VG扭曲算子在期权定价中的应用.pdf

2023,43A(5):15751584http:/VG-f3?d?A1,2141j(1?Hn?410205;2OSUO?H-:?410205):|VCg,uVGJ?VG-f.3VG?.,y?UVG-f?d3?e?d.O(JL,UVG-f?d?O(.c:?d;VG;VG-f.MR(2010)Ka:60G51;91G20;a:O211.6zI:A?:1003-3998(2023)05-1575-101?d?xd?2?|d?1d,?d?xd?KJ?U.?xy z?:u,AO?x?U,d?KJ?.d,?d?xd?K,k-?ny.?xd?V-CA?d,?dd?d|?K.3?xn,VC?xd.?CX“Lx,Wang1|ChoquetdHX;XeHX;=Z0gSX(x)1dx+Z0gSX(x)dx,SX(x)X?),g,V-f.XJXK,KHX;zHX;=Z0gSX(x)dx.uIO?,Wang1J?Cg(u)=(1(u)+),(1.1)(u)IO?.?C=k?5,?U?n?L).?C?5Buhlmann?n.3;Black-Scholes?.,vF:2020-11-24;?F:2023-02-22E-mail:78:?Hp?8-:8(19A267,20A485)!?Hg,7(2022JJ30202)?Hg,78(2023JJ30196)Supported by the Scientific Research Fund of Hunan Provincial Education Department(19A267,20A485),the Natural Science Foundation of Hunan Province(2022JJ30202)and the Natural ScienceFoundation of Hunan Province(2023JJ30196)1576n?Vol.43 A?=(r)T,|?Cy,U(1.1)O?dTBlack-Scholes.Labuschagne?2,Badescu?3?C3?d?A.Hamada?4?(JL:3?e,U?CO?dUBlack-ScholesO?d;3?,U?CO?d%.,?,7K?A?0,0,KZl(,)?VG,P Z VG(,).duVG,?XeVGL.2.2Vm(,F,(Ft)t0,P)?m4LZ=Zt,t 0k(,)?VGL,XJZtv(1)Z0=0 a.s.,(2)ZtkO-O,(3)Zt VG(t,/t,t,t).ZtLXe/,Zt=t+Gt+WGt,W=Wt,t 0,G=Gt,t 0OIOK$(1,)?L,W,G.w?,3CLGte,Ztt+Gt,?2Gt?K$.d,Zt?LfZt(x)=Z0fN(x;t+g,g2)f(g;t,)dg,No.5?:VG-f3?d?A1577fN(x;t+g,g2)=1p2g2exp(x t g)22g2,x R,f(g;t,)=t/(t/)gt/1exp(g/),g 0.w,fZt(x)?-d(?pd.y?,fZ1(x)=2e(x)21/(1/)2|x|q22+21/0.5K1/0.5|x|q22+22,K(x)=12Z+0u1exp(12x(u1+u)du,x 0.PC=1,G=1q1422+122 12,M=1q1422+122+12,Madan?12?VGL?A?n?|(,0,FVG(dx),=C(G(eM 1)M(eG 1)MG,FVG(dx)=CeGx|x|1dx,x 0.3VG-f?!uVGJVG-f,VG-fVGC?K.3.1?Z VG(,),V-fg;,(x)=FZ?F1Z(x)+?(3.1)VG-f.?Ck?,VG-fk5.?,VG-frZ?x m|,?-#Z?.?h1(x)Y!4O?K,h(x;h1,a)=(h1(x)a)+,x R,a 0.?h(x;h1,a)?,B?d.w,h(x;h1,0)=h1(x).e?VG-fVGC?K.n 3.1?Z VG(,),h1(x)Y!4O?K,X=h(Z;h1,a),KHX;Z+0g;,(SX(x)dx=Eh(Z+;h1,a).(3.2)1578n?Vol.43 Ayx 0,w,kh(Z;h1,a)x=h1(Z)a+x.qZY.C,?SX(x)=P(X x)=P(h(Z;h1,a)x)=P(Z,=?(3.2).ln3.1?yw?,CZ?YN4O,IZ?&E.d,n3.12Xe,yL?C.n 3.2?C Z?FZY!N4O?,h1(x)Y!4O?K,X=h(Z;h1,a).KHX;Z+0FZF1Z(SX(x)+dx=Eh(Z+;h1,a).(3.3)5 3.13CZ?u?:N?e,Hama-da?4?(3.3).Godin?7uNIG?-f?5.n 3.2IFZY!N4O,d2?Hamada4?(J,5.4?d?!VG-f3?d?A.b?|k?,x?,dLBt=exp(rt);,?,dLXt=X0eZt,0 t T,(4.1),x|r?X0r 0,X0 0;Z=Zt,t 0,T36?Vm(,F,F=(Ft)t0,T,P)?L evyL,Fv.|?.?,?3?d?.XJU,IO,?d?d,KU?d?n?dO.3L evy?.,?d(z13).?g?L evyL?,?TL?L.rSchoutens13?(Jo(Xen.n 4.13?.(4.1),?Zt(,)?VGL,K1dK!?FT?w?(XT K)+3?Q e?|dEQerT(XT K)+,(4.2),Zt3Qe(,r+1ln(1 122)?VGL.No.5?:VG-f3?d?A15793?.(4.1),?Zt36?Vm(,F,F=(Ft)t0,T,P)?(,)?VGL.d2.2,ZT VG(T,/T,T,T).-h1(x)=X0ex,KXT=h(ZT;h1,0).|n3.1,kHXT;=Eh(ZT;h1,0)=X0eEeZT=X0ee(i)T,(i)=lnEPeX1=1ln(1 122).Nw?,XJ-=(i)r)T,(4.3)KkHXT;=X0erT.,3VG-fg;T,/T,T,Te,?ux|.?yOI?Xen.n 4.1?X VG(,),PY=aX+b,a 6=0.KY VG(|a|,a,a+b).