyÞ denote the(uncomputable) maximum-likelihood estimator (MLE). Let

lðmÞT ;DðyÞ ¼

XnT ;D

i¼1

log pðmÞðD;X iDjX ði�1ÞD; yÞ

be the mth order approximate likelihood function. Denote byðmÞT ;D � arg maxy2YlðmÞT ;DðyÞ. Let

IT ;DðyÞ � Ey½_lT ;DðyÞ_lT ;DðyÞ

T� be the information matrix. To simplify notations, _lT ;DðyÞ

indicates derivative with respect to y and the dependence of byT ;D and byðmÞT ;D on T and Dwill not be made explicit when there is no confusion. Let iDðyÞ � Ey½ðq=qyÞlog pðD;XDjX 0; yÞ � ðq=qy

TÞ log pðD;XDjX 0; yÞ�.

The asymptotics of byðmÞT ;D is obtained in two steps. We first establish that the differencebetween byðmÞT ;D and the true MLE by is negligible compared to the asymptotic distribution of

9Linear PDEs can be solved using, for example, the method in chapter 6 of Carrier and Pearson (1988).


by. The asymptotic distribution of byðmÞT ;D can then be deduced from that of by for the purposeof statistical inference.Under Assumption 6, it can be shown10 that the true MLE byT ;D satisfies

I1=2T ;Dðy0ÞðbyT ;D � y0Þ ¼ GT ;Dðy0Þ

�1ST ;Dðy0Þ þ opð1Þ (12)

as T !1, uniformly for all DpD, where GT ;DðyÞ � �I�1=2T ;D ðyÞ€lT ;DðyÞI

�1=2T ;D ðyÞ and ST ;DðyÞ �

I�1=2T ;D ðyÞ_lT ;DðyÞ. GT ;Dðy0Þ

�1ST ;Dðy0Þ ¼ Opð1Þ as T !1 uniformly for DpD under verygeneral conditions including but not restricted to stationarity, see for example, Aıt-Sahalia(2002) and Jeganathan (1995). kIT ;Dðy0Þk�1=2 captures the magnitude of the statistical noise inthe true MLE. The next theorem shows that the difference between the approximate and thetrue MLE is negligible compared to kIT ;Dðy0Þk�1=2 when the sampling interval is small.

Theorem 3. Under Assumptions 1–7, there exists a sequence D�T !T!1

0 so that for any fDT g

satisfying DTpD�T ,

kI1=2T ;DTðy0ÞðbyðmÞT ;DT

� byT ;DTÞk ¼ opðDm

T Þ

as T !1.

Therefore, the asymptotic distribution of byðmÞT ;DTinherits that of the true MLE. For

statistical inference, we provide a primitive condition for stationarity in Proposition 2 belowand derive under stationarity the asymptotic distribution of the true MLE (hence theasymptotic distribution of the approximate MLE). Under non-stationarity, primitivecondition of sufficient generality is unavailable for the asymptotic distribution of the trueMLE to the author’s knowledge, with the exception of Ornstein–Uhlenbeck model (see Aıt-Sahalia, 2002). Therefore under non-stationarity, the asymptotic distribution of the trueMLE needs to be derived case by case in practice using methods in for example Jeganathan(1995). Note that the main result of the paper—that the approximate MLE inherits theasymptotic distribution of the true MLE as the sampling interval shrinks (Theorem 3)—applies irrespective of stationarity or the asymptotic distribution of the true MLE.

Proposition 2. The jump-diffusion process X in (1) is stationary if there exists a non-negative

function f ð�Þ : Rn ! R such that for x in the domain of X,

limjxj!1

ABf ðxÞ ¼ �1 and supx

ABf ðxÞo1,

where AB is the infinitesimal generator defined in (2).

This condition is easy to verify. As an example, consider the process X with mðxÞ ¼ �kx

for some k40, sðxÞ ¼ sffiffiffixp

, constant jump intensity, and constant jump distribution withfinite variance. We can see this process is stationary by letting f ðxÞ ¼ x2 in Proposition 2because ABf ðxÞ ¼ �2kx2

þ oðx2Þ when jxj ! 1.

Theorem 4. Under Assumptions 1–7, there exists D40 such that byT ;D is consistent as T !1

for all DpD. Further, if X process is stationary, byT ;D has the following asymptotic distribution:

ðnT ;DiDðy0ÞÞ1=2ðbyT ;D � y0Þ ¼ Nð0; Ip�pÞ þ opð1Þ

as T !1, uniformly for DpD.

10See the proof of Theorem 4.


Corollary 3. Under Assumptions 1–7, there exists a sequence D�T !T!1

0 so that for any fDT g

satisfying DTpD�T ,

ðnT ;DTiDTðy0ÞÞ

1=2ðbyðmÞT ;DT

� y0Þ ¼ Nð0; Ip�pÞ þ opð1Þ þ opðDmT Þ

as T !1.

Therefore, byðmÞT ;D inherits the asymptotic property of byT ;D if the sampling interval becomessmall. The remainder term in Corollary 3 is of order opð1Þ. It is explicitly decomposed into twoparts. The first part opð1Þ is the usual approximation error of large sample asymptoticsðT !1Þ. The second part opðDm

T Þ results from using an approximate likelihood estimator.Even a low order (m) of approximation yields the same asymptotic distribution as the trueMLE.This benefits from the use of in-fill asymptotics (small D) in additional to large sampleasymptotics (large T). A small D asymptotics is required because the correction term of theapproximate density is analytic at D ¼ 0 only for univariate diffusions or reducible multivariatediffusions. For these processes, asymptotics can be derived by letting T !1 and letting thenumber of correction terms m!1 holding D fixed as in Aıt-Sahalia (2002). However, forirreducible multivariate diffusions (see Aıt-Sahalia, 2006) and general jump diffusions, analyticityat D ¼ 0 may not hold. Therefore, the likelihood expansion (7) is to be interpreted strictly as aTaylor expansion and the asymptotics hold for small D holding fixed the order of expansion m.In practice, D is not zero and T and m are both finite. The numerical examples in Fig. 1 showthat even the first order likelihood approximation can be very accurate for weekly data. Theaccuracy improves rapidly when the order of approximation increases.

4. Realignment risk of the Yuan

The currency of China is the Yuan. Since 1994, daily movement of the exchange ratebetween the Yuan and the US Dollar (USD) is limited to 0.3% on either side of the exchangerate published by People’s Bank of China, China’s central bank. Since 2000, the Yuan/USDrate has been in a narrow range of 8.2763–8.2799 and is essentially pegged to USD. A recentexport boom has led to diplomatic pressures on China to allow the Yuan to appreciate. Whatis the realignment probability of the Yuan implicit in the financial market? What factors doesthis probability respond to? These are the questions to be addressed in the rest of the paper.

4.1. Data

11Da

1 mon

Ironically, what began as a protection against currency devaluation has now becomethe chief tool for betting on currency appreciation, particularly in China—where anexport boom has led to diplomatic pressure to allow the Yuan to rise in value.

— ‘‘Feature—NDFs, the secretive side of currency trading’’

Forbes, August 28, 2003

We obtained daily non-deliverable forward (NDF) rates traded in major off-shorebanks. The data, covering February 15, 2002–December 12, 2003, are sampled by WM/Reuters and are obtained from Datastream International.11

tastream daily series CHIYUA$, USCNY1F and USCNY3F are obtained for the spot Yuan/USD rate,

th NDF rate and 3 month NDF rate. All the data are sampled by WM/Reuters at 16:00 h London time.


An NDF contract traded in an off-shore bank is the same as a forward contract exceptthat, on the expiration day, no physical delivery of the Yuan takes place. Instead, profit(loss) based on the difference between the NDF rate and the spot exchange rate at maturityis settled in USD. The daily trading volume in the offshore Chinese Yuan NDF market isaround 200 million USD.12

Fig. 2 plots the spot Yuan/USD exchange rate, the one-month NDF rate and the three-month NDF rate. A feature of the plot is the downward crossing over spot rate of bothforward rates near the end of year 2002. In early 2002, both forward rates are above thespot rate; however, since late 2002, both forward rates have been lower than the spot rates.This reflects the increasing market expectation of the Yuan’s appreciation, which isconfirmed by the estimation results in Section 4.5.For a currency that can be freely exchanged, the forward price is pinned down by the

spot exchange rate and domestic and foreign interest rates (the covered interest rate parity)and does not contain additional information. Due to capital control, the arbitrageargument leading to the covered interest rate parity does not hold for the Yuan. The ‘‘CIPimplied rate’’ in Fig. 2 is the three-month forward rate implied by the covered interest rateparity if the Yuan were freely traded.13 Both the level and the trend of the CIP implied ratediffers from that of the NDF rate. Therefore, the NDF rate does provide additionalinformation which will be used in the following study.

