book11-3-vhomepage.fudan.edu.cn/jgying/files/2012/02/book11-3-v.pdf · 1 " ( 0 # 2 & 5 12....
TRANSCRIPT
789:6;zWe ?xOYL-L2012 6 " 6 f
? Ekwh,K? wh,Kggwh+g,K? nna&9&3a\. wh,Kgiy^0k`` A:Jswh^a2#, [3sF, #9,K3p, Hi!, j~* !,K9s. %, ,K7A5A:Js9~*SSYMVSS, 4FBas, Fwh,KK Ba#5Hf.wh7+, wh,Ki5Jo$6nk, aokT 1952 Markowitz atJ.> , 7okT8, 1973 Black-Scholes 3T!Sv. qZi0BJ6Jgi, Markowitz atJ.> VD`` iBZv;Ji>s, 4 Black-Scholes v>sDlg Ito A:J> , \M#\bA, wh,KZ\N^g M1_M+Ka6Ji, j Delta 1_ VaR.
qf> D i" K I OBsV\ 1" * I xBVx 9" I il>Zi 22" I il>Zi II 28" / I *8xn; 35" d I !^%FD%Z 47" z I x%F2 57" I exotic options 66" Q I -26Z 73" Æ I ^g 80" ÆK I -2'a: VaR 89LX 96S-5 97
ii
# L J PC tW1. i<%: wh,KBnZi, qegJon6J: (1) Markowitz tLJ.> ; (2) Black-Scholes 3T!Sv; (3) M+K>. )wFO<%^, b
Capinski Zastawniak wh,Ka% (^), B%,KL3g, i^32, lijkp; 4o Wilmott whsZa% (^), ,Kgl2ana2; i6U4FTuk7 John Hull Options, Futures and other derivatives a%,7gZ, ^:q^x, Blh~whsZl_\!k7, siJ2, 3MsU6g ; *^k J% Delbaen & Schachermayer OB,Ka% (x) EnU,K, ,K9gF1whT;RK#(. dpxZ+KQm#(x, w.:qs, B2%:q&733YP. qeZvuwh,KwhsZA iBa29`.
2. ,>HK: wh,Kr+yH,K 1874 War-
las aH> : R!Msx n ar,, 4upDLr,SlÆ,, B n ar,uD/p n nZ,SlDanIK, FiVHqSlm,^5LV n− 1 n)K, Warlas R an9/: yupy^yDL. B\-SanZ, w an n−1 Z n−1 )K_, a35, Bn_s, sFaSlT. [Bq,), ` Arrow Debreu ) 1954 y3$!>p SlM.
1
1 "(0#2& 2
3. wh: whHanvA, TwhMHK. whM=WanO3Z, wMHL. ^?o, p3m=WCKwhM, s^p3mCKMH. whM'8!#[, Rgat1`, hatM. TwhM_[HKOaQE*.HKO whMr,\Nr,F3, rC3Y!A, 7A5whM3Y!A. smTwhMDaa7yT, A:9uk#F. )FB2SP, qwhMman_? iwhMDtLSl|^G_, B_JX', )Ua;+9 H, 9DtLYMV3AA:A, rA:J> 3T, 4F83L\MA:J> .
4. wh,K83: ) 1979 Ross B4AdDtL!Sa> ,T, > +lq, ,KM^3Sl, aQ 1990 , Dalang, Morton Willinger MMaI=tL!S9!>, Bwh,Ka> 8eDnq. G2``KOkBn7;U, 4^^``KOSP, G) 1980ManCJ.u>)a9!>M^7 M_, G31wh,K^2rb7p MSl4ZqM. $ik ) 20 MpA:#\anS 8, MME"A*A:vJZss)A, HLs,!>i=, 38H)wh,K^?y, wh,K^brPa, $ik sm) 2010 t,KOz2s6a/h#g, B^g,KO^an.
1 "(0#2& 3
5. _: ts2h~whM? OÆ2whL,?whw:Hw:k3k? GtN57vuh~whMA .
6. tL: tLan?`, i8!, ir,, iSMV. Bq HJWJ;: MtL, jSM, MtL, j2, t>s*. MLs4Sl3CA3,w, ~*Sl9C, FiHM, t>s*7A2Vn!, Fi5mM, sm4^#[xBA:)KiB`` .
7. M: Man~l, v sA:A. 0k\+KTMSYymaZ9, j A:)KI7A+qI(I/q) WM, sHA:)K(<4/SZ+an+K, )_tL%^, ^ A:)K/nwt``WM, sBpa!Z+s5anv7wZ+. sm, MaaA:A+K, Vaa .
