谷 偉, 崔俊交

(中南財(cái)經(jīng)政法大學(xué) 統(tǒng)計(jì)與數(shù)學(xué)學(xué)院, 武漢 430073)

?

擴(kuò)散過程模型估計(jì)效率問題研究

谷 偉*, 崔俊交

(中南財(cái)經(jīng)政法大學(xué) 統(tǒng)計(jì)與數(shù)學(xué)學(xué)院, 武漢 430073)

考慮擴(kuò)散過程模型的一種基于偏微分方程的估計(jì)方法,該方法通過數(shù)值求解與擴(kuò)散模型相關(guān)聯(lián)的偏微分方程(PDEs),獲得轉(zhuǎn)移密度函數(shù)的近似解,把Hurn等和陳暉等所采用的方法進(jìn)行了對(duì)比,并與轉(zhuǎn)移密度的閉端解和Euler法獲得的近似解進(jìn)行比較,同時(shí),比較了這幾種方法在模型參數(shù)識(shí)別方面的效率.比較結(jié)果說明Hurn法對(duì)于對(duì)數(shù)似然和的近似效果均優(yōu)于陳暉法,且對(duì)于模型參數(shù)的估計(jì)效果要優(yōu)于Euler法和陳暉法,且Hurn法對(duì)于模型參數(shù)的識(shí)別能力要好于Euler法.

極大似然估計(jì)法; Euler法; 轉(zhuǎn)移密度函數(shù)

近年來,隨著我國(guó)利率市場(chǎng)化的不斷推進(jìn),針對(duì)利率模型(特別是短期無風(fēng)險(xiǎn)利率模型)的研究,已引起許多學(xué)者的廣泛關(guān)注,如:Merton模型、Vasicek模型、CIR模型以及CKLS模型等.這些模型均是考慮通過對(duì)相應(yīng)的隨機(jī)利率模型設(shè)定不同的漂移項(xiàng)和擴(kuò)散項(xiàng)來獲得的,然而,其“逆向問題”,即利用利率的離散觀測(cè)值,估計(jì)隨機(jī)利率模型中的未知參數(shù),還有待于構(gòu)造適當(dāng)?shù)膮?shù)估計(jì)數(shù)值算法進(jìn)行深入的探索和研究.

極大似然估計(jì)是其中較流行的一種參數(shù)方法,該方法的關(guān)鍵問題是如何獲得轉(zhuǎn)移密度函數(shù).一般來說,有4種近似轉(zhuǎn)移密度函數(shù)的方法:(i) 數(shù)值求解擴(kuò)散過程滿足的Fokker-Planck偏微分方程,進(jìn)而獲得相應(yīng)的轉(zhuǎn)移密度函數(shù)近似值(Lo(1988)[1];Jensen和Poulsen(2002)[2];Hurn等(2007)[3];陳暉等(2004)[4]);(ii)Hermite多項(xiàng)式近似法(Ait-Sahalia(2002)[5], (2008)[6]);(iii)模擬極大似然估計(jì)法(Pedersen(1995)[7];Brandt和Santa-Clara(2002)[8];Durham和Gallant(2002)[9]);(iv) 精確模擬極大似然估計(jì)法(Beskos等(2006)[10]).本文考慮利率擴(kuò)散過程模型的一種基于偏微分方程的估計(jì)方法,該方法通過數(shù)值求解與擴(kuò)散模型相關(guān)聯(lián)的偏微分方程(PDE),獲得轉(zhuǎn)移密度函數(shù)的近似解.本文通過大量的數(shù)值試驗(yàn)把Hurn等(2007)和陳暉等(2004)所采用的方法進(jìn)行了對(duì)比,并與轉(zhuǎn)移密度的閉端解和Euler法獲得的近似解進(jìn)行比較,從而得到與陳暉不同的結(jié)論,對(duì)于Vasick模型,Euler法對(duì)轉(zhuǎn)移密度的近似程度最高,但Hurn法對(duì)模型參數(shù)估計(jì)的偏差要小于Euler法.對(duì)于CIR模型,Hurn所考慮的方法對(duì)轉(zhuǎn)移密度函數(shù)的近似程度最好,且對(duì)模型參數(shù)估計(jì)的偏差最小.同時(shí),比較了這幾種方法在CKLS模型參數(shù)識(shí)別方面的效率,Hurn方法的效率最高.