4.2. Setup

The spot exchange rate S in USD/Yuan is assumed to follow a pure jump process

dSt

St�

¼ Jut dUt þ Jd

t dDt,

where U and D are Poisson processes with arrival rate lu;t and ld;t. The jump size Jut (J

dt ) is

a positive (negative) i.i.d. random variable. Therefore, U and D are associated with upward(the Yuan appreciates) and downward (the Yuan depreciates) realignments, respectively.The time-varying realignment intensities are parametrized as lu;t ¼ lue

Gt� and ld;t ¼

lde�Gt� where Gt follows

dGt ¼ �kGt dtþ sdW t þ Kt dLt

for some k;s40. W is a Brownian motion, Kt�Nð0; s2K Þ, L is a Poisson process witharrival rate Z. W, L and K are assumed to be independent of other uncertainties in themodel.This specification parsimoniously characterizes how the realignment intensity varies

over time. When G40, lu;t4lu and appreciation is more likely than long-run average.When Go0, ld ;t4ld and depreciation is more likely. The process G mean reverts to 0 atwhich point appreciation and depreciation intensities equate their long-run averages.Under regularity conditions, the rate Fd of a NDF with maturity d satisfies

0 ¼ EQ e�R d

0rs dsðSd � FdÞ

��S0;G0

� �,

12From ‘‘Feature—NDFs, the secretive side of currency trading’’, Forbes, August 28, 2003.13Three-month eurodollar deposit rate and three-month time deposit rate for the Yuan are used. They are

obtained from series FREDD3M and CHSRW3M in Datastream International.

ARTICLE IN PRESS

Jan02 Apr02 Jul02 Oct02 Jan03 Apr03 Jul03 Oct03 Jan04

8.1

8.12

8.14

8.16

8.18

8.2

8.22

8.24

8.26

8.28

8.3

Date

Yuan / U

SD

spot rateone-month NDF ratethree-month NDF rateCIP implied 3m NDF rate

Fig. 2. Spot and NDF rates.


where r is the short-term USD interest rate and Q is a risk-neutral equivalent martingalemeasure.

We make a simplifying assumption that the realignment risk premium is zero whichallows us to identify the risk-neutral probability Q with the actual probability. Risk-neutral probability consists of actual probability and risk premium. We will mostly beinterested in changes in realignment probability. This assumption is therefore anassumption of constant risk premium and setting it to zero just makes the notationsimpler. Incorporating time-varying risk-premium requires writing down a formal modelwhich depends on the different parametrization of its time variation. To avoid the resultsbeing driven by potential misspecification of the time variation of risk premium, we takethe first order approximation by using a constant risk premium. To compute the forward

price, we also assume that e�R d

0rs ds

and Sd are uncorrelated. The main advantage of thisassumption is that it avoids potential misspecification of interest rate dynamics in light ofthe recent finding that available term structure models do not fit time series variations ofinterest rates very well (see Duffee, 2002). We will use one-month forward rate in the

estimation. When the time-to-maturity d is small,R d

0 rs ds ¼ r0dþ oðdÞ, i.e. changes in

interest rates is of smaller order relative toR d

0 rs ds. The above two assumptions also make

the empirical analysis easy to replicate.Under these two assumptions, the forward pricing formula simplifies to

Fd ¼ E½Sd jS0;G0�. (13)

We have performed a robustness check of the model specification. Let spd ¼ Fd � S bethe spread of a d-maturity forward rate over the spot rate. Let bspd be the spread implied by(13) using the estimated parameters. The following regression

spd ¼ ad þ bd � bspd þ ed


produced estimates bad close to 0 and estimates bbd close to 1 for all maturities d longer thanone month, which would be the case if the model is correctly specified. The joint hypothesisthat bbd ¼ 1 for all d is not rejected. The detailed robustness check results are availableupon request.

4.2.1. Term structure of the forward realignment rate

Let qðtÞ denote the probability of no appreciation before time t.14 When the Poisson

intensity is a constant l, the probability of no jumps before t is e�lt which converges to one

as t! 0. Under the current setup, qðtÞ ¼ E0½e�R t

0lu;t dt� which is a stochastic version for the

probability of no jumps.15 Because the jump intensity is positive, qðtÞ decreases with t and

we can find a function f ðtÞX0 such that qðtÞ ¼ e�R t

0f ðtÞ dt

. It can be verified that f ðtÞ ¼

�q0ðtÞ=qðtÞ and that, for sXt, the probability of no realignment before s conditioning on norealignment before time t (denoted by qðsjtÞ) satisfies

qðsjtÞ ¼ e�R s

tf ðtÞ dt

.

We call the function f the forward realignment rate. Knowing the term structure of f ðtÞ (i.e.,its variation with respect to t) gives complete information regarding future realignmentprobabilities.16

This paper uses one factor G to model the evolution of the realignment intensity. Thissimple setup already affords a rich set of possibilities for the term structure of f. Fig. 3 plotsthree such possibilities including decreasing, increasing and hump-shaped term structure.With a decreasing term structure, the market perceives an imminent realignment.

Conditioning on no immediate realignment, the chance of a realignment decreases over time.An increasing term structure has the opposite implication: realignment probability is consideredsmall and reverts back to the long-run mean. When the term structure is hump-shaped, themarket could be expecting a realignment at a certain time in the future and, if nothing happensby then, the chance of a realignment in the further future is perceived to be smaller.

4.2.2. Approximation of the forward price

The forward price in (13) does not have a closed form. We will approximate the forwardprice with its third order expansion around time-to-maturity d ¼ 0 using the infinitesimalgenerator A of the process fS;Gg, i.e.,

F d S0 þX3i¼1

di

i!Aif ðS0;G0Þ, (14)

where f ðSd ;GdÞ � Sd . Since a NDF with short maturity (one month) will be used in theestimation, the approximation is expected to be good. We have verified using simulationthat such approximation has an error of no more than 1� 10�4. The accuracy is sufficient

14The term structure of downward realignment rate can be defined in direct parallel. However, it is suppressed

since the current interest is in the appreciation of the Yuan.15This can be shown using a doubly stochastic argument in Bremaud (1981) by first drawing flu;tg and then

determining the jump probability conditioning on flu;tg.16Readers familiar with duration models will recognize this is the hazard rate of the Yuan’s realignment. This

concept is also parallel to the forward interest rate in term structure interest rate models and to the forward

default rate in credit risk models (see Duffie and Singleton, 2003).

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0.1

0.15

0.2

0.25

0.3

Year

Fo

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Downward sloping term structure

0 1 2 3 4 50.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Year

Fo

rwa

rd R

ea

lign

me

nt R

ate

Upward sloping term structure

0 1 2 3 4 50.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

Year

Forw

ard

Realig

nm

ent R

ate

Hump-shaped term structure

Fig. 3. Possible shapes of the term structure.


for the application since the forward price data reported by Datastream International havejust four digits after the decimal point.

4.2.3. Identification

Let gðSdÞ ¼ Sd . The forward price (13) can be rewritten as

Fd ¼ S0 þ E

Z d

0

LgðSsÞds

��S0;G0

� �,

where L is the infinitesimal generator of the process S. It can be calculated that

LgðStÞ ¼ ½eGt�luEðJ

ut Þ þ e�Gt�ldEðJ

dt Þ�St. (15)

Therefore, lu, EðJut Þ, ld and EðJd

t Þ are not separately identified from the forward ratealone. Intuitively, a higher intensity and a larger magnitude of realignment have the sameimplication for the forward rate.