8. MtL: MtLaTt>s*LV8>2V,[M#1, M1, t>", LV&I,Mys), Hqia3I. MtLa!n, 3wB, )fkSl A(0) 2V, j n.i ? A(1), k A(1)−A(0) B, #1SA(1)−A(0)
A(0)VBn.B`, ZVB`. M2V=2a, jo:wSNZ=, ja8a 3 100 2V) 3eNZ 100 , )fSl A(0), ^%V:B
1 "(0#2& 4NZ=, j)fa8 100 %V, sw5 3NZ2t?. =3b, &a\. [B`)', #135, B`)'3z, 4F#1!. B`1y``, CjjMtLB` r, aNZap?)VD 1000(1 + r)−1 M. qw5Bnlm.
9. \8%<`: \8%<`anbHT+, swF5B`~mB`, LB`\8%<`, j\8%<`g~mB`, kLB`[,, 8!y)'S, =P, LB`K,. )wh,K^, q38\8%<`, si \8%<`g^)B`^, ^5qi B`LB`.
10. <pX<: q^anatnZ, #!X#,Fi:<pX:GIX&im", JAtsmN-, )5S, <pXrij>s, rQ; )TI?, :GIXA:Z< (j47%$) lijyv;J3>, 9^22. p> siM, GIX<<pX<!I-.
11. atJ.> : any`` 9_TMVatMarkowitz, H MVA:)K, anatAJ.oaMVatC, Vat, [G!mMnM, .RzjiJ.(, >AatAa!J.nrgFMrat,G8B\J.3za, \N/O"aN, V1%t. Gwp M0kJ.atan\"e.
1 "(0#2& 5
12. OBRz: OBTmMn?:2. M1A9vA, bOB:2&2P=-, smMOBan5SRz, ^Gan9E4Rz, sLsz2,a)ij	wF MOB:2, Ksjm, whMij82Bk4 , ^Kij4 8ÆOB:2m3C. anR OBRzHKO Miller, G) 1958 Tv7SS49 LBR : jJnv73SSa\, kHq)SS^a\, P/ijy,anOB:2. 7A5, `h+YFdKHi|4\FdJKH. 3aq8, 2whY,Sl&i\OBRz3Y!, smOBRzWMwhHK9RzPa.
13. Black-Scholes: 3P , wh,KCK;7 Black Sc-
holes ) 1973 8,3T!Sv, )~*SlNE47%$RzP, Gq75<p M3T!San,w, 4^yM Ito v, v yMSZ+iB,!>B24`+!>, 4FBnv)MYZ^9+, Chicago 3TijF^K) 1973 WD, ÆYMV> TZMF, pmwh,KYMVMa7P=8373, W7M3,K> ukky.
14. YMV: kYMV? YMVSSHFb9tL!, MslN^3, 383T. Zxe\5%HN%F/KOphw (ÆX1w, 4v, B!!) K,>&. H)>"mNZ, 4^r,3FbtL, 4Sl)33Y!. [BKSSAsXiesr,Sl_)4_), j)r!Xr,MSg
1 "(0#2& 6r!S, kSSK, =P4SS[. xBKOKOSbOf'8 )[-2Zx>&. 3TE1~*4V, T)r!Xir!Slzfa~*TB, 3C1, iD;. 338&m , 3#FR>. sTB4Em , 3T)ijDNZ, sm3T!SS.
15. YMVlm: YMV3,KO0!&+ 3, HsMDL4Ls). HDLy^$M^, Cjnv7s)aY!Æa ol3, Hi)MsfaKao3, B\3MHn?, sao3`0k\aOÆ, $soz:4pv7|3llM, ÆrPY!. FiYMVs)ihA:A, ^hsM/$u7M. jI&Jw, YMVihM, ^iDzM. Cj, Rz8 100 oa^, aÆpoifap^, a#9on!:S 130 a^, )Msif on! 3 110 a^38, GJnJ. (1) 8fkap^, (2) Da),iaÆpMwfk 10 p^38. on!, jj3: 130 o, kaaJ.|3 30 pBl, 7aJ.|3300 pBl, jon! 80 oa^, kaaJ.ÆG.B 20 po, 47aJ.2ÆGM, MDz3V4.
16. _: GtN5a'YMVA h.
17. a:: a::T:2, a:T2)MsO>OB:2>, a:s)ssMA:A, 7A5mA:Ama:. a:eBL, whMszKa:As), a
1 "(0#2& 7:A"3ks)ÆM)r1, OB:2-r)rT. YMV[Cl2a:AFB. Ls, ata:mqN-, 2atMa:, 2a:^Zat.