1 估計(jì)方法

考慮如下一維擴(kuò)散過程模型[11-12]

dXt=μ(Xt;θ)dt+σ(Xt;θ)dWt, 0≤t≤T,

(1)

其中,θ是待估參數(shù)向量,漂移項(xiàng)μ和擴(kuò)散項(xiàng)σ是非線性函數(shù),Wt是標(biāo)準(zhǔn)布朗運(yùn)動(dòng).假定擴(kuò)散過程X在時(shí)間點(diǎn)0=t0

若已知擴(kuò)散過程X的轉(zhuǎn)移密度p(xt|xs;θ)(s

(2)

其中,p(Δ,xi|xi-1;θ)是兩連續(xù)觀測(cè)值xi-1和xi間的轉(zhuǎn)移密度函數(shù).則θ的估計(jì)值可通過最大化(2) 式,或最小化 (3) 式來獲得

(3)

事實(shí)上,X的轉(zhuǎn)移密度函數(shù)通常都是未知的,如何獲得p(Δ,xi|xi-1;θ),則成為一個(gè)關(guān)鍵問題.本文考慮了Euler估計(jì)法、基于偏微分方程的估計(jì)算法等方法近似p(Δ,xi|xi-1;θ).

1.1Euler法

Euler法是其中最簡(jiǎn)單的一種極大似然函數(shù)的近似方法,在時(shí)間區(qū)間[ti-1,ti]上,對(duì)方程(1)進(jìn)行Euler離散

(4)

其中,序列{εi-1}相互獨(dú)立且均服從標(biāo)準(zhǔn)正態(tài)分布.則可得xi-1和xi間的轉(zhuǎn)移密度函數(shù)為:

pEuler(Δ,xi|xi-1;θ)= (2πνi-1)-1/2exp{-(xi-mi-1)2/(2νi-1)},

(5)

其中,mi-1=xi-1+μ(xi-1;θ)Δ,vi-1=Δσ2(xi-1;θ),pEuler表示在Euler法下轉(zhuǎn)移密度函數(shù)的近似值.

1.2 基于偏微分方程的估計(jì)方法

假定p(Δ,xi|xi-1;θ)≡p(ti,xi|ti-1,xi-1;θ),并記p(t,x)=p(t,x|ti-1,xi-1;θ),Karatzas and Shreve(1992)[13]指出轉(zhuǎn)移密度函數(shù)p(t,x)是以下Fokker-Plank偏微分方程的解

(6)

且滿足初值條件

p(ti-1,x)=δ(x-xi-1),

(7)

其中,δ(·)為Dirac delta函數(shù), 如何處理Dirac delta初值函數(shù)成為一個(gè)棘手的問題.

式(6)可改寫為:

pt(t,x)=a(x)p(t,x)+b(x)px(t,x)+c(x)pxx(t,x),

(8)

1.2.1 陳暉采用的方法 采用Crank-Nicolson差分對(duì)(8)進(jìn)行離散,記ai=a(yi),bi=b(yi),ci=c(yi),則可得如下格式

(9)

實(shí)際算法實(shí)現(xiàn)中,還要對(duì)格式(9)限定邊值條件p0=pN=0,且選擇均值為xi-1+μ(xi-1;θ)Δt,方差為Δtσ2(xi-1;θ)的正態(tài)分布作為初值函數(shù)的近似(Jensen和Poulsen(2002)[2]).

1.2.2 Hurn采用的方法 Hurn等(2007)[3]指出(6)還應(yīng)該滿足如下邊值條件

(10)

記μi=μ(yi),σi=σ(yi),則可將(6)離散為:

(11)

事實(shí)上,(11)也是通過Crank-Nicolson差分法對(duì)(6)進(jìn)行離散獲得的.