In principle, the parameters lu, EðJut Þ, ld and EðJd

t Þ are identified from the time series ofexchange rates. However, in the sample period from February 2002 to September 2003, norealignments took place. To overcome this problem, we collected all the realignment eventssince the creation of the Yuan in 1955 (see Appendix D). In the past forty-eight and a halfyears, there have been two appreciations for the Yuan: 9% in 1971 and 11% in 1973 withan average of 10%. There have been four depreciations: a roughly 28% depreciation in1986, a 21% and a 10% depreciation in 1989 and 1990, respectively; and finally a 33%depreciation in 1994. The average of these depreciations is 23%.

Therefore, we will set lu ¼2

48:5, EðJut Þ ¼ 0:1, ld ¼

448:5, EðJd

t Þ ¼ �0:23. I.e., we willconsider realignment risk with an average magnitude of 10% appreciation and 23%depreciation. This is a normalization that will not affect the results later. To see this, giventhe prevailing anticipation of Yuan’s appreciation, the term eGt�luEðJ

ut Þ corresponding to

appreciation will dominate in (15). Suppose the true value of luEðJut Þ ¼ v and we instead

set luEðJut Þ ¼ cv, then eGt�luEðJ

ut Þ ¼ eGt�v ¼ eGt��log c cv. This amounts to a level effect of

magnitude log c on estimates of fGtg which will not affect conclusions on time-variations ofGt which are our focus.

4.3. Iterative maximum-likelihood estimation

Let y � fk;s; Z;sKg with y0 being the true parameter value. Starting from an initialestimate yðmÞ, we can invert the forward pricing formula (14) to get bGðmÞ ¼ GðyðmÞÞ. The


vector bGðmÞ ¼ fbGðmÞ0 ; bGðmÞD ; bGðmÞ2D ; . . . ;bGðmÞT g estimates the process G at different points in time.

(To save notation, G is used for both the process G and the function inverting the forward

pricing equation. The dependence of GðyðmÞÞ on observed spot and forward rates is not

made explicit.) Knowing bGðmÞ,we can apply maximum-likelihood estimation to get another

estimate of the parameter yðmþ1Þ ¼ HðbGÞ. The process G does not admit a closed-formlikelihood. Fortunately, the approximate likelihood introduced in the first part of the

paper can be applied to estimate yðmþ1Þ. In particular, a third-order likelihoodapproximation will be used. This defines an iterative estimation procedure.

yðmþ1Þ ¼ HðGðyðmÞÞÞ. (16)

Let y�T ;D denote the uncomputable MLE if we can observe the process G, i.e.,

y�T ;D ¼ HðGðy0ÞÞ. y�T ;D is uncomputable because the process G is not directly observed by

the econometrician. The following proposition shows that the iterative maximum-likelihood estimation procedure converges and the resulting estimator approaches theefficiency of y�T ;D. As a reminder, d is the maturity of the forward contract (one month

here), D is the sampling interval (daily here) and T is the sample period (February2002–December 2003).

Proposition 3. Given D and T, there exists a positive function MðdÞ !d!0

0 so that as d ! 0,with probability approaching one, the mapping HðGð:ÞÞ is a contraction with modulus MðdÞ

and the iterative maximum-likelihood estimation procedure converges to an estimate byT ;D.Further,

kbyT ;D � y0kp1

1�MðdÞky�T ;D � y0k.

This proposition ensures we can approach efficiency feasible when the process G isobservable. Theorem 3 ensures y�T ;D approaches the efficiency of the MLE using the exactlikelihood function. Therefore, the iterative MLE asymptotically achieves the efficiency ofthe true MLE estimator which is uncomputable both because the process G is unobservableand because the likelihood function is unavailable in closed form. When byT ;D is estimated,bG ¼ GðbyT ;DÞ estimates the time series of the process G.

4.4. Estimation result

To invert G from the forward pricing formula, forward rate of one maturity suffices. Thedata contain forward rates of nine different maturities: one day, two days, one week, onemonth, two months, three months, six months, nine months and one year. Proposition 3demands the use of the shortest maturity. However, we did not take Proposition 3 literallyand use the forward price with the shortest maturity (one day) because, when the time-to-maturity is too small, the realignment risk becomes negligible and the price can be prone tonoises not modelled here. In the sample, the forwards with extremely short maturitiesbehave very differently from other forwards. The forward contracts with maturities onemonth or longer are highly correlated (correlation coefficient above 0.95) and any one ofthem captures most of the variations of other forwards. Therefore, the one-month forwardrate will be used in the estimation, striking a balance between Proposition 3 and the short-term noises. Given that we are mostly concerned with the realignment risk in the next

ARTICLE IN PRESS

Table 1

Parameter estimates

k s Z sK

0:2545ð0:042Þ

0:5159ð0:012Þ

31:414ð10:38Þ

0:2393ð0:240Þ


several years, this choice is more sensible than the one-day forward rate. We applied theiterative MLE procedure on the spot exchange rate and the one-month NDF rate. Theestimates are in Table 1. The standard errors in the parentheses are computed fromTheorem 4. A statistically significant Z indicates the presence of jumps. The jump standarddeviation sK is harder to estimate due to the infrequent nature of jumps.

The future realignment probability can be assessed from the term structure of the forwardrealignment rate. Using the parameter estimates and the filtered G, this term structure onDecember 12, 2003, the last day in the sample, is recovered and plotted in Fig. 4.17

Interestingly, the term structure is hump-shaped and peaks at about six months fromDecember 12, 2003. The market considers the Yuan likely to appreciate in 2004 and,conditioning on no realignment in that period, realignment in the further future isperceived less likely.

4.5. What does the realignment probability respond to?

Fig. 5 plots the daily estimates of process bG. The realignment intensity can varydramatically over a short period of time. In this section, we infer from the estimates whatinformation the realignment intensity, especially its jumps, responds to. Such informationhelps identify the factors influencing exchange rate realignment.

To identify when a jump takes place, we computed the standard deviation over one day ofthe process G should there be no jumps. Days over which G changed by more than five suchstandard deviations are singled out. They contain at least one jump with probability close toone by Bayes rule. There are 24 jumps identified in the sample period using this method.

To find out what these jumps respond to, we collected new releases on these jump days.Eight of these jumps are found to coincide with economic news releases. These eight daysare marked in Fig. 5 by dots and the jump magnitudes are in the parentheses. Therealignment intensity is proportional to the exponential of G, so a 0.01 increase in Gtranslates into about a one percentage increase of the intensity. On some of these jumpdays, the realignment intensity swings by as much as 50%. News releases on the eight jumpdays are listed in Table 3 in Appendix E. The jumps can be seen to coincide with newsreleases on the Sino–US trade surplus, state-owned enterprise reform, Chinese governmenttax revenue and, most importantly, both domestic and foreign government interventions.Given the tightly managed nature of the Chinese Yuan, it is not surprising that themarket’s second-guess of the Chinese government’s intention plays a dominant role.Diplomatic pressures from foreign governments are also important.

To illustrate how the term structure of the forward realignment rate responds to thenews release, Fig. 6 plots the term structures on August 28, August 29 and September 2 in

17Simulation with 100,000 sample paths is used to compute the term structure. Each week is discretized with 50

points in between using the Euler Scheme.

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0 1 2 3 4 50.1

0.15

0.2

0.25

0.3

0.35

0.4

Year(s) from December 12, 2003

Fo

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Fig. 4. Term structure of forward realignment rate.

Jan02 Apr02 Jul02 Oct02 Jan03 Apr03 Jul03 Oct03 Jan040.5

1

1.5

2

2.5

3

3.5

Date

Gam

ma

3/8/02 (-0.21)

1/6/03 (0.35)6/13/03 (0.18)

8/29/03 (0.52)

9/2/03

(-0.17)

9/23/03 (0.42)

10/7/03 (0.37)

10/8/03

(-0.64)

Fig. 5. Filtered G and eight of its jumps.

J. Yu / Journal of Econometrics 141 (2007) 1245–12801266

2003. On August 29, 2003, before the visit of the US Treasury Secretary, the JapaneseFinance Minister publicly urged China to let the Yuan float freely and the state-ownedenterprises in China showed improved profitability. On the same day, the level of theforward realignment rate almost doubled relative to the previous day. More interestingly,the peak of the term structure moved from ten months in the future to six months in thefuture, indicating the market is anticipating the Yuan to appreciate sooner. Four dayslater, after the diplomatic pressure receded, the peak of the term structure moved back toroughly eight months in the future and the level of the term structure dropped.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.16

0.18

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0.22

0.24

0.26

0.28

0.3

0.32

Year(s) later

Fo

rwa

rd R

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lign

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nt R

ate

08/28/03

09/02/03

08/29/03

Fig. 6. Changes of the term structure on jump days.