18. 9!>: iE Brown %$tLSl< Black-Scholes vwE, aq84^2#T<, BZ1976 Ross OB!S> , Bn6&) 1979-1981 Harrison, Kreps, Pliskaa5nq, 4r3VtL!SJn9!>, a!>B\,), MOBRzSs)C+S``C+Æ)BnC+9tL3sn`M2Vn`, V5, 9tLSl)7C+. 7!>5YMVnatMtL SC+za. )BJn!>, bYMVSl&H) 3eSS3s. 5HfBJn!>, qp , >s>st!S9!>. 9m, Ross Cox, Rubinstein a7) 1979 R MZ7(3T<, QsMJn9!>, .Fy^8!!>, XXmMT, i GIX Black-Scholes v. p9!> 8, whH^YMV!S)ENQ4FZ, 3Dui;DaH> .
19. X\!S: YMV2a2\, 338"PXij, 3T3a\, Hnlpwh:y !, 3Ti\OB>G:!S.Eipz?M. s1&h3T:y35, jh aK3T[aKM1Hq353m". Hq3s aKMII, ^5M!S. [, 3TR>\an:yD
1 "(0#2& 8`NZ, kan&9, BKni\\aNZ3TIbat\)FbtLsaat . ji, Bn!SVM!S, BK3TVX\. B^tL!S79!>, H5M!Ss)^J#Wa,!>WD. HxBrA:J> .
20. whM: )wh_gan[z, pEowhw:r, EiHYangTO%, HpÆBO|3B3QCK5I. 3wh_whTj83, qy^Jwh_3T _, p-IA|3(p8_r2htPÆ, whTwhMb!. )aq (0L_aTa) 9gtJMB2, &L5&pwhM^O>OB:2, 6Q3l9\, j)2_A&p2\a 8Bg+ij^O>OB:2,B2>mkÆV, [^mkB-, 5: 1J, /Z). )O>vQ-a9\:2,^GKW4Ga^;. Jj, )Mwaa/.
# + J yCWy1. whMstL: MtLMtL. MtLTt>s*7LV, wh:yv78>2V. MtL<,~*, [+il!/wiBr,. )gm, q3N-tLSlSS, 7A5SSSl3K.
2. ~*r,: <MtL~*, ~*aaijFijwhL,. MVijF\N2~*, Taa~*Vn~, n~SlAX)'Vn~. x)~*MJTV~T, HJ ,A~*SlPTL, jo+q26, x FTSE100, eeT,, ^sMT,. r,aT+w',AL,, HqSluD)'4XA:A, qBk)38Msijr,, 438Ms !r,.
3. CM: CM4/ , Yq3OÆ3, sM73nUA:. eMa2(C, z2,a&3siS93Ce3, B)h~Jn~*<: 7(< Black-Scholes <^> s3, sBJn<MarkovA,H5)Ce3, FS9&gT))~S^M. C(jm, ÆzBn, Ut+,V8q)ZECM. %,%kjCM,273nUS, j4:(k.
(a) 9wJ: 91'~*74HwhL, 2s<H3C, 4^0aHLBJ, +'~*9
2! 10Ts<, iB+UK)s<.
(b) G(J: 91~*A SlHJ3C3Sl, T (1) O"e, B)WsN ; (2) NU~E; (3) M"; (4) e$/"; (5) #1C+, ,S N Vs:V,4C; (6) FT+, !m100× ~S − nVS
nVgS − nVS ;
(7) 4Q|; (8) ef"e, Hi)anes r29.
4. _: G.%vuaaMC9, #yMJ.
5. whYs: YMVaa4SSb44HMtLSSMV. YMVwh_k7L,, Hi bM1_\MtLM. YMV83~a, )HgwhMbW6JM, Hwh,KgT1+. wqvuEaLYMV.
6. 3: 3T3T!Xi)r!Slzf7Ah MtL3p, T!XVije (7A 3e), r!SlV3Sl, \NJ F . y?zftLV32b, y?h tLV3!b. )3>"ij,mwNZ. jMtL) 3eSl Sn, k\32batA6B Sn−F , \!bA6B F −Sn, &A:)K, 3^MtLaa.
2! 11
7. 3Sl: 3mwNZ, Fi3SlYb, kH3iY!Æ? Rz3^MtL)0 Sl S, 3e 1, 3XMtLB` r, jMtLm\W, kH3Sl F y^
F = (1 + r)S. Mj:
(a) j F g (1 + r)S, ki>"aK3!b, [)~* S ?fkMtLaK\ niSl F hp\2batA, [+pt>?BB (1+ r)S, jmnw3 F − (1 + r)S. =P, j F (1 + r)S, ki>"aK32b, [)h!aKMtL S ?, skt>, n, pt> tw (1 + r)S, i3Sl F fkMtL+pFA, jmnw3 (1 + r)S − FswJaI-y,n?:2nUmMn?:2, VOB:2. \N , OB)anzKW&atAM3, 7A5OB:2satAeB>4P=-, OB3s), 3Sly^ (1 + r)S. Ba!S9\VOB!S9. Ha6&h Slg|tL, fkSl|tL4p^6B.
