期权希腊字母

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1、CalculationsofGreeksintheBlackandScholesFormulaClaudioPacatiMay15,20131Non-dividendpayingstockIntheBlackandScholesmodelthepriceofanEuropeancalloptiononanon-dividendpayingstockisC=SN(d)KerN(d);(1)12whereSisthestock'spriceatvaluationdate,Kisthestrikeprice,risthe(constant)spotrate,=Ttisthe

2、timetomaturity,Ttheexpiry,tthevaluationdateandlogS+(r+12)d=Kp2;(2)1logS+(r12)pd=Kp2=d;(3)21whereisthestock'svolatility.Theorem1.Thegreeksforthecalloptionare:@Cdelta:
C==N(d1);@S@2CN0(d)KerN0(d)12gamma:C==p=p;@S2SS2@CSN0(d)N0(d)rp1rp2theta:C==rKeN(d2)=KerN(d2)+

3、;@t22@Crrho:C==KeN(d2);@r@Cppvega:V==SN0(d)=KerN0(d):C12@InordertoprovethetheoremwecollectsomecommoncalculationsinthefollowingLemma1.ItholdsSN0(d)KerN0(d)=0;(4)12@d1@d21==p;(5)@S@SSp@d1@d2==;(6)@r@r@d2@d1=p;(7)@t@t2@d1@d2p=:(8)@@1Proof.Firstofall,werememberthat01x2=

4、2N(x)=pe:2Statement(4)holdsifandonlyifSN0(d)Sd2d2SN0(d)=KerN0(d)()er=2()log+r=12:120KN(d1)K2Noticethattherighthandsideofthelastconditionisd21d211ppS1212=(d1+d2)(d1d2)=(2d1)=log+(r+)222K22S=log+rKandthiscompletestheproofof(4).Theproofsoftheotherstatementsarestraightforwardca

5、lculations.Proofoftheorem1.Forthedelta,wehavethat@C0@d1r0@d2
C==N(d1)+SN(d1)KeN(d2)@S@S@S@d10r0=N(d1)+SN(d1)KeN(d2)by(5)@S=N(d1)by(4).(9)Using(9)and(5)thegammais@2C@
@dN0(d)C01p1C=2==N(d1)=:@S@S@SSBy(4)itcanbealsowrittenintheformKerN0(d2)r0SKeN(d2)C=p=p:SS2Thethetais@C0@d

6、1rr0@d2C==SN(d1)rKeN(d2)KeN(d2)@t@t@t@dKerN0(d)r10r0p2=rKeN(d2)+SN(d1)KeN(d2)by(7)@t2rSN0(d1)=rKeN(d2)pby(4)2rKerN0(d2)=rKeN(d2)pby(4)2N0(d)rp2=KerN(d2)+:2Fortherhowehave@C0@d1rr0@d2C==SN(d1)+KeN(d2)KeN(d2)@r@r@rr@d10r0=KeN(d2)+SN(d1)Ke

7、N(d2)by(6)@r=KerN(d)by(4).22Finally,thevegais@C0@d1r0@d2VC==SN(d1)KeN(d2)@@@pr0@d10r0=KeN(d2)+SN(d1)KeN(d2)by(8)@p=KerN0(d)by(4)2p=SN0(d)by(4).1Considernowaforwardcontract,withstrikeKandmaturityT,i.e.withpayoattimeTgivenbyF(T