Economic theory is needed to gain further insights on the Yuan’s realignment. Most ofthe existing theory on exchange rate is about depreciation, unlike the case of the Yuanwhere the market is expecting an appreciation. The results here lend empirical evidence tomodelling realignment where a currency is under pressure to appreciate.

5. Conclusion

This paper provides closed-form likelihood approximations for multivariate jump-diffusion processes. For a fixed order of approximation, the maximum-likelihoodestimator (MLE) computed from this approximate likelihood achieves the asymptoticefficiency of the true yet uncomputable MLE as the sampling interval shrinks. It facilitatesthe estimation of jump-diffusions which are very useful in analyzing many economicphenomena such as currency crises, financial market crashes, defaults, etc. Theapproximation method is based on expansions from Kolmogorov equations which arecommon to Markov processes and therefore can potentially be generalized to other classesof Markov processes such as the time-changed Levy processes in Carr et al. (2003) and thenon-Gaussian Ornstein–Uhlenbeck-based processes in Barndorff-Nielsen and Shephard(2001). The question then is to find an appropriate leading term for these processes toreplace (6).

The approximation technique is applied to the Chinese Yuan to study its realignmentprobability. The term structure of the forward realignment rate is hump-shaped and peaksat six months from the end of 2003. The implication is that the financial market isanticipating an appreciation in the next year and, conditioning on no realignment beforethen, the chance of realignment is perceived to be smaller in the further future. SinceFebruary 2002, the realignment intensity of the Yuan has increased fivefold. Therealignment probability responds quickly to news releases on the Sino–US trade surplus,


state-owned enterprise reform, Chinese government tax revenue and, most importantly,both domestic and foreign government interventions.

Acknowledgments

I thank Yacine Aıt-Sahalia for helpful discussions throughout this project. I amespecially grateful to Peter Robinson (the Editor), an associate editor and two anonymousreferees for comments and suggestions that greatly improved the paper. I also thankLaurent Calvet, Gregory Chow, Martin Evans, Rodrigo Guimaraes, Harrison Hong, BoHonore, Robert Kimmel, Adam Purzitsky, Helene Rey, Jacob Sagi, Ernst Schaumburg,Jose Scheinkman, Wei Xiong and participants of the Princeton Gregory C. ChowEconometrics Research seminar and Princeton Microeconometric Workshop for helpfuldiscussions. All errors are mine.

Appendix A. Assumptions

To save notation, the dependence on y of the functions mð:; yÞ, sð:; yÞ, V ð:; yÞ, lð:; yÞ; nð:; yÞand pðD; yjx; yÞ will not be made explicit when there is no confusion.

Assumption 1. The variance matrix V ðxÞ is positive definite for all x in the domain of theprocess X.

This is a non-degeneracy condition. Given this assumption, we can find, by Choleskydecomposition, an n� n positive definite matrix V1=2ðxÞ satisfying V ðxÞ ¼ V 1=2ðxÞ

½V 1=2ðxÞ�T.

Assumption 2. The stochastic differential equation (1) has a unique solution. Thetransition density pðD; yjxÞ exists for all x, y in the domain of X and all D40. pðD; yjxÞ iscontinuously differentiable with respect to D, twice continuously differentiable with respectto x and y.

Theorem 2–29 in Bichteler et al. (1987) provides the following sufficient conditions forAssumption 2 to hold: (1) jump intensity lð:Þ is constant, (2) mð:Þ and sð:Þ are infinitelycontinuously differentiable with bounded derivatives, (3) the eigenvalues of V ð:Þ is boundedbelow by a positive constant, and (4) The jump distribution of Jt has moments of all orders.

Assumption 3. The boundary of the process X is unattainable.

This assumption implies that the transition density pðD; yjxÞ is uniquely determined bythe backward and forward Kolmogorov equations.

Assumption 4. nð:Þ, mð:Þ, sð:Þ and lð:Þ are infinitely differentiable almost everywhere in thedomain of X .

Assumption 5. The model (1) describes the true data-generating process and the trueparameter, denoted y0, is in a compact set Y.

Let IT ;DðyÞ � Ey½_lT ;DðyÞ_lT ;DðyÞ

T� be the information matrix. To simplify the notation, _lT ;DðyÞ

is used for derivative with respect to the vector y. Let GT ;DðyÞ � �I�1=2T ;D ðyÞ€lT ;DðyÞI

�1=2T ;D ðyÞ and

ST ;DðyÞ � I�1=2T ;D ðyÞ_lT ;DðyÞ:


Assumption 6. IT ;DðyÞ is invertible and lT ;DðyÞ is thrice continuously differentiable withrespect to y 2 Y. There exists D40 such that
kIT ;DðyÞk�1!p0 as T !1, uniformly for y 2 Y and DpD; Uniformly for y 2 Y and DpD, G�1T ;DðyÞ and ST ;DðyÞ converge to GDðyÞ and SDðyÞ thatare bounded in probability as T !1; kI�1=2T ;D ðyÞ l
:::

T ;DðeyÞI�1=2T ;D ðyÞk is uniformly bounded in probability for y, ey 2 Y and DpD.

The last condition in this assumption is stronger than the usual one which only requires

uniform boundedness in probability for ey in a shrinking neighborhood of y. This stronger

version is used to prove the asymptotic property of byðmÞ. If the process is stationary, this

condition will automatically hold because kl�1=2T ;D ðyÞ l

:::

T ;DðeyÞI�1=2T ;D ðyÞk will converge in

probability to a constant ZðD; y;eyÞ which is bounded for parameters in the compact set Yand DpD (see Eq. (A.49) in Aıt-Sahalia, 2002 for the stationary diffusion case), so no costis incurred from strengthening this condition.

Assumption 7. For any to1, Mt � sup0psptkX sk is bounded in probability.

Appendix B. Proofs

Proof of Proposition 1. The result in this proposition is standard and the proof is thereforeomitted and is available upon request.

Derivation of Condition 2. It is clear that this initial condition does not affect the result inTheorem 1 which is satisfied by any solution to the backward equation.

Knowing Cð�1Þðx; yÞ ¼ 12½ðq=qxÞCð�1Þðx; yÞ�TV ðxÞ½q=qxCð�1Þðx; yÞ�,Z

pðD; yjxÞdy

¼

ZD�n=2 exp �

Cð�1Þðx; yÞ

D

� �Cð0Þðx; yÞdyþ oðDÞ

¼

ZD�n=2 exp �

½ðq=qxÞCð�1Þðx; yÞ�TV ðxÞ½ðq=qxÞCð�1Þðx; yÞ�

2D

�Cð0Þðx; yÞdyþ oðDÞ

¼ ð2pÞn=2Zð2pDÞ�n=2 expð�wTw=ð2DÞÞCð0Þðx;w�1x ðwÞÞ

jdetV ðxÞ1=2ðq2=qxqyTÞCð�1Þðx;w�1x ðwÞÞjdwþ oðDÞ

by a change of variable formula. w�1x ð:Þ is defined as the inverse function ofV ðxÞ1=2ðq=qxÞCð�1Þðx; :Þ and satisfies w�1x ð0Þ ¼ x. It can be verified as in the proof ofTheorem 1 that this change of variable is legitimate.ð2pDÞ�n=2 expð�wTw=ð2DÞÞ converges to a dirac-delta function at 0 implies, as D! 0,Z

pðD; yjxÞdy! ð2pÞn=2Cð0Þðx;xÞ detV ðxÞ1=2q2

qxqyTCð�1Þðx; yÞ

��y¼x

��1

.


Requiring the right-hand side equals 1 gives Condition 2, noticing from Eq. (11) that

q2

qxqyTCð�1Þðx; yÞ

��y¼x

¼ �V ðxÞ�1: &

The proofs to Theorems 1 and 2 are very similar. We, therefore, prove Theorem 1.Corollaries 1 and 2 are obtained by explicitly solving the equations in Theorems 1 and 2.

