Black-Scholes-模型_beamer

Black-Scholes-.‘°d2010c1217F2010c1217F‘°d1/26ÆS‡:O£ÙK$Ä(Brownianmotion)‘ÅÈ©(stochasticintegrals)ItoÚn(Itolemma)Black-Scholes-..bgK]Ý]üÑÃ@|½dBlack-Scholes-îªÏ½dúªBlack-Scholes-.$^2010c1217F‘°d2/26ÙK$ÄESµ‘ÅL§(;F)§(R;B)ü‡ÿݘm§IR0L«˜‡«m½lÑꊧ·‚¡˜XŒÿNfStgt2IS:I!RIþ‘ÅL§I:=T=f0=t0t1:::tNg,lёÅL§I:=[0;T];ëY‘ÅL§éu˜‡½!,NS(!):I!R¡´»éu˜‡½t2I,NSt:!R¡tž‘ÅCþ2010c1217F‘°d3/26ÙK$ʇ‘ÅL§¡ÙK$ħXJ§÷v±e^‡(1)£Ð©^‡¤W0=0(2)£ÕáOþ¤éurstu,WuWtÚWsWr´ƒpÕá(3)£Oþ©Ù¤WtWsN(0;pts)(4)£ëY´»¤éu?¿s2[0;T]Ú,ÑUé,jtsjž§jWt(!)Ws(!)j¤á5µ(2)`²WuWt†utÝk'§†u;t¤? ˜Ã'2010c1217F‘°d4/26ÙK$ÄÙK$Ä´»3?¿:ÑØUéžm¦y²µŠâÙK$Ľ§·‚Wt+
tWt
tN(0;1
t)‰½?¿«mA=(a;b),
t!0ž§ProbWt+
tWt
t2A=Zba
tp2expfx22

tgdx=Zb
ta
t1p2expfx22gdx!02010c1217F‘°d5/26ÙK$Ä&E(ÙK$Ä6(filtration)Œ½ÂfFWtgt0;FWt:=fWs;0stgWt´·A3fFWtgt0þ‘ÅL§§=éu8t2TWt2FWt;Wt3FWtþŒÿÞ~µ‰½¯‡A1=fWsa;s9g,A2=fW10bg,A12FW9A22FW10;A262FW9ϏFW9* [0;9]«mþ´»§A2u)3t=10ž2010c1217F‘°d6/26ÙK$ÄÙK$Ä´y²µ0st,E[WtjFs]=E[(WtWs)+WsjFs]=E[WtWsjFs]+E[WsjFs]=Ws:2010c1217F‘°d7/26ÙK$ÄÙK$ÄäMarkov5Ÿy²µ0st,Šâ½ÂWtWsN(0;pts),^xÚyO“WsÚWt§yx—ݼêf(yx)=Z111p2(ts)
e(yx)22(ts)d(yx))x‰½ž§y^‡—ݏf(yjx)=Z111p2(ts)
e(yx)22(ts)dy¤±y©Ù†xk'§=E[WtjFs]=E[WtjWs]2010c1217F‘°d8/26‘ÅÈ©ëY‘ÅL§ŒwŠlёÅL§4ål©§T=f0=t0t1:::tNg;
t=ti+1ti8i=0;:::;N1:·‚½ÂStnSt0:=N1Xi=0(i;Si)(ti+1ti)+N1Xi=0(i;Si)(Wti+1Wti)(1)i2[ti;ti+1]´«m[ti;ti+1]¥?¿˜:y3‡Ä¯K´§
t!0žª£1¤4§l½Â‘ÅÈ©2010c1217F‘°d9/26‘ÅÈ©1˜‘Ú,éu?¿!2,=?¿´»§·‚Œ^iùÈ©£Riemanintegral¤½ÂZT0(u;Su(!))du:=limN!1N1Xi=0(i;Si)(ti+1ti):´ÙK$ÄWt´»Ã?Œéžm¦§iùÈ©Ø·^§¤±·‚I‡Ú\L24VgXJlimN!1E[(XNX)2]=0;·‚¡XNL24X,LˆªXNL2!X:2010c1217F‘°d10/26‘ÅÈ©·‚Œ±y²N1Xi=0(i;Si)(Wti+1Wti)kL24éÙK$Ä‘ÅÈ©Œ½ÂZT0(u;Su)dWu:L2=limN!1N1Xi=0(i;Si)(Wti+1Wti):ù§ª£1¤4‘ÅÈ©ZT0dSu=ZT0(u;Su(!))du+ZT0(u;Su)dWu;½‡©LˆªdSu=(u;Su(!))du+(u;Su)dWu:2010c1217F‘°d11/26‘ÅÈ©éÙK$Äü‡­‡È©éš‘żêÈ©ZT0(u)dWu:L2=limN!1N1Xi=0(i)(Wti+1Wti):EPN1Xi=0(i)(Wti+1Wti)#=0)EPZT0(u)dWu=0VPN1Xi=0(i)(Wti+1Wti)#=N1Xi=02(i)(ti+1ti)N!1!ZT02(u)du)VPZT0(u)dWu=ZT02(u)du:¤±43§ZT02(u)dWuN0;ZT02(u)du:2010c1217F‘°d12/26‘ÅÈ©éÙK$Äü‡­‡È©é‘ÅþÈ©ZT0WudWu:L2=limN!1N1Xi=0Wi(Wti+1Wti)=12(W2TW20)+12T;Ù¥§i=(1)ti+ti+1;i=1;:::;N1;2[0;1],4†iÀJk'=0,i=ti§=Ito‘ÅÈ©ZT0WudWu=12(W2TW20)12T:2010c1217F‘°d13/26‘ÅÈ©y²g´µÐm$Ž§N1Xi=0Wi(Wti+1Wti)=12(W2tNW20)12N1Xi=0(Wti+1Wti)2