假定p(t,x)的解落在區(qū)間[y0,yN]或區(qū)間[y0,∞]上,漂移項(xiàng)和擴(kuò)散項(xiàng)滿足μ(y0)≥0,σ(y0)=0,可以認(rèn)為在邊界點(diǎn)y0處沒有累積任何密度,則可取p(t,y0)=0.然而在邊界點(diǎn)yN處,通過離散邊值條件(10),且令(11)中i=N-1,可得如下格式

(12)

其中,

若取p(t,y0)=0(否則,可獲得類似(12)的結(jié)果),則離散格式(11)的起始迭代為:

(13)

結(jié)合(11)～(13),最終,可得如下矩陣形式的差分格式

ALpi+1=ARpi,

2 估計(jì)轉(zhuǎn)移密度似然函數(shù)的效率

考慮OU和CIR模型,值得一提的是這兩個(gè)模型均存在轉(zhuǎn)移密度函數(shù)的閉端解,便于進(jìn)行Euler法和基于偏微分方程法估計(jì)效率的比較分析.

Vasicek(1977)[14]提出了Vasicek模型(或OU模型),滿足如下隨機(jī)微分方程

dXt=θ1(θ2-Xt)dt+θ3dWt,X0=x0,

(14)

該方程的轉(zhuǎn)移密度函數(shù)的閉端解為:

pVasicek(Δ,x|xi-1)=φN(x;m(Δ,xi-1),v(Δ,xi-1)),

(15)

Cox, Ingersoll和Ross(1985)[15]提出了CIR模型,滿足如下隨機(jī)微分方程

(16)

(17)

此外,為比較估計(jì)方法對(duì)模型參數(shù)的估計(jì)偏差,定義了ARE:

(18)

表1結(jié)果說明,對(duì)于OU模型,Euler法近似的對(duì)數(shù)似然和的效果最好,陳暉等所采用的估計(jì)法的效果最差,如果取較小的空間步長(zhǎng)Δx其近似效果可以得到改善.對(duì)比陳暉等和Hurn等所采用的方法的估計(jì)效果,發(fā)現(xiàn)Hurn法均優(yōu)于陳暉法.從表3可以看出,精確極大似然估計(jì)法(EML)、Euler法、Hurn法對(duì)于OU模型的參數(shù)估計(jì)的均值差別不大,而陳暉法除對(duì)θ1的估計(jì)效果較差外,對(duì)于θ2和θ3的估計(jì)效果相對(duì)較好.從ARE的值來看,Hurn法所獲得的ARE值要小于Euler法,這說明Hurn法對(duì)OU模型參數(shù)估計(jì)的偏差要小于Euler法.

表2結(jié)果說明,對(duì)于CIR模型,陳暉法和Hurn法近似的對(duì)數(shù)似然和的效果均優(yōu)于Euler法,且空間步長(zhǎng)越小,近似效果越好,而Hurn法的近似效果又均優(yōu)于陳暉法.另外,從表4可以看出,Hurn法對(duì)CIR模型參數(shù)的估計(jì)值最接近EML法,且估計(jì)偏差也要小于陳暉法和Euler法.

總的說來,Hurn法對(duì)于對(duì)數(shù)似然和的近似效果均優(yōu)于陳暉法,且對(duì)于模型參數(shù)的估計(jì)效果要優(yōu)于Euler法和陳暉法.實(shí)際應(yīng)用中,在計(jì)算需求允許的條件下,對(duì)于基于偏微分方程的估計(jì)法,盡量取較小的空間步長(zhǎng),以獲得精確的估計(jì)結(jié)果.