Proof of Theorem 1. To get CðkÞðx; yÞ and DðkÞðx; yÞ, we replace pðD; yjxÞ with (6) in thebackward Kolmogorov equation (3), match the terms with the same orders of Dk and theterms with the same orders of Dk exp½�Cð�1Þðx; yÞ=D� and set their coefficients to 0. Most ofthe steps are careful bookkeeping, only the following two points need to be addressed.
How to expandR
CpðD; yjxþ cÞnðcÞdc in increasing orders of D?
Prove that, for fixed y, wBð:; yÞ is invertible in a neighborhood of x ¼ y and its inversefunction w�1B ð:Þ is continuously differentiable.
NoticeZC

pðD; yjxþ cÞnðcÞdc! nðy� xÞ as D! 0

implies the expansion ofR

CpðD; yjxþ cÞnðcÞdc does not involve terms in the form of

Dk exp½�Cð�1Þðx; yÞ=D�. Therefore, we first group terms with the same orders ofDk exp½�Cð�1Þðx; yÞ=D�, set their coefficient to 0 and get


2

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cð�1Þðx; yÞ

� �(17)

which is the equation for Cð�1Þðx; yÞ in Theorem 1. Take the derivative of wBðx; yÞ andnotice ðq=qxÞCð�1Þðx; yÞjx¼y ¼ 0,

qqxT

wBðx; yÞ

��x¼y

¼ ½V1=2ðyÞ�Tq2

qxqxTCð�1Þðx; yÞ

��x¼y

¼ V ðyÞ�1=2

by Eq. (11). By the inverse mapping theorem (Section 8.5 in Hoffman, 1975), there is anopen neighborhood of x ¼ y, so that wBðx; yÞ, for fixed y, maps this neighborhood 1–1 andonto an open neighborhood of 0 in Rn and its inverse function is continuouslydifferentiable.Now, knowing wBðx; yÞ is invertible, we will expand

RC

pðD; yjxþ cÞnðcÞdc in increasingorders of D. First, for a given kX0, by (17)Z

C

D�n=2 exp �Cð�1Þðxþ c; yÞ

D

� �CðkÞðxþ c; yÞnðcÞdc

¼

ZC

D�n=2 exp �wT

Bðxþ c; yÞwBðxþ c; yÞ

2D

� �CðkÞðxþ c; yÞnðcÞdc. ð18Þ

Let C1 � C be an open set containing y� x on which wBðxþ :; yÞ is invertible for the givenx and y. Denote by W 2 Rn the image of C1 under wBðxþ :; yÞ. As shown previously, W is


open and contains 0. As D! 0, the last integral differs exponentially small fromZC1

D�n=2 exp �wT

Bðxþ c; yÞwBðxþ c; yÞ

2D

� �CðkÞðxþ c; yÞnðcÞdc

¼

ZW

D�n=2 exp �oTo2D

� �gkðx; y;oÞdo

by a change of variable. The function gkðx; y;oÞ is defined in the theorem. Using themultivariate differentiation notation in Section 2.3, we Taylor expand gkðx; y;oÞ aroundo ¼ 0 to getZ

W


� �X1r¼0

1

r!

Xjsj¼r

os qs

qusgkðx; y; uÞ

��u¼0

do

¼X1r¼0

1

r!

Xjsj¼r

qs

qosgkðx; y;oÞ

��o¼0

ZW


� �os do

X1r¼0

1

r!

Xjsj¼r

qs

qosgkðx; y;oÞ

��o¼0

Dr=2

ZRn

exp �oTo2

� �os do,

where the last term follows by first changing the variable o=ffiffiffiffiDp

and then ignore terms thatare exponentially small as D! 0. We note that, to decide which terms are exponentiallysmall, we need not worry about the change of limit operators of the form: limD!0

P1r¼0 toP1

r¼0limD!0 simply because only finite number of terms in the expansionP1

r¼0 will be usedto compute the approximate transition density pðmÞðD; yjxÞ for fixed m. I.e., the notationP1

r¼0 here does not mean convergence of the infinite sum, it simply represents a Taylorexpansion from which only a finite number of terms will be used.

Let Mns � ð2pÞ

�n=2 RRn exp½�wTw=2�ws dw denote the sth moment of n-variate standard

normal distribution. Mns ¼ 0 if any element of s is odd. The last expression therefore equals

¼ ð2pÞn=2X1r¼0

Dr

ð2rÞ!

Xs2Sn

2r

qs

qwsgkðx; y;wÞ

��w¼0

Mns (19)

with Sn2r defined as in Section 2.3.

Note, we have implicitly assumed y� x 2 C. In the case y� xeC, Assumption 4 impliesnð:Þ and its derivatives of all orders vanish at y� x which implies the right-hand side of (19)is 0. It is also clear that (18) is exponentially small in D if y� xeC. Therefore, (19) is stillvalid.

Now, knowing the above expansion,ZC

D�n=2 exp �Cð�1Þðxþ c; yÞ

D

� �X1k¼0

CðkÞðxþ c; yÞDknðcÞdc

¼ ð2pÞn=2X1k¼0

DkX1r¼0

Dr

ð2rÞ!

Xs2Sn

2r

qs

qwsgkðx; y;wÞ

��w¼0

Mns

¼ ð2pÞn=2X1k¼0

DkXk

r¼0

1

ð2rÞ!

Xs2Sn

2r

qs

qwsgk�rðx; y;wÞ

��w¼0

Mns .


This last expression, expanded in increasing orders of Dk, can be used to match otherterms in orders of D. This completes the proof. &

Proof of Theorems 3 and 4. The true MLE sets the score to 0. Therefore,

_lT ;Dðy0Þ ¼ �€lT ;DðeyÞðbyT ;D � y0Þ

for some ey in between y0 and byT ;D.

I1=2T ;Dðy0ÞðbyT ;D � y0Þ ¼ I

1=2T ;Dðy0Þ½�€lT ;Dð

eyÞ��1 _lT ;Dðy0Þ

¼ � ½I�1=2T ;D ðy0Þ€lT ;Dð

eyÞI�1=2T ;D ðy0Þ��1I�1=2T ;D ðy0Þ_lT ;Dðy0Þ

¼ � ½I�1=2T ;D ðy0Þ€lT ;Dðy0ÞI

�1=2T ;D ðy0Þ�

�1½I�1=2T ;D ðy0Þ_lT ;Dðy0Þ� þOpð1Þ

¼ G�1D ðy0ÞSDðy0Þ þOpð1Þ

as T !1 uniformly for DpD and y0 2 Y by Assumption 6. We have now proved theconsistency of byT ;D.

Knowing that byT ;D will asymptotically be in an I�1=2T ;D ðy0Þ-neighborhood of y0, by

repeating the previous argument, we have

I1=2T ;Dðy0ÞðbyT ;D � y0Þ ¼ � ½I

�1=2T ;D ðy0Þ€lT ;Dð

eyÞI�1=2T ;D ðy0Þ��1I�1=2T ;D ðy0Þ_lT ;Dðy0Þ

¼ � ½I�1=2T ;D ðy0Þ€lT ;Dðy0ÞI

�1=2T ;D ðy0Þ�

�1½I�1=2T ;D ðy0Þ_lT ;Dðy0Þ� þ opð1Þ

¼ G�1D ðy0ÞSDðy0Þ þ opð1Þ.

The asymptotic distribution of byT ;D now follows from the limiting distributions of GD and

SD. In the stationary case, GDðy0Þ is non-random and it follows as in Theorem 4 that

ðnT ;DiDðy0ÞÞ1=2ðbyT ;D � y0Þ ¼ Nð0; Ip�pÞ þ opð1Þ.

We next investigate the stochastic difference (see Robinson, 1988) between byðmÞT ;DTandbyT ;DT

.