(4.4)yIyY?A?k(2.1)?/.Eexp(iuY)=Eexp(iu(aX+b)=eiu(a+b)?1 iu(a)+12(|a|)2u2?1.?LY VG(|a|,a,a+b).e?n3VG?.,dVG-f?d?u?e?d.n 4.23?.(4.1),XJZt(,)?VGL,KHerT(XT K)+;=EQerT(XT K)+,(4.5)Q?,d(4.3).y-h1(x)=erTX0ex,Kh(ZT;h1,erTK)=erT(XT K)+.dn3.1,Hh(ZT;h1,erTK);=Eh(ZT;h1,erTK)=erTE(X0eZT K)+.PYT=ZT.du3SVPe,ZT VG(T,/T,T,T),dn4.1,3SVPe,YT VG(T,/T,T,T ).,EQerT(XT K)+=erTEQ(X0eZT K)+.dn4.1,3?Qe,ZT VG(T,/T,T,(r+1ln(1 122)T).N?y,T=(r+1ln(1 122)T.d,YT3VPe?,ZT3VQe?.l?kHerT(XT K)+;=erTE(X0eYT K)+=erTEQ(X0eZT K)+=EQerT(XT K)+.d,n?(.1580n?Vol.43 A5?!Godin?7?,5go?d?.(B-S?.!Mertona*?.!NIG?.VG?.)?,?n-f(Ou?!NIGVG)d?O(5.?C,Godin?7O?XeS1.T,r,K?.?,?dXT=S0eZT;S2.OXT?dSXT(x);S3.|?C(1.1),?H(XT;)=S0erT;S4.Ow?CT=(XT K)+?dSCT(x);S5.O?derTH(CT;)=erTR+0g(dSCT(x)dx;S6.O?.e?nderTH(CT;)?.duNIG!VG-fO?,?I?O?,?aq,p2?.n,3?o?.,?d?e(.0,(.+.?3O?L,7kminXT 0,maxXT +.,3O,I?“O.?);?K.?e5?1dC?)?K.1eK maxXT,KCT 0,dSCT(x)=1,x 0,0,x 0.5?n-f?g(x)=F?F1(x)+?/,pFIO?!NIGVG?.?n-f?derTH(CT;)=erTZ+0g(dSCT(x)dx=erTZ+0F?F1(0)+?dx=0.X?1dKu?umaxXT,?.e,n-f?u1.2?K minXT,w?derTH(CT;)=erTZminXT0g(dSCT(x)dx+erTZ+minXTg(dSCT(x)dx.uklimK0+erTH(CT;)=erTminXT+X0.,?,?.?nd31dK 0+?udX0.d,?1dK 0+,n-f?u exp(rT)minXT/X0.5 5.1(-f?.d,u?J?,?n-f?O(5vkoS.u?,e?31d l?dX0?.No.5?:VG-f3?d?A15815.1B-S?.3?.(4.1),-Zt=(0.52)t+Wt,WtIOK$.k,|X0=30,r=0.08,T=0.5,=0.15,=0.2,?2000?dXT.g,|4q,?OONIGVG?,(JXe =34.6066,=4.3150,=0.0469,=0.3793,=0.0458,=0.0739,=0.1052,=0.0488.?,|n-fOw?d?(uB-S?d),O(J1.l1w?,?1d3?dNC,dn-f?dO(.1B-S?.n-fd?5.2Mertona*?.Merton14?a*?.Xt=X0exp(122(eY+0.52Y 1)t+Wt+Ntj=1lnYj),Nt?PoissonL,WtIOK$,Ynn1?CS?,lnYj N(Y,2Y),Yn,Nt,Wtn.3?e,Merton14?FT!1dK?w?dCtCt EQer(XT K)+=Xn0e()nn!CBS(,Sn,n,K),=T t,2n=2+n2Y,Sn=X0expnY+n2Y2 exp(Y+2Y2)+,CBS(,S,K)3;Black-Scholes?.,?m,dS,IO?,1dK?w?d,=CBS(,S,K)=S(lnSK+(r+122)Ker(lnSK+(r 122).1582n?Vol.43 AB-S?.?aq,X0=20,r=0.05,T=0.5,=0.15,=0.1,=1,Y=0.2,Y=0.1,k?2000?dXT,?|4q,?O?NIGVG?O =20.5889,=17.0237,=0.1289,=0.1278,=0.1474,=0.79653,=0.1224,=0.4938.dn-f?d?2.w?NIGdVGd?O(,?Cd?.2Merton?.n-fd?5.3NIG?.3?.(4.1),?Zt(,)?NIGL.X0=30,r=0.05,T=0.5,=5,=2,=0.2,=0.5?,?2000?dXT.|4q,?O?VG?=0.2536,=0.3124,=0.3302,=0.1486.3NIG?.n-fd?No.5?:VG-f3?d?A15833?dn-f?d?(u?ew?nd).3NIG?.,w?,NIGdVGd?O(,?Cd?.5.4VG?.?.(4.1)?Zt(,)?VGL.X0=50,r=0.05,T=0.5,=0.1,=0.2,=0.15,=0.2?,?2000?dXT.d(4.3),=0.0310.|?d,?O=0.0323.NIG?4q,?O =19.2556,=7.1597,=0.1258,=0.1972.4?dn-f?d?(u?ew?nd).3VG?.,E,w?,NIGdVGd?O(,?Cd?.4VG?.n-fd?n?(J,uy:.1,3?,?CdO(?;3?,?Cd?.Hamada?4?(J.1?,?,VGdNIGdO(.d,VG-fJ?