8. 38: 383#9. 338)ijFij, Na!/i$vxM, 38!nV>r?, V (marking to market). 38)ijmSS, AsX_), r,uLZr,Sl_), s4Z38LSS, B2ZNZ. jS 100 B` 5%
2! 12a338S 105 , )7VS:Y 120 ,ka38Sy^ 126 , khy^NZpf 21 .M$vas9NZ7A3NZ,Wx, ijFL/ska!,KtwMw, ja3LiMw, 382/>, Q-, Mw;"^?U6Va. Fi3VDkijeSl, ^XSl7S3#)l, 4383zDkijeSl, ^DkBn3XSl, sCÆ`n?, ^s[3^XSl/$4/>, 3xM, 4389m.
9. _: GtN5a'7Al38ijFh.
10. iFO3S: 3\NO3S. j5atA 3 n!h 100 p^, G>aKon! 338!b. Msma\38r,, j!l, Msm, iJ.#|38r,3>O3, jQl, ViF1_ (cross hedge), Sy^>2tKÆ?BO3`S, FO3`T\38,KM[tL,KPXC. a&9i 1:1 C3S, Ls, BnC3<, s, [8Sl38Sl`2MaZ, 38Sl)'8Sl)'53nUaZ. BnC2tÆ? ∆S ∆F J-O3S3!8Sl)'38Sl)', σS σF J-Hq1y+qI, ρ Hq#,, aBnSm| 1. (z hO3S`. O3SA\tL2b38!b, )O
2! 133S3!tLSS)'∆S − h ·∆F,#=, 1!btL2b38, SS)'h ·∆F −∆S. aI-, SS)'I
v = σ2S + σ2
F − 2hρσSσF ,h = ρ
σS
σF, v w /S, BnSVNO3`. JA)'5nUaZ, Bn` 1.
11. C: v7OÆHe) 3 n!zf 100 pPb!\N. )3 n!nPb!\NSl)'+qI 0.032, O3SQ\N38Sl)'+qI 0.040Hq#, 0.8. smNO3`
0.8× 0.032
0.040= 0.64.a838 42,000 Pb, smv7y^zf
0.64× 1, 000, 000
42, 000= 15.2C 15 83>O3S.
12. _: G38Mz%t, Hq3sn/$`,HqSl)'+qIiB#,?
2! 14
13. 3T: VlaaYMV3T, 3T2a2\, Z3T3T!Xi)r!Slzf7h MtLTB, zfV:3T, h V3T. H3N-3#FR>, 43Ti3R>. jatazfaK 3Xn T!Sl K :3T, [j n MtLSlg K, ataBe, R>3T, 6B (Sn −K), =PjMtLSl K, kataBe, [J.3R>3T, B\ata6B 0. ,KR,w, zf:3Tata6B
(Sn −K)+ = max(Sn −K, 0).\zf3Tata6B(K − Sn)
+ = max(K − Sn, 0).Hq&b4 n tLSl Sn, ) n A:.
14. :: !an:3T !an3T, 73JA;9, /s3, sJAF&!, A)B3a\, aK:3TA)SSLs!, s> s~*Slmst, 4aK3TSSy!,s~*Slt. jfMaKR>Sl K 3e n :3Th aK\<,3T, kBnat) 3eSS(Sn −K)+ − (Sn −K)− = Sn −K.Hi\)b\a~~*~* e−rnK 3wW, B5) t :3TSl C 3TSl P y^i|
C − P = St − e−r(n−t)K.
2! 15mvV:3T/S (put-call parity).
15. kl37A38`SSr,Sl"AÆ,, 3T3.B),K9sZz3, "ASyE"ASZ.
16. o3T: o:3T;9, 3Pi)o3Ti) 3XbXR>. 0k,wo3TSSÆ? [o:3TSSy^Q:3T, satayiJ.) 3XR>3T. Sata3 rzBn,j3J.)tLSlw 3XgR>3TÆ? Ls[ata3 B, sGF9i,H3OÆ3, G3OÆkg. qweMo:3TLs3r, HqS?y^a\.
17. YMVSS: YMV) 3SSatLSl!, j33TSStL 3Sl Sn !, 7A5SnÆ,,o3TSSFXtLSl S1, S2, · · · , Sn!, 7A5 S1, S2, · · · , Sn Æ,. Bq5 3 n YMVSS Vn S1, · · · , Sn C, 7A5an 3en YMVaK, HSS (S1, · · · , Sn) Æ,.