_lT ;DTðbyðmÞT ;DT

Þ � _lðmÞT ;DTðbyðmÞT ;DT

Þ

¼ _lT ;DTðbyðmÞT ;DT

Þ

¼ _lT ;DTðbyT ;DT

Þ þ €lT ;DTðyÞðbyðmÞT ;DT

� byT ;DTÞ

¼ €lT ;DTðyÞðbyðmÞT ;DT

� byT ;DTÞ

for some y between byT ;DTand byðmÞT ;DT

. Therefore,

I1=2T ;DTðy0ÞðbyðmÞT ;DT

� byT ;DTÞ

¼ ½I�1=2T ;DTðy0Þ€lT ;DT

ðyÞI�1=2T ;DTðy0Þ��1I

�1=2T ;DTðy0Þ½_lT ;DT

ðbyðmÞT ;DTÞ � _lðmÞT ;DT

ðbyðmÞT ;DTÞ�

¼ ðG�1D ðy0Þ þOpð1ÞÞI�1=2T ;DTðy0Þ½_lT ;DT

ðbyðmÞT ;DTÞ � _lðmÞT ;DT

ðbyðmÞT ;DTÞ�.

We will show later that there exists a sequence fD�T g so that for any sequence fDT g

satisfying DTpD�T , _lðmÞT ;DTð�Þ incurs a relative error of order opðDm

T Þ in approximating _lT ;DTð�Þ,

uniformly for the parameters in the compact set Y. Assuming this holds now, we have, for


some ey between byðmÞT ;DTand y0,


� byT ;DTÞ ¼ opðDm

T ÞI�1=2T ;DTðy0Þ_lT ;DT

ðbyðmÞT ;DTÞ

¼ opðDmT ÞI�1=2T ;DTðy0Þ½_lT ;DT

ðy0Þ þ €lT ;DTðeyÞðbyðmÞT ;DT

� y0Þ�

¼ opðDmT Þ½I

�1=2T ;DTðy0Þ_lT ;DT

ðy0Þ þOpð1ÞI1=2T ;DTðy0ÞðbyðmÞT ;DT

� y0Þ�

¼ opðDmT Þ½S þOpð1ÞI

1=2T ;DTðy0ÞðbyðmÞT ;DT

� byT ;DTþ byT ;DT

� y0Þ�,

where S is the probability limit of I�1=2T ;DTðy0Þ_lT ;DT

ðy0Þ under Py0 as T !1.

Therefore,

ðI � opðDmT ÞÞI

1=2T ;DTðy0ÞðbyðmÞT ;DT

� byT ;DTÞ ¼ opðDm

T Þ½Opð1Þ þOpð1ÞI1=2T ;DTðy0ÞðbyT ;DT

� y0Þ�

¼ opðDmT Þ

which proves Theorem 3.Finally, we show the existence of fD�T g, which was used in proving the theorem.

Uniformly for y in the compact set Y, pðmÞðD;X iDjX ði�1ÞD; yÞ is an expansion of

pðD;X iDjX ði�1ÞD; yÞ with relative error Dme1ðeDi;X iD;X ði�1ÞD; yÞ for some eDipD, i.e.,

pðmÞðD;X iDjX ði�1ÞD; yÞ is equal to pðD;X iDjX ði�1ÞD; yÞð1þ Dme1ðeDi;X iD;X ði�1ÞD; yÞÞ. Simi-

larly, ðq=qyÞpðmÞðD;X iDjX ði�1ÞD; yÞ is an expansion of ðq=qyÞpðD;X iDjX ði�1ÞD; yÞ with relative

error Dme2ðbDi;X iD;X ði�1ÞD; yÞfor some bDipD.Now we bound the functions e1 and e2. Let fqtg and fxtg be two sequences of positive

numbers converging to 0. Under the assumption that MT is bounded in probability for anyTo1, X t for all toT is in a compact set with probability 1� qT and we can alwayschoose the sequence fD�T g converging to 0 fast enough so that for any fDT g satisfying

DTpD�T , ke1k and ke2k for observations inside this compact set are bounded by xT ,

uniformly for y 2 Y. With this choice of fD�T g, _lT ;DTðbyðmÞT ;DT

Þ � _lðmÞT ;DTðbyðmÞT ;DT

Þ equals

opðDmT Þ_lT ;DTðbyðmÞT ;DT

Þ. &

Proof of Corollary 3. This follows from Theorems 3 and 4 and that


� y0Þ ¼ I1=2T ;DTðy0ÞðbyðmÞT ;DT

� byT ;DTÞ þ I

1=2T ;DTðy0ÞðbyT ;DT

� y0Þ: &

Proof of Proposition 2. By the non-negativity of f ð�Þ,

�f ðX 0ÞpE½f ðX tÞ� � f ðX 0Þ

¼

Z t

0

E½ABf ðX sÞ�ds.

limjxj!1ABf ðxÞ ¼ �1 implies for jxj4r, ABf ðxÞ is bounded above by a constant �Cr

where Cr !1 as r!1. Let supxABf ðxÞ ¼ ko1. Therefore,

�f ðX 0Þp�Z t

0

CrPðjX sj4rÞdsþ kt.


1

t

Z t

0

PðjX sj4rÞdsp�f ðX 0Þ

tCr

�k

Cr

p�k

Cr

! 0 as r!1.

Stationarity of X now follows from Theorem III.2.1 in Has’minskiı, 1980. &

Proof of Proposition 3. Let Gð:Þ ¼ HðGð:ÞÞ. First, we show, with probability approachingone as d ! 0, kG0koMðdÞ for some MðdÞ !

d!00. By Eq. (13), the forward pricing satisfies

F dðS;GÞ ¼ S þ df 1ðGÞ þ12d2f 2ðG; yÞ þ oðd2Þ

for some function f 1 and f 2. In particular, f 1 ¼ lueGEðJuÞ þ lde

�GEðJdÞ. Notice, theparameter y does not affect f 1. Therefore, the function G from inverting this pricingformula satisfies

qGqy¼ �

qf 2ðG; yÞ=qyqf 1ðG; yÞ=qG

d þ oðdÞ.

Given G, the next-step iterative estimator satisfies ðq=qyÞLðG; yÞ ¼ 0. Therefore,

qqG

H ¼ �ðq2=qyqGÞL

ðq2=qy2ÞL.

y is in a compact set and with probability approaching one, G stays in a compact set forgiven D and T. Therefore, ðq=qGÞH is bounded, ðqG=qyÞ is OðdÞ. It is clear then we can findMðdÞ !

d!00 so that kG0koMðdÞo1 with probability approaching one as d ! 0: Therefore,

HðGð:ÞÞ is a contraction. The convergence of the iterative MLE procedure now followsfrom the contraction mapping theorem (see Theorem 3.2 in Stokey et al., 1996). Now, withprobability approaching one as d! 0,

kbyT ;D � y0kpkbyT ;D � y�T ;Dk þ ky�T ;D � y0k

¼ kHðGðbyT ;DÞÞ �HðGðy0ÞÞk þ ky�T ;D � y0k

pMðdÞkbyT ;D � y0k þ ky�T ;D � y0k

and

kbyT ;D � y0kp1

1�MðdÞky�T ;D � y0k: &

Appendix C. Extensions of the approximation method

C.1. When some state variables do not jump

The state variables in the main part of the paper jump simultaneously. However, it issometimes of interest to consider cases where a subset of the state variables jump whileothers do not.18 In this case, the solution to the Kolmogorov equations no longer has anexpansion in the form of (6). However, the methodology still applies and all we need isanother form of leading term in the expansion. This new leading term, together with theclosed-form approximation, will be provided below.

18As an example, one could model an asset price as a jump diffusion with stochastic volatility where the

volatility process follows a diffusion.


Let the model setup be the same as Section 2.1 except that the state variable X can bepartitioned into two subsets.

X n�1 ¼X C

n1�1

X Dn2�1

0@ 1A,

where X D can jump and X C cannot.To simplify the algebra, only the following case is considered where the variance matrix

can be partitioned as

V ðX Þ ¼V 11ðX

CÞ 0

0 V22ðXDÞ

!.

This condition simplifies the algebra below without losing any intuition on the newleading term. It can be relaxed, though details are suppressed.

The transition density in this case has an expansion in the form of

pðD; yjxÞ ¼ D�n=2 exp �Cð�1Þðx; yÞ

D

� �X1k¼0

CðkÞðx; yÞDk

þ D�n1=2 exp �Dð�1ÞðxC ; yCÞ

D

� �X1k¼1

DðkÞðx; yÞDk.

Intuitively, when jump happens, X C has to diffuse to the new location while X D canjump to the new location thus the new form of leading term. (See also Section 2.2). Thisconjecture can be substituted into the Kolmogorov equations and solve for the unknowncoefficient functions.