#?dg.z1 Wang S.A class of distortion operators for pricing financial and insurance risks.Journal of Risk andInsurance,2000,67(1):15362 Labuschagne C,Offwood T.Pricing exotic options using the Wang transform.North American Journal ofEconomics and Finance,2013,25(2):1391503 Badescu A,Cui Z,Ortega J.A note on the Wang transform for stochastic volatility pricing models.FinanceResearch Letters,2016,19:1891964 Hamada M,Sherris M.Contingent claim pricing using probability distortion operators:Methods frominsurance risk pricing and their relationship to financial theory.Applied Mathematical Finance,2003,10(1):19475 Wang S.CAT bond pricing using probability transforms.Geneva Papers:Etudes et Dossiers,2004,278:19291584n?Vol.43 A6 Kijima M,Muromachi Y.An extension of the Wang transform derived from B uhlmanns economic premiumprinciple for insurance risk.Insurance:Mathematics and Economics,2008,42(3):8878967 Godin F,Mayoral S,Morales M.Contingent claim pricing using a normal inverse gaussian probabilitydistortion operator.Journal of Risk and Insurance,2012,79(3):8418668 Godin F,Lai V,Trottier D.A general class of distortion operators for pricing contingent claims withapplications to CAT bonds.Scandinavian Actuarial Journal,2019,2019(7):5585849 Debora D,Georg C.The distortion principle for insurance pricing:Properties,identification and robustness.Annals of Operations Research,2020,292:77179410 Madan D B,Seneta E.The Variance-Gamma model for share market returns.Journal of Business,1990,63(4):51152411 Madan D B,Milne F.Option pricing with V.G.martingale components.Mathematical Finance,1991,1(4):395512 Madan D B,Carr P P,Chang E C.The variance gamma process and option pricing.Review of Finance,1998,2(1):7910513 Schoutens W.L evy Processes in Finance:Pricing Financial Derivatives.Chichester:John Wiley and Sons,200314 Merton R C.Option pricing when underlying stock returns are discontinuous.Journal of Financial Eco-nomics,1976,3(1/2):125144Application of VG Distortion Operator in Option Pricing1,2Yao Luogen1Liu Huan1Chen Qiqiong(1School of Science,Hunan University of Technology and Business,Changsha 410205;2Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computing,Changsha 410205)Abstract:Using the idea of probability transformation,VG distortion operator is proposed based onthe distribution of VG.It is shown that option prices obtained by VG distortion operator are consistentwith option prices obtained under the mean correcting martingale measure in the VG model.Thenumerical results show that option prices obtained by VG distortion operator are very accurate.Key words:Option pricing;VG distribution;VG distortion operatorMR(2010)Subject Classification:60G51;91G20