金融市场风险度量方法的发展

291201202CHINESEJOURNALOFENGINEERINGMATHEMATICSVol.29No.1Feb.2012:1005-3085(2012)01-0001-22¤1,2,2(1-710064;2-710049):VaRCoherent:VaRCoherent:AMS(2000)90B50:F830:A1¯±°[1;2]()VaRCoherent278:2010-06-18.:(19792)..¤:(70531030;70971109;11001031)(CHD2009JC158).22921952Markowitz{(MV)XE[X]XVar(X)=E[(X¡E[X])2][3]Var(X)MarkowitzVar(X)[4]MV[5,6]|[7]()()(){[7]MV[8;9]{King[10]{[4]MVMVSharpe[11;12]Stone[13]MV[11-13][14]MVKonnoYamazaki[15]E[jX¡E[X]j]|MV[16]E£max(0;T¡X)¤¡®E£max(0;X¡T)¤;T®T13[17,18]·(X)=E£(X¡EX)3¤=E£(X¡EX)2¤3=2;(Skewness)[19][20]Konno[21]{{Roy1952[22]Roy(Roysafety¯rsttechnique)Markowitz[23]Markowitz(below-targetsemi-variance,SVt)SVt=E£max(0;T¡X)¤2:TEX(Below-meansemi-variance,SVm)SVm=E£max(0;EX¡X)¤2:(partialvariance)(semi-variance)HamzaJanssen[24]N®;¯=®E£min(0;EX¡X)¤2+¯E£max(0;EX¡X)¤2;®¯HamzaJanssenBawa[25]Fishburn[26](lowerpartialmoment,LPM)LPM(®;T)=E£max(0;T¡X)¤®:®LPM®1LPM®=1®1®LPMLPMLPMLPM(0;T)429[27]MonteCarlo[28]t[29][30]LPM(0;T)®®[31]tLPM(®;T)®=1[31]LPMKijimaOhnishi[32]ÂXÂKijimaOhnishi¾:X!R(R1)¾(X)¸0(R2)¾(¸X)=¸¾(X);¸¸0(R3)¾(X1+X2)·¾(X1)+¾(X2)(R4)¾(¸+X)=¾(X);¸¸0(R1)(R2)¸¸(R3)(R4)c¡·0·c+f(x)=8:c+x;x¸0;c¡x;x0;(1)¾k(X;f)¾k(X;f)=©E£fk(X¡EX)¤ª1=k;k¸1:c+=1;c¡=¡1k=1;2;1¾k(X;f)()[33]l1max1·i·nE£jxi¡E[xi]j¤;xiXic+=0;c¡=¡1LPM()Satchell[34]15(exponentiallyweightedmeansquarerisk)©(!;T)=E£!(X)(X¡T)2¤;T!(X)©(!;T)TT!(X)T=E[X]!(X)=1©(!;T)X·E[X]!(X)=1X¸E[X]!(X)=0©(!;T)!(X)=e¡µ(X¡T)©(!;T)|3{(mean-riskanalysis)(stochasticdominance)Hardy[35;36](majorizationtheory)RothschildStiglitz[37]Fishburn[26][38,39]XPXF(1)XF(1)X(´)=Z´¡1PXd»=P(X·´);8´2R:629kF(k)XF(k)X(´)=Z´¡1F(k¡1)X(»)d»;8´2R:kXkY´2RF(k)X(´)·F(k)Y(´)´F(k)XF(k)YXk¡1YXkY{()[40]Bawa[41][42]MVPorter[43]{OgryczakRuszczy¶nski[44]{{E[max(0;EX¡X)2]1=2{E[max(0;EX¡X)]GotohKonno[45][44]{{{OgryczakRuszczy¶nski[46]k±kX=E[max(0;EX¡X)k]1=kE[X]¡¸±kX(¸2(0;1))k[47]{[48](expectedshortfall,ES){ES17{4VaR1994(value-at-risk,VaR)[49]VaRVaRVaRJ.P.MorganRiskMetrics[50;51]VaR2001VaR[52]VaRk2(0;1)VaRk1¡kkVaRk=¡F¡1X(k);F¡1XXVaR(riskofruin)VaRkFXRockafellarUryasev[53]VaRkF¡1X(k)VaRk=inf©¡F¡1X(k)ª:VaR[54,55]VaR[55;56]MonteCarloVaRMonteCarloMonteCarloJamshudianZhu[57](scenario)MonteCarlo829[58]±-()±-GARCH°-±-GARCH±-GARCHGARCHGARCHEGARCHEGARCHrt=¹+´t;´tj­t¡1»N(0;ht);lnht=®+¯lnht¡1+Áhj´t¡1jpht¡1¡(2=¼)1=2i+°´t¡1pht¡1;rt¹­t¡1t¡1®;¯;Á°htÁ°ht´t¡1[59]VaR[60]VaR[61]VaR[62]VaRPaVXiFi(xi)x¤i®xix¤i=F¡1i(®)H¤(x¤1;x¤2;¢¢¢;x¤n)(x¤1;x¤2;¢¢¢;x¤n)PaVVaRPaV[63-65]VaRVaR[66]VaRVaRVaRVaRVaRVaR[67]VaR[68]VaRVaR()VaRKritzmanRich[69]VaRVaR19VaRVaRVaRVaRVaRVaRVaR[70-72]VaRVaRMVVaRVaRVaRVaRNomoreVaR[73]VaRCoherent5CoherentVaR[74-77]ArtznerCoherent[75;76](­;F;P)XUXUR½:U!RCoherent½(A)½(X+a)=½(X)¡a;8X2U;a2R(B)½(X+Y)·½(X)+½(Y);8X;Y2U(C)½(¸X)=¸½(X);8X2U;¸¸0(D)X·Y)½(X)¸½(Y);8X;Y2UX2U;½(X+½(X))=0()(B)(C)(B)½(nX)·n½(X);n=1;2;¢¢¢(C)(B)(C)½Coherent[75;77]Coherent2()CoherentX1029CoherentXVaRArtzner1999(tailconditionalexpectation,TCE)[76]VaR(tailVaR)1A(a)Ax(®)x(®)X®x(®)=q®(X)=inf©x2R:P[X·x]¸®ª;x(®)=q®(X)=inf©x2R:P[X·x]®ª;E[X¡]1X¡XTCETCETCE®=TCE®(X)=¡E£XjX·x(®)¤®XTCE;TCE®=TCE®(X)=¡E£XjX·x(®)¤®XTCE:ArtznerTCETCE®TCE®¸TCE®[78]TCE®Coherent(worstconditionalexpectation,WCE)[76]E[X¡]1WCE®=WCE®(X)=¡inf©E[XjA]:A2F;P[A]®ª®XWCEE[X¡]1WCE®WCEXWCEWCE®¸TCE®WCEUryasev[79]VaR(conditionalvalueatrisk,CVaR)CoherentE[X¡]1CVaR®=CVaR®(X)=inf©E[(X¡s)¡]=®¡s:s2Rª®XCVaRP[X·x(®)]=®;P[Xx(®)]0XCVaR®TCE®[80;81]F®(y;³)=³+(1¡®)¡1E©[f(X;y)¡³]+ª;f(X;y)y³CVaRCVaR®(y)=min³F®(y;³):CVaRF®(y;³)miny2YCVaR®(y)=min(y;³)2Y£RF®(y;³);YyXF®(y;³)CVaRCVaR()CVaRCoherent[82-84]111(tailmean,TM)(expectedshortfall,ES)[81]E[X¡]1TM®(X)=®¡1¡E[X1X·x(®)]+x(®)¡®¡P[X·x(®)]¢¢®XTMES®=ES®(X)=¡TM®(X)®XESTMESX®TM,ESCVaRES®(X)=WCE®(X)=TCE®(X)=TCE®(X)CoherentESWCE[85]AcerbiTasche[81]ESCoherentCoherentCoherentGianin[86]g-(g-expectations)Coherentg-CoherentLehrer[87](partially-speci¯edprobability,PSP)CoherentCoherentArtznerCoherentJouiniMeddeb[88](d;n)CoherentRdRnFischer[89]Rockafellar[90]LpCoherent½(X)=EX+¯°°(X¡EX)+°°p;p¸1;¯¸0:[91]LpCoherentCVaRCoh-erent[92]p2[1;1)®2(0;1)½®;p½®;p(X)=mins2R³k(X¡s)¡kp®¡s´:[89,90]p½®;p(X)Lpspp=1½®;1CVaR®s¤¡VaR®k(X¡s)¡kpXs¡ss0½®;p(X)X®2(0;1)Coherent1229LPMCoherentCoherentBen-TalTeboulle[93]½®;¯(X)=min´2R©´+®E(X¡´)+¡¯E(X¡´)¡ª;®1;0·¯1:®¯Rockafellar[90]pCoherentCoherent[94]p2[1;1);0·a·1½a;p:Lp(Q)!R½a;p(X)=a°°(X¡EQ[X])+°°1+(1¡a)°°(X¡EQ[X])¡°°p¡EQ[X]:½a;p(X)[89,90][89]½a;p(X)(X¡EQ[X])+1-(X¡EQ[X])¡p-|EQ[X]CoherentRockafellar[90]1pap½a;p(X)VaRCoherent[95]CVaRCoherentCoherentCoherent[95]ESVaR[96]ESVaRESVaRCoherent1136Artzner[76]Delbaen[78]CoherentCoherentCoherentCoherent[97,98][99]FÄollmerSchied[100]FrittelliRosazzaGianin[101]Coherent(B)(C)(E)(E)½(¸X+(1¡¸)Y)·¸½(X)+(1¡¸)½(Y);8X;Y2U;¸2[0;1]½(0)=0½(X)XLÄuthiDoege[102]FÄollmerSchied[100]½(X)=inf©m2RjEP[e¡m¡X]·1ª=logEP[e¡X]½(X)=supQ³EQ[¡X]¡(px0)1=p°°°dQdP°°°q´;q=p=(p¡1):CoherentCoherent(A)Artzner[76]()[103]CoherentRockafellar[90]CVaR(PCVaR)[104]Dhaene[103][105]1429(weightedexpectedshortfall,WES)(­;F;P)XE[X¡]1®WESWES®(X)=®¡1¡w(x®)x®¡P[X·x®]¡®¢¡E[w(X)X1fX·x®g]¢;x®=inffx2R:P[X·x]¸®gw(x)xx·0w(x)x0w(x)[105]w(x)exp(¡¸x)(¸¸0);exp