18. YMV!S: YMV)>! ( 0) SlYMV!S. 37A38) 3en", #FR>, 4)>"3NZ, Y!3Sl3_F). 43TTB3m , Fia!Z , Fi3TT!Sl3b, bH) 0 SkS.
19. A:L,: SS3CL,iVA:L,. A:L,2
2! 16a2\, MtLA:L,, YL,[^A:L,. ZQA:L, Las Vegas *M9 :, ja2b,wonmm$a25ZU on,x, ::q /h a!3+?, ^k^m. BnA:L,0k!S? ^5, y^#2t?mao? \N9yz,!_, SlA:L,3s1. z,!_!S>ACPb4z\, sz,!_!Ss)M, pz,!_ , M\KW=, M9-e, B^v7a!13_ >, v7^3la\K/a, j5`8w, jM,WB. z,!_!SZ, WM.
20. YMV!S6&z,!_!S6&nU3, YMVSSb4Gr9A:L,, Fis)3z,!_!S9. z,!_!SÆH*M7A;9*Mv7, 433[Mt>MVv7T _. T _99/ataRuanij7A,!/I, 4HwF3[M, VQT I.
21. M!S: jan,!3Tv7, Hh aK3T, h aK-b, ) 3XH#Fp3TzfA#y1, jGk^3, 3TanM. FiH&9Q>I,\at`6#y1, tL:, BK1<3T1, BF3TX\, ^5QI\atMtL.63Tn. BnI3T!S.
22. X: <pX n = 0, 1, 2, · · · , anXO+iy, J,
2! 17V, !, iB4HbX. q5 n − 1 n n n.. z 0 SS A0 MtL) n SS An, AnanY!EA:1,O.
rn :=An − An−1
An−1 n n.B`, 1y,
1
1 + rn=
An−1
An n n.sv, n a(?#a1
1+rn(?. z Sn nMtLSl, Sn anA:)K,j,2aMtL, Sn ianA:*K. Sn : n ≥ 1anA:HO. qRztLSlyK, C Sn > 0. \
Kn :=Sn − Sn−1
Sn−1MtL) n n.n`, ^nA:)K.
23. at: zMs2aMtL, twiabat)2aMtLspqJat, B\YtihM.kq3ra2? 3iaZ93e23>atÆ? ik\+q3J.Æ?7qevu Markowitz atJ.> . ) n \x KMtL y KMtLatA n 9\
Vn = xSn + yAn. (2.1),1 (x, y) Vat7 portfolio, \j2aMtL,B x i*K. )L~*M78!M, x, y a#
2! 18FJ,, M,Ks(, qB x, y ib,. xK, q5atAfk, 72bbt, P/5atAh!, 75!bbt. #y, y K, 5atAs* (8!2b), P/5atA~* (8!!b). M3iG!b.
24. 9\1P: n− 1 at (x, y)
Vn−1 = xSn−1 + yAn−1 (2.2)) n 9\Vn = xSn + yAn. (2.3)Bi n .at (x, y) 9\)'
Vn − Vn−1 = x(Sn − Sn−1) + y(An −An−1). (2.4)
25. atn`: ;9, at (x, y) n`KV =
Vn − Vn−1
Vn−1
.
26. 3snM: A:)K,K3sIJnb,xPH. MtLn,K3sV3sn, 4+qI (I/q) NVM, sHx)nA:Az/Z+anT+. jMtLn`3s+qIµn = E[Kn],
σn =√E(Kn − E[Kn])2.
2! 19
27. >AJ.: )N Jn3at), ngs<, nA:)K, E9l, 43snMJn,x,Hql#1ij. 9Rz, jJnatn#, kM/r; jM#, knzr.
28. C: zMtL A(0) = 100, A(1) = 110, ~*Sl S(0) = 80 FP(S(1) = 100) = 0.8, P(S(1) = 60) = 0.2.Rz) 10,000 o, f 50 ~~* 60 K2V, k 1,tLSSV1 = 50S1 + 60A1 =
11600, j~*:;
9600, j~*.katnKV =
50(S1 − S0) + 60(A1 −A0)
V0=
0.16, j~*:;
−0.04, j~*.at3snE[KV ] = 0.16× 0.8 + (−0.04)× 0.2 = 0.12C 12%, atM
√var(KV ) = 0.08,klMS\Nm1lm, V#1lm.
29. JatL: zJatL, 4 n SlJ- S1(n) S2(n).) 0 J-at x1, x2, B\, batSSV (0) = x1S1(0) + x2S2(0).
2! 20
x1, x2 V=attL,K, mC=attLSS, smq!mTbw1 =
x1S1(0)
V (0), w2 =
x2S2(0)
V (0),HqJ-at)JatLsatTb, 73rQ. jtL3h!, kTb&KF&) 0,1 PX, CKlmsTb, jtLih!, kTbK[, i|
w1 + w2 = 1.LsVanw+. kukTb? wat^,
x, y tLSl, 4tLSl3a\, j-~*n~ 1000 o, 4Æ+~*Vn~ 50 o, Fi)~*,Klm3z, ^3(, 4#P, ukTbr(, Hx)yatikCJ# TMtLs.