It turns out the functions CðkÞðx; yÞ are exactly the same as those in Theorems 1 and 2.This is intuitive since a change in jump behavior will not affect terms for the diffusion part.The other coefficient functions are characterized as

Dð�1ÞðxC ; yCÞ ¼1

2

qqxC

Dð�1ÞðxC ; yCÞ

� �TV 11ðx

CÞq

qxCDð�1ÞðxC ; yCÞ

� �,

Dð1Þðx; yÞ ¼ lðxÞnðy� xÞ �Dð1Þ LBDð�1Þ �n1

2

h i�

qqx

Dð�1Þðx; yÞ

� �TV ðxÞ

qqx

Dð1Þðx; yÞ

� �,


1þ k

ABDðkÞ þ ð2pÞn2=2lðxÞPkr¼0

1

ð2rÞ!

Ps2S

n22r

Mn2s

qs

qwsgk�rðx; y;wÞ

��w¼0

�Dðkþ1Þ LBDð�1Þ �n1

2

h i�

qqx

Dð�1Þ� �T

V ðxÞqqx

DðkÞðx; yÞ

� �8>>>>><>>>>>:

9>>>>>=>>>>>;for k40,


where

gkðx; y;wÞ �

CðkÞxC

w�1B ðwÞ

!; y

!nðw�1B ðwÞ � xDÞ

detq

qðxDÞT

wBðxD; yDÞ

��xD¼w�1

BðwÞ

��

; wBðxD; yDÞ � ½V

1=222 ðx

DÞ�Tq

qxDHðxD; yDÞ.

The function H satisfies HðxD; yDÞ ¼ 12½ðq=qxDÞHðxD; yDÞ�TV22ðx

DÞ½ðq=qxDÞHðxD; yDÞ�.Fixing yD, wBð:; yDÞ is invertible in a neighborhood of xD ¼ yD and w�1B ð:Þ is its inversefunction in this neighborhood. (For the ease of notation, the dependence of w�1B ð:Þ on yD isnot made explicit.)

C.2. State-dependent jump distribution

Now let us consider a process characterized by its infinitesimal generator AB on abounded function f with bounded and continuous first and second derivatives

ABf ðxÞ ¼Xn

i¼1

miðxÞqqxi

f ðxÞ þ1

2

Xn

i¼1

Xn

j¼1

vijðxÞq2

qxiqxj

f ðxÞ

þ lðxÞZ

C

½f ðxþ cÞ � f ðxÞ�nðx; cÞdc.

The jump distribution nðx; :Þ is now state-dependent. The solution of the Kolmogorovequations admits an expansion in the form of (6). It turns out that the coefficient functionsCðkÞðx; yÞ and DðkÞðx; yÞ in this case are characterized by Theorem 1 and Corollary 1, withthe exception that nð:Þ is replaced everywhere by nðx; :Þ.

C.3. Multiple jump types

Consider a process whose infinitesimal generator AB, on a bounded function f withbounded and continuous first and second derivatives, is given by

ABf ðxÞ ¼Xn

i¼1

miðxÞqqxi

f ðxÞ þ1

2

Xn

i¼1

Xn

j¼1

vijðxÞq2

qxiqxj

f ðxÞ

þX

u

luðxÞ

ZC

½f ðxþ cÞ � f ðxÞ�nuðx; cÞdc.

This is a process with multiple jump types with different jump intensity luð:Þ and jumpdistribution nuðx; :Þ. The solution of the Kolmogorov equations also admits an expansionin the form of (6) where the coefficient functions CðkÞðx; yÞ and DðkÞðx; yÞ are given as

0 ¼ Cð�1Þðx; yÞ �1

2

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cð�1Þðx; yÞ

� �and Cð�1Þðx; yÞ ¼ 0 when y ¼ x,

0 ¼ Cð0Þ LBCð�1Þ �n

2

h iþ

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cð0Þðx; yÞ

� �,


0 ¼ Cðkþ1Þ LBCð�1Þ þ ðk þ 1Þ �n

2

h iþ

qqx

Cð�1Þðx; yÞ

� �TV ðxÞ

qqx

Cðkþ1Þðx; yÞ

� �þ

Xu

luðxÞ � LB

" #CðkÞ for non-negative k,

0 ¼ Dð1Þ �X

u

luðxÞnuðx; y� xÞ,

0 ¼ Dðkþ1Þ �1

1þ kABDðkÞ þ ð2pÞn=2

Xk

r¼0

1

ð2rÞ!

Xu

luðxÞXs2Sn

2r

Mns

qs

qwsgu;k�rðx; y;wÞ

��w¼0

24 35for k40,

where gu;kðx; y;wÞ � CðkÞðw�1B ðwÞ; yÞnuðw�1B ðwÞ � xÞ=j detðq=qxTÞwBðx; yÞjx¼w�1

BðwÞj, wBðx; yÞ

� ½V 1=2ðxÞ�Tðq=qxÞCð�1Þðx; yÞ: Fixing y, wBð:; yÞ is invertible in a neighborhood of x ¼ y

Table 2

History of the Yuan’s exchange rate regime

Date Exchange rate regime Official

rate

March 1, 1955 The Yuan was created and pegged to USD 2.46

August 15, 1971 A new official rate against the USD was announced 2.267

February 20, 1973 The official rate (the ‘‘Effective Rate’’) was realigned 2.04

August 19,

1974–December 31,

1985

The Effective Rate was pegged to a trade-weighted basket of fifteen

currencies with undisclosed compositions. The rate was fixed almost daily

against that basket

1.5–2.8

January 1, 1986 The Effective Rate was placed on a controlled float

July 5, 1986 The Effective Rate was fixed against the USD until December 15, 1989 3.72

November 1986 A second rate, Foreign Exchange Swap Rate determined by market force,

was created

3.72

The Foreign Exchange Swap Rate quickly moved to 5.2 in one month

December 15, 1989 The Effective Rate was realigned 4.72

November 17, 1990 The Effective Rate was realigned 5.22

April 9, 1991 The Effective Rate start to adjust frequently depending on certain

economic indicators

December 31, 1993 The Effective Rate was 5.8 5.8

The Foreign Exchange Swap Rate was 8.7

January 1, 1994 The Effective Exchange Rate and the swap market rate were unified at the

prevailing swap market rate

Daily movement of the exchange rate of the Yuan against the USD is

limited at 0.3% on either side of the reference rate as announced by the

PBC

ARTICLE IN PRESS

Table 3

News release on the jump days

Date Events News source

3-8-2002 1. State-owned enterprises, which contribute 60 percent of China’s tax

revenue, see a profit drop of 1.4% last year

ChinaOnline

2. China plans to incur 80 billion Yuan, a quarter of its planned 2003

deficit, on restructuring SOEs

InfoProd

1-6-2003 1. China’s foreign trade increased 21 percent over 2002 with surplus

expanding to USD 27 billion for the first 11 month of 2002

Business Daily

2. China’s tax revenue rose 12.1% year-over-year with GDP estimated

to grow by 8%

China Daily

6-13-2003 Goldman Sachs forecasts an appreciation of the Yuan to 8.07 by the

year-end based on the view that China was preparing for a change in

currency policy, as SARS and deflationary pressures recede and the

dollar weakens

AFX News

8-29-2003 1. Japanese Finance Minister urges China to let the Yuan float freely

before meeting US treasury secretary

Japan Economic

Newswire

2. China’s state-owned enterprises posted a profit of 212 billion Yuan

from Jan to July, up 47% year-over-year

AFX News

9-2-2003 US Treasury Secretary wraps up first day of talks in China, no

comments on the Yuan

AFX News

9-23-2003 G-7 requests China to move rapidly towards more flexible exchange

rates

South China

Morning Post

10-7-2003 The People’s Bank of China is considering revaluing the Yuan by

about 30 percent over the next five years, subject to a final decision

from the State Council led by Premier Wen Jiabao

Jiji Press Japan

10-8-2003 China’s Premier Wen Jiabao suggested he would resist foreign pressure

to revalue the Yuan

Xinhua News

J. Yu / Journal of Econometrics 141 (2007) 1245–12801278

and w�1B ð:Þ is its inverse function in this neighborhood. (For the ease of notation,

the dependence of w�1B ð:Þ on y is not made explicit.)