|{z}()+N1Xi=0(WiWti)2|{z}()+N1Xi=0(Wti+1Wi)(WiWti)|{z}()(2)2010c1217F‘°d14/26‘ÅÈ©y²g´£Y¤µ|^ÙK$Ľ§کÙMomentOŽª§XN(0;)E[Xk]=(k1)2E[Xk2]k=2n0k=2n+1gOŽªmýˆ‘4()!12T()!N1Xi=0(iti)=T()!0y²L§¥Œ­‡(Ø(dWt)2=dt;dtdWt=02010c1217F‘°d15/26ItoÚn£˜‘ÙK$Ĥ½nµItoÚn£˜‘ÙK$Ĥf(t;x1;:::xn)´[0;T]RnþëY¼ê§…Ù˜dÚdêëYfXi(t)gt,i=1;:::;n´n‡˜‘‘ÅL§§÷vdXi(t)=i(t;Xi(t))dt+i(t;Xi(t))dWt;=§‚Ñ´dӘÙK$ĤÚå‘ÅL§§@of(t;X1(t);:::Xn(t))´˜‡‘ÅL§§÷vdf=ftdt+nXi=1fXidXi+12nXi=1nXj=1fXifXjdXidXj;Ù¥§fxL«@f@x,dXidXj=ijdt:y²µ‡Ñ¤ÆSN§Ñ2010c1217F‘°d16/26ItoÚn£˜‘ÙK$Ĥ~1µdSt=Stdt+StdWtS0=s)´St=s
exp(122)t+Wt;…ÏŠE[St]=set:2010c1217F‘°d17/26ItoÚn£˜‘ÙK$Ĥ~1µy²µXt:=lnSt,$^ItoÚn§dXt=1StdSt121S2t(dSt)2=Stdt+StdWt121S2t2S2tdt=(122)dt+dWt)St=s
exp(122)t+Wt:)E[St]=set:2010c1217F‘°d18/26ItoÚn£˜‘ÙK$Ĥ~2µdS1t=rS1tdt+1S1tdWt;S10=adS2t=rS2tdt+2S2tdWtS20=b¦µdS1tS2t.Šâ~1(ØS1t=a
exp(r1221)t+1WtS2t=b
exp(r1222)t+2Wt)S1tS2t=ab
exp12(2221)t+(12)Wt:2010c1217F‘°d19/26ItoÚn£˜‘ÙK$Ĥ~2(Y)dS1tS2t=1S2tdS1tS1tS2tdS2t1(S2t)2dS2tdS1t+S1t(S2t)3(dS2t)2=S1tS2t(rdt+1dWt)S1tS2t(rdt+2dWt)S1tS2t12dt+S1tS2t22dt)dS1tS2t=S1tS2t[2(21)dt+(12)dWt]:2010c1217F‘°d20/26Black-Scholes-..b½|ÃÞëY´|Çr´~êü«yúxÝ]B(t;T)=expfr(Tt)g)dB=rBdt¦ºxÝ]§vkù|dSt=Stdt+StdWt2010c1217F‘°d21/26Black-Scholes-.Šâ~1(J)St=St0
exp(122)t+Wt:lnStSt
tN(122)
t;2
t)E[St]=St0et:²L¿ÂÙK$Ä´»3?¿:ÑØUéžm¦ÙK$ÄMarkov5ŸIto‘ÅÈ©2010c1217F‘°d22/26Black-Scholes-.îªCalltžd‚Calle[S;K;t;T]=f(t;St)$^ItoÚn§df(t;S)=ft(t;S)dt+fS(t;S)dS+12fSS(t;S)(dS)2=ftdt+fS(Sdt+SdWt)+12fSSS22dt=[ft+SfS+12fSSS22]dt+SfSdWt:t=Tžf(T;S)=maxf0;STKg2010c1217F‘°d23/26Black-Scholes-.½|ä5§¤±îªCallŒ±dBÚS|¤gK]Ý]üÑftgt2[0;t]=f1t;2tgt2[0;t]E›.´˜‡‘ÅL§=(1;2):[0T]!R2ŒL§1t,2t3FtþŒÿ²L¿ÂµÝ]ûüÄutžƒc&EdŠ(Value)L§V(t):=1tB(t;T)+2tSt;8t2[0;T]gK]Ý]üÑ,Ý]|ÜdŠCzd|ÇÈ\Ú¦d‚CzÚå§vk]7’ÑÚÝ\§dV(t)=1tdB(t;T)+2tdStV(T)=maxf0;STKg:2010c1217F‘°d24/26Black-Scholes-.ŠâÃ@|½dK8t2[0;T],f(t;St)=V(t)df(t;St)=dV(t))f(t;St)=1tB(t;T)+2tSt(ft+SfS+12fSSS22)dt+SfSdWt=1trBdt+2tSdt+2tSdWt2=fS1B=f(t;S)2S=fS
fS1trB+2tS=ft+SfS+12fSSS22)ft+SrfS+12fSSS22rf=0(3)f(T;S)=maxf0;STKg(4)2010c1217F‘°d25/26Black-Scholes-.) ‡©§§=ª£3¤Calle[S;K;t;T]=StN(d1)Ke(Tt)rN(d2)d1=2=lnStKe(Tt)r122(Tt)pTtŠâPutcallparity,Pute[S;K;t;T]=Calle[S;K;t;T]S+Ke(Tt)r=Ke(Tt)rN(d2)StN(d1):2010c1217F‘°d26/26