30. atn: z)JatLsatnJ- K1 K2, katnKV = w1K1 + w2K2,s
KV =V (1)− V (0)
V (0)
=x1(S1(1)− S1(0)) + x2(S2(1)− S2(0))
V (0)
= w1S1(1)− S1(0)
S1(0)+ w2
S2(1)− S2(0)
S2(0)
= w1K1 + w2K2.
2! 21)qzTb w1, w2 <<,, HqJ.atJ..
# J jm?A^[j1. 3sn`: at3sn`JatLs3satn`PT/
E[KV ] = w1E[K1] + w2E[K2].eswon3sn`im, µV , µ1 µ2,
µV = w1µ1 + w2µ2.n`IX'a2,
E[KV − E[KV ]]2 = E[w1(K1 − E[K1]) + w2(K2 − E[K2])]
2
= w21var(K1) + w2
2var(K2) + 2w1w2cov(K1, K2).j σ2V , σ
21 σ2
2 J-, K1, K2, KV I, [uk#,ρ1,2 =
cov(K1, K2)
σ1σ2,k
σ2V = w2
1σ21 + w2
2σ22 + 2w1w2ρ1,2σ1σ2.s |ρ1,2| ≤ 1, Fim w1, w2 E[, k
σV ≤ w1σ1 + w2σ2,at+qIiJaMtL+qIPX.
2. 3snM: tLI7A+qI σV >sMaa+K, k\_)Tb, 3sMiµV = w1µ1 + w2µ2;
22
$ 45/3 23
σV =√
w21σ
21 + w2
2σ22 + 2w1w2ρ1,2σ1σ2;
w1 + w2 = 1.B\, i-S w1, w2, sm µV σV /O"σ2V = (µ1 − µ2)
−2[(σ21 + σ2
2 − 2σ1σ2ρ1,2)µ2V
− 2µV (µ2σ21 + µ1σ
22 − σ1σ2(µ1 + µ2)ρ1,2)
+ (µ22σ
21 + µ2
1σ22 − 2σ1σ2µ1µ2ρ1,2)]s σV > 0, FiHV/O"aN, F4
µ2σ
21 + µ1σ
22 − σ1σ2(µ1 + µ2)ρ1,2
σ21 + σ2
2 − 2σ1σ2ρ1,2,
σ1σ2
√1− ρ21,2
√σ21 + σ2
2 − 2σ1σ2ρ1,2
.
3. M3s/w: anat3snz+, +qI++)an/w+s, Bn+/wVM3s/w,7A σ-µ /w. Q, jq JatLMn (σ1, µ1) (σ2, µ2) Jn+)M3s/ws, kT'il>-2P~WK$%;``0$7; :ukV, rlT~W7,U`$ HrK). Vl, µ1 = µ2 , /O"j'aW/>1/dQ"; |ρ1,2| = 1, Hj'JW!=wiFQ". j K1 M, k σ1 = 0 4 µ1 MtLB` r, B^JWQ"σV =
∣∣∣∣µV − r
µ2 − r
∣∣∣∣ σ1.
4. nnMtL: z na3MtL, n`J- K1, · · · , Kn,atKV = w1K1 + w2K2 + · · ·+ wnKn,
$ 45/3 244^ wi atC, w1 + · · · + wn = 1, \, j3iGh!, w1, · · · , wn &E[. W µi i atL3sn`, CA:*K (K1, · · · , Kn) 3IGC = (cov(Ki, Kj))1≤i,j≤n,4^
cov(Ki, Kj) = E[(Ki − µi)(Kj − µj)].a35, G C K!, eEtLn"A#.
5. atJ.: jaty3!, iatMtL!, kqysO>#1at. B Markowitz atJ.> , Hb41MnZJ. qRz, 1Jn3at, j3sn#, katayJ.Ml/nat; =P, jM#, katayJ.3snlgat. ,KV35, Mn/wsat (σ, µ) a!(H: (σ1, µ1) (σ2, µ2) ,JnatMn, k(σ1, µ1) < (σ2, µ2)Fz σ1 > σ2 F µ1 < µ2, Cat 2nr, Mr/.^5, )I3s/ws, anatl, Hsjatr, >Aata[J.. K0'il>*T:, :\U='il>gmT 1<'il>. )Gat^, F1atUTVT:N7Al>F.
6. M: )whU, ManNY l, MkM 2z, 0^95Hf, M^3s
$ 45/3 25#, Yx)K, )Bngm^, q2oR M, ^m+q!m.
7. 1%t: p! n ^MtL, Hat1%ta353za, HaWpjU sjO". jMn, sjgMgn. 0k)1%tsJ.b4naM. atA*n=, ^gMgn, 42atA*b, ^Mn.
8. I/': Wm = (µ1, · · · , µn), w = (w1, · · · , wn).s+ T ,Gm[. katn` KV 3s+qIJ-
µV = µ(w) = mwT ,
σV = σ(w) =√wCwT , (3.1)I σ2(w) w anK!7o<. kl σ2
V w Æ,an n | w, +anÆzo. !\Waw1 + w2 + · · ·+ wn = 1, w wsanQ/w, swBnÆos)BnQ/wsQS+ w (7A"), H[/S. y Lagrange Y,9iijI? )
w =1C−1
1C−11Tw /S, 4^ 1 ,F< 1 >*K. BanJT/S, j^3iGh!I-, BnJT/S3a!)GN, L6/S2EN/ .
$ 45/3 26
9. 3sn"s/I: U6atM/Slm3z, s2ata.3a!*M/nat. Vlanl6S, atALan!3sn µ, &p4^ at^J.anM/at, ^5? σ2(w) /S? B#? σ2(w) )Jn!\Wa1wT = w1 + w2 + · · ·+ wn = 1,
mwT = w1µ1 + w2µ2 + · · ·+ wnµn = µ (3.2)/S. BJn"A+Wa, (y LagrangeY,9 )w =
∣∣∣∣∣∣1 1C−1mT
µ mC−1mT
∣∣∣∣∣∣1C−1 +
∣∣∣∣∣∣1C−11T 1
mC−11T µ
∣∣∣∣∣∣mC−1
∣∣∣∣∣∣1C−11T 1C−1mT
mC−11T mC−1mT
∣∣∣∣∣∣
(3.3)
w /S, 4^ m 3sn`>*Km = (µ1, · · · , µn).
10. I?Z: WG(w, x, y) = wCwT − x(mwT − µ)− y(1wT − 1),4^ x, y Lagrange Y,.
∂G
∂wi= 0 n nZ
2wC − xm− y1 = 0,
$ 45/3 27s w =
1
2xmC−1 +
1
2y1C−1. (3.4)rJn!\Wa JnZ
1 = w1T =1
2xmC−11T +
1
2y1C−11T ,
µ = wmT =1
2xmC−1mT +
1
2y1C−1mT .ms x, y k (3.4) (3.3).
11. BLss)Jn*K a,b Æw = aµ+ b,4F3
a1T = 0, b1T = 1.kσ2 = (aµ+ b)C(aµ+ b)T
= (aCaT )µ2 + 2bCaT · µ+ bCbT ,)M3s/wsi|s)Z (σ, µ) /O"aN. B\1%tHfM, HBW/O"sN, atAy^p4^J.at. s4^nn,at&, ^5mHrat, 44Hbanat&3.
# J jm?A^[j II
1. Markowitzv: FGatn`M -3s (σV , µV ))M3s/ws, 3, VGN, )iGh!, I-Z, n = 2 , BaN/O", n ≥ 3 , an/O"WN, 4%tswR /M/O", FiBnNl=janvb, V Markowitz v. H%tsNW7A1%t. )3Gh!I-, I-X'2, GNojNw1 + · · ·+ wn = 1, w1 ≥ 0, · · · , wn ≥ 0) (3.1) !mw (σ(w), µ(w)) P+!X. B, LH/MO"3Æ Lagrange Y,9.
2. C: ^onMV, 3snm = (0.1, 0.15, 0.2),3IG
C =
0.0784 −0.0067 0.0175
−0.0067 0.0576 0.0120
0.0175 0.0120 0.0625
./Iat
w =1C−1
1C−11T= (0.316, 0.439, 0.245),4M - 3s
(σV , µV ) = (0.162, 0.146).
28
) 45/3 II 29)p!3sn` µ Wa, /I)w = (1.578− 8.614µ, 0.845− 2.769µ,−1.422 + 11.384µ)iw
σ(w) =√wCwT =
√0.237− 2.885µ+ 9.85µ2.
3. tM": j+an!n` r MtL, kqip1%t> > sanat. BF3sn µ / r at (σ, µ) 3#^, 1 µ > r at(σ, µ), HMtLGm (0, r) (σ, µ) Q"s,[jJnat (σ, µ) (σ′, µ′), HqJ- (0, r) GmQ"% 3, swWQ"[rJ., B\qVD> 1%tsn, ÆH (0, r) GmQ"FQ"sw, B[ (0, r) F1%tO"#EWQ", Vi7, >AatAy^J.BW"sat.
4. 0k>tM": Gm (σ(w), µ(w)) (0, r) Q"4`k(w) :=
µ(w)− r
σ(w),4^ σ(w) =
√wCwT , µ(w) = wmT . p,Kj+, qVD)+Wa w1 + · · ·+wn = 1 > ÆBn4`z w iM. +y Lagrange Y,9, W
f(w, t) = k(w) + t(1wT − 1).L(,∂f
∂wi=
µi
σ(w)− (mwT − r)
∑nj=1 σi,jwj
σ(w)3− t = 0,
) 45/3 II 30'Zµiσ(w)2 − (mwT − r)
n∑
j=1
σi,jwj − tσ(w)3 = 0. (4.1)J%Yi wi [1 i LmwTσ(w)2 − (mwT − r)wCwT − tσ(w)3 = 0,i
t =r
σ(w). H1 (4.1)
µiσ(w)2 − (mwT − r)n∑
j=1
σi,jwj − rσ(w)2 = 0, (4.2)5V*K,wmσ(w)2 − (mwT − r)wC − rσ(w)21 = 0.^5
(µ(w)− r)wC = (m− r1)σ(w)2, (4.3)sms)N, c Æw = c · (m− r1)C−1, (4.4)CFLTb w (m − r1)C−1 *, B\(yWa w1 = 1WaI? w.
5. C: +BswCv^onMMV, (anMMV,n` r = 5%, qiI?MtL, ?(m− r1)C−1 = (0.293, 1.341, 2.061),
) 45/3 II 31s w Bn*KN,, 4 w JKP 1, Fii w =
1
0.293 + 1.341 + 2.061(0.291, 1.341, 2.061) = (0.079, 0.363, 0.558).
6. MoS: p!MB` r > 0, )M3s/ws (0, r) 1%t#EQ"tM", swp ME,atTb w, H1yatVil>, HM3s(σ(w), µ(w)),5 (σM , µM). B\tM"Z
µ = r +µM − r
σMσ.nan>AatA&2)tM"sJ.at. %a(B`, 7(V-2QF(risk premium), H1atAl[M|33s1.
7. betasv: wq3an!atn KV MtLn KM PX, 1y (KM , KV ) + 3, \N\193O>N", V977A#a7. O>α, β Æ
E[(KV − (α + βKM))2]/. B</7Y, ij? βV =
cov(KV , KM)
σ2M
= ρV,M · σV
σM;
αV = µV − βV µM .
) 45/3 II 32qVsw βV p!at V (7AnMV) beta Rk.
beta svp!tLnAsJnM>3)'T+. PH"Zy = βV x+ αV .W
εV = KV − (αV + βVKM),kσ2V = var(εV ) + β2
V σ2M ,4^a(=II, JpM; 7(_M73JpM. 0ksÆ? qi i atLn`5
Ki = αi + βiKM + εi,4^ εi3sTF!#'DA:^$, H3sn µi =
αi + βiµi, Iσ2i = β2
i σ2i + var(εi).katn`
KV =∑
i
wiKi =∑
i
αiwi +KM
∑
i
βiwi +∑
i
εiwi,HMσV =
√σ2M
∑
i,j
wiwjβiβj +∑
i
w2i var(ε
2i ).j wi = 1/n, kwa( n MJMT, ^5AstLa;1P, He`-.
) 45/3 II 33
8. MoS: qOÆM2satAatMtL4p#yMoS, LsMV2p_M7A3JpM#yMoS, 32pJpMboS, sHqi\Jpat-eM.
9. ( beta sv: p (4.2) (µi − r)σ2
M = (µM − r)∑
j
σi,jwj,kµi = r + βi(µM − r), (4.5)4^
βi =
∑j σi,jwj
σ2M
=cov(Ki, KM)
σ2M i aMtL beta sv. )^, (4.5) b, )|I3sn beta svI-, HipatA8 BnMV!Sg+Hf9. Vl, mi 1bat V ,
µV = r + βV (µM − r),4^ βV swF!m beta svβV =
cov(KV , KM)
σ2M
,KatMMz/, µV −rβV
= µM −r MMoS, C[ 1 MMQ31`, ^V Treynor `. , at V Mn (σV , µV ) MtL (0, r) G"4`kV =
µV − r
σV
) 45/3 II 34zS)MtL (σM , µM) iw , HqSρ = kV /kM = βV · σM
σV
=cov(KV , KM)
σV σM
= ρV,M ,^5, H<atnMtLn#,.
# 0 J +9yo<_I A:J)wh^y)3T!Sw,7vuQZ7(3T<, ukOB, at, 3T, , 1_wh`,Kx), .SC+iB,`+,K`> wh^lm, g6& Doob 9!>, \7K, Ky^9s>sMlm. )mhwh,KP&R?zOCti, 3X\,Fi3iqgCwh, HV>&v._l
• OB• 3T• whtBa• SC+• 1_• X\!S
35
+ -!. 36
1. aA : Ito[~, Gp3m GrywhU. +BaI 19009h~,KO H. Poincare 0 Bachelier 0 , B' pnj+ M Brown %$me``u+, B)\N A. Einstein t8'h~ +M 8 , d Bachelier )JQ, sa>s4dt, aQ 60 8ab7 , Fi) wh,KT;