Appendix D. History of the Yuan’s exchange rate regime19

Table 2 documents the history of the Yuan’s exchange rate regime since its creation.20

Appendix E. News release on jump days

Table 3 documents the news releases that coincide with the jumps in G.

19Courtesy of the Economics Department at the Chinese University of Hong Kong.20Official rate is quoted in Yuan/USD.


References

Aıt-Sahalia, Y., 1996. Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527–560.

Aıt-Sahalia, Y., 2002. Maximum likelihood estimation of discretely sampled diffusions: a closed-form

approximation approach. Econometrica 70, 223–262.

Aıt-Sahalia, Y., 2006. Closed-form likelihood expansions for multivariate diffusions. Working paper.

Aıt-Sahalia, Y., Kimmel, R., 2007. Maximum likelihood estimation of stochastic volatility models. Journal of

Financial Economics 83, 413–452.

Aıt-Sahalia, Y., Yu, J., 2006. Saddlepoint approximations for continuous-time Markov processes. Journal of

Econometrics 134, 507–551.

Aıt-Sahalia, Y., Mykland, P., Zhang, L., 2005. How often to sample a continuous-time process in the presence of

market microstructure noise. Review of Financial Studies 18, 351–416.

Andersen, T., Benzoni, L., Lund, J., 2002. An empirical investigation of continuous-time equity return models.

Journal of Finance 57, 1239–1284.

Andersen, T., Bollerslev, T., Diebold, F., 2006. Roughing it up: including jump components in the measurement,

modeling and forecasting of return volatility. Review of Economics and Statistics, forthcoming.

Bakshi, G., Ju, N., 2005. A refinement to Aıt-sahalia’s (2002) ‘‘maximum likelihood estimation of discretely

sampled diffusions: a closed-form approximation approach’’. Journal of Business 78, 2037–2052.

Bakshi, G., Ju, N., Ou-Yang, H., 2006. Estimation of continuous-time models with an application to equity

volatility. Journal of Financial Economics 82, 227–249.

Bandi, F., Nguyen, T., 2003. On the functional estimation of jump-diffusion models. Journal of Econometrics

116, 293–328.

Bandi, F., Phillips, P., 2003. Fully nonparametric estimation of scalar diffusion models. Econometrica 71, 241–284.

Bandi, F., Russell, J., 2006. Volatility. In: Handbook of Financial Engineering. Elsevier, Amsterdam,

forthcoming.

Barndorff-Nielsen, O.E., Shephard, N., 2001. Non-Gaussian ornstein-uhlenbeck-based models and some of their

uses in financial economics. Journal of the Royal Statistical Society Series B 63, 167–241.

Barndorff-Nielsen, O.E., Shephard, N., 2004. Power and bipower variation with stochastic volatility and jumps.

Journal of Financial Econometrics 2, 1–48.

Bates, D., 1996. Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options.

Review of Financial Studies 9, 69–107.

Bichteler, K., Gravereaux, J.-B., Jacod, J., 1987. Malliavin Calculus for Processes with Jumps. Gordon and

Breach Science Publishers, London.

Brandt, M., Santa-Clara, P., 2002. Simulated likelihood estimation of diffusions with an application to exchange

rate dynamics in incomplete markets. Journal of Financial Economics 63, 161–210.

Bremaud, P., 1981. Point Processes and Queues, Martingale Dynamics. Springer, Berlin.

Carr, P., Geman, H., Madan, D., Yor, M., 2003. Stochastic volatility for levy processes. Mathematical Finance

13, 345–382.

Carrasco, M., Chernov, M., Florens, J.-P., Ghysels, E., 2007. Efficient estimation of general dynamic models with

a continuum of moment conditions. Journal of Econometrics, in press, doi:10.1016/j.jeconom.2006.07.013.

Carrier, G.F., Pearson, C.E., 1988. Partial differential equations, 2nd ed. Academic Press, New York.

Duffee, G., 2002. Term premia and interest rate forecasts in affine models. Journal of Finance 57, 405–443.

Duffie, D., Glynn, P., 2004. Estimation of continuous-time Markov processes sampled at random time intervals.

Econometrica 72, 1773–1808.

Duffie, D., Singleton, K., 2003. Credit Risk. Princeton University Press, Princeton, NJ.

Duffie, D., Pan, J., Singleton, K., 2000. Transform analysis and asset pricing for affine jump diffusions.


Elerian, O., Chib, S., Shephard, N., 2001. Likelihood inference for discretely observed nonlinear diffusions.


Eraker, B., Johannes, M., Polson, N., 2003. The impact of jumps in volatility and returns. Journal of Finance 58,

1269–1300.

Gallant, A.R., Tauchen, G., 1996. Which moments to match? Econometric Theory 12, 657–681.

Gourieroux, C., Monfort, A., Renault, E., 1993. Indirect inference. Journal of Applied Econometrics, S85–S118.

Hansen, L.P., Scheinkman, J., 1995. Back to the future: generating moment implications for continuous-time

Markov processes. Econometrica 63, 767–804.

http://10.1016/j.jeconom.2006.07.013


Has’minskiı, R.Z., 1980. Stochastic Stability of Differential Equations. Sijthoff & Noordhoff.

Hoffman, K., 1975. Analysis in Euclidean Space. Prentice-Hall, Englewood Cliffs, NJ.

Huang, Y., 2003. Selling China. Cambridge University Press, Cambridge.

Huang, X., Tauchen, G., 2005. The relative contribution of jumps to total price variance. Journal of Financial

Econometrics 3, 456–499.

Jeganathan, P., 1995. Some aspects of asymptotic theory with applications to time series models. Econometric

Theory 11, 818–887.

Jensen, B., Poulsen, R., 2002. Transition densities of diffusion processes: numerical comparison of approximation

techniques. Journal of Derivatives 9, 18–32.

Johannes, M., 2004. The statistical and economic role of jumps in continuous-time interest rate models. Journal of

Finance 59, 227–260.

Kessler, M., Sørensen, M., 1999. Estimating equations based on eigenfunctions for a discretely observed diffusion

process. Bernoulli 5, 299–314.

Kristensen, D., 2004. Estimation in two classes of semiparametric diffusion models. Working paper.

Ledoit, O., Santa-Clara, P., Yan, S., 2002. Relative pricing of options with stochastic volatility. Working paper.

Liu, J., Longstaff, F.A., Pan, J., 2003. Dynamic asset allocation with event risk. Journal of Finance 58, 231–259.

Merton, R., 1992. Continuous-Time Finance. Blackwell Publishers, London.

Pan, J., 2002. The jump-risk premia implicit in options: evidence from an integrated time-series study. Journal of

Financial Economics 63, 3–50.

Pedersen, A., 1995. A new approach to maximum-likelihood estimation for stochastic differential equations based

on discrete observations. Scandinavian Journal of Statistics 22, 55–71.

Piazzesi, M., 2005. Bond yields and the federal reserve. Journal of Political Economy 113, 311–344.

Protter, P., 1990. Stochastic Integration and Differential Equations. Springer, Berlin.

Revuz, D., Yor, M., 1999. Continuous Martingales and Brownian Motion, third ed. Springer, Berlin.

Robinson, P.M., 1988. The stochastic difference between econometric statistics. Econometrica 56, 531–548.

Rudin, W., 1976. Principles of Mathematical Analysis, third ed. McGraw-Hill, New York.

Schaumburg, E., 2001. Maximum likelihood estimation of levy type SDEs. Ph.D. Thesis, Princeton University.

Singleton, K., 2001. Estimation of affine asset pricing models using the empirical characteristic function. Journal

of Econometrics 102, 111–141.

Stokey, N., Lucas, R., Prescott, E., 1996. Recursive Methods in Economic Dynamics. Harvard University Press.

Sundaresan, S., 2000. Continuous-time methods in finance: a review and an assessment. Journal of Finance 55,

1477–1900.

Varadhan, S.R.S., 1967. On the behavior of the fundamental solution of the heat equation with variable

coefficients. Communications on Pure and Applied Mathematics 20, 431–455.

White, H., 1982. Maximum likelihood estimation of misspecified models. Econometrica 50, 1–26.

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