表1 Euler法和基于偏微分方程法估計(jì)OU模型轉(zhuǎn)移密度對(duì)數(shù)似然函數(shù)和的誤差比較,基于200個(gè)模擬數(shù)據(jù)序列

Tab.1 Measures of relative error in the calculation of log-likelihood for OU model based on 200 data sets,compare Euler method with the PDE methods

MREAAREMSREEuler:Δ=1/121.73391E-053.46331E-073.30786E-12陳暉法:Δx=0.0022.44892E-031.21032E-043.54117E-08Δt=1/120,Δx=0.0013.46305E-043.12746E-051.31881E-09Δx=0.00052.61783E-042.36075E-057.63422E-10Hurn法:Δx=0.0025.84601E-049.37847E-055.82997E-09Δt=1/120,Δx=0.0013.12343E-042.87751E-051.07249E-09Δx=0.00052.31755E-041.95545E-055.73030E-10

表2 Euler法和基于偏微分方程法估計(jì)CIR模型轉(zhuǎn)移密度對(duì)數(shù)似然函數(shù)和的誤差比較,基于200個(gè)模擬數(shù)據(jù)序列

表3 Euler法和基于偏微分方程法估計(jì)OU模型參數(shù)的平均值和ARE(%)(括弧內(nèi)數(shù)值),基于200個(gè)模擬數(shù)據(jù)序列

表4 Euler法和基于偏微分方程法估計(jì)CIR模型參數(shù)的平均值和ARE(%)(括弧內(nèi)數(shù)值),基于200個(gè)模擬數(shù)據(jù)序列,其中參數(shù)真值為θ1=0.2;θ2=0.08;θ3=0.1

3 參數(shù)識(shí)別的效率比較

本節(jié)考慮CKLS[17]模型的參數(shù)識(shí)別問題,它是一個(gè)最基本的利率過程模型,國(guó)內(nèi)外很多實(shí)證研究都是基于該模型進(jìn)行的.其具體形式為:

(19)

其中,利率的均值回復(fù)速度θ1,利率的均值回復(fù)水平θ2,利率的波動(dòng)系數(shù)θ3,粘性系數(shù)θ4為4個(gè)待估參數(shù),該模型的轉(zhuǎn)移密度函數(shù)無法解析表達(dá).

令式(19)中x0=0.1,θ1=0.2,θ2=0.08,θ3=0.03,θ4=0.5,對(duì)(19)采用Milstein算法進(jìn)行離散獲得1 000個(gè)數(shù)據(jù),其時(shí)間序列圖可見圖1.表5列出了用這1 000個(gè)數(shù)據(jù)獲得的CKLS模型的參數(shù)估計(jì)結(jié)果,其中時(shí)間步長(zhǎng)Δ=1,Hurn1表示在Hurn法中取Δt=0.02,空間步長(zhǎng)Δx=0.0005,Hurn2表示在Hurn法中取Δt=0.02,空間步長(zhǎng)Δx=0.0001.從估計(jì)結(jié)果上來看,Hurn法比Euler法具有更強(qiáng)的參數(shù)識(shí)別能力,且在CKLS模型中更關(guān)心θ2的識(shí)別和估計(jì)精度,對(duì)于Hurn法,其對(duì)θ2的識(shí)別能力會(huì)隨著空間步長(zhǎng)的變小而變強(qiáng).

圖1 CKLS模型模擬出的1 000個(gè)數(shù)據(jù)的時(shí)間序列圖

表5 CKLS模型的參數(shù)估計(jì)結(jié)果

Tab.5 Parameter estimation results of CKLS model

θ1θ2θ3θ4Euler估計(jì)0.18650.08140.01570.2709EulerSD0.01850.00140.00550.1390EulerT值10.107960.15922.84491.9485Hurn1估計(jì)0.20530.08140.02120.3511Hurn1SD0.02260.00140.00730.1366Hurn1T值9.096660.16972.90692.5713Hurn2估計(jì)0.21010.08140.02030.3339Hurn2SD0.02270.00130.00700.1366Hurn2T值9.242961.40972.90732.4441

4 結(jié)論

本文采用基于偏微分方程的估計(jì)法對(duì)利率擴(kuò)散模型進(jìn)行參數(shù)估計(jì),并和Euler法進(jìn)行了對(duì)比.從數(shù)值實(shí)驗(yàn)結(jié)果來看,Hurn法對(duì)于對(duì)數(shù)似然和的近似效果均優(yōu)于陳暉法,且對(duì)于模型參數(shù)的估計(jì)效果要優(yōu)于Euler法和陳暉法,且Hurn法對(duì)于模型參數(shù)的識(shí)別能力要好于Euler法.

[1] Lo A W. Maximum likelihood estimation of generalized Ito processes with discretely sampled data[J]. Econom Theo, 1988, 4:231-247.

[2] Jensen B,Poulsen R. Transiton densities of diffusion processes: numerical comparison of approximation techniques[J]. J of Deriv, 2002, 9:18-32.

[3] Hurn A S, Jeisman J,Lindsay K A. Transitional densities of diffusion processes: a new approach to solving the Fokker-planck equation[J]. J of Deriv, 2007, 14:86-94.

[4] 陳 暉, 謝 赤, Crank N. 差分下利率擴(kuò)散模型估計(jì)效率研究[J]. 系統(tǒng)工程, 2004, 22(6):44-48.

[5] Ait S Y. Maximum likelihood estimation of discretely sampled diffusions: a closed form approximation approach[J]. Econometrica, 2002, 70:223-262.

[6] Ait S Y. Closed-form likelihood expansions for multivariate diffusions[J]. Annal of Stat, 2008, 36(2):906-937.

[7] Pedersen A R. A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations[J]. Scand J Stat, 1995, 27:385-403.

[8] Brandt M,Santa-Clara P. Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets[J]. J Financ Econ, 2002, 63:161-210.

[9] Durham G B,Gallant A R. Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes[J]. J Bus Econ Stat, 2002, 20(3):297-316.

[10] Beskos A. Exact and computationally efficient likelihood based estimation for discretely observed diffusion processes[J]. J Roy Stat Soci Series B, 2006, 68:333-382.

[11] 谷 偉. 銀行間同業(yè)拆借市場(chǎng)利率擴(kuò)散模型及實(shí)證[J].統(tǒng)計(jì)與決策, 2013, 23:177-179,.

[12] 谷 偉, 肖 雯. 利率期限結(jié)構(gòu)模型改進(jìn)極大似然估計(jì)效率研究[J].統(tǒng)計(jì)與信息論壇, 2013, 28(8):27-31.

[13] Karatzas I,Shreve S. Brownian Motion and Stochastic Calculus[M]. Second ed. New York: Springer-Verlag, 1992.

[14] Vasicek O. An equilibrium characterization of the term structure[J]. Journal of Financial Economics, 1977, 5:177-188.

[15] Cox J C,Ingersoll J E, Ross S A. A theory of the term structure of interest rates[J]. Econometrica, 1985, 53:385-407.

[16] Kloeden P,Platen E. Numerical Solution of Stochastic Differential Equations[M]. Berlin: Spinger-Verlag, 1992.

[17] Chan K C, Karolyi G A,Longstaff F A, et al. An empirical comparison of alternative models of the short-term interest rate[J]. Journal of Finance, 1992, 47:1209-1227.

An efficiency study on estimation of diffusion processes models

GU Wei， CUI Junjiao

(School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073)

An maximum likelihood estimation algorithm based on partial differential equations (PDEs) is provided to estimate one dimensional diffusion processes models. Crank-Nicolson difference scheme is used to obtain the numerical solution of the corresponding PDEs, from which the transition probability density function (PDF) is obtained. Methods considered by Hurn et al and Chen Hui et al are provided, and we compare the approximations by PDE method with the closed-form density and the approximations by Euler method. Meanwhile, the efficiencies of parameter identifications of the models are considered. We conclude that the Hurn’s method is the best one of the three methods, and the Hurn’s method is better than the Euler method in identity ability of the parameter identification.

maximum likelihood estimation method; Euler method; transition probability density function

2014-06-28.

國(guó)家自然科學(xué)基金項(xiàng)目(11401591);教育部留學(xué)回國(guó)人員科研啟動(dòng)基金資助項(xiàng)目(2013693);中央高?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金資助(2014143);研究生教育教學(xué)理論研究項(xiàng)目(2014JY05).

1000-1190(2015)01-0001-06

F832

A

*E-mail: wei-gu@znufe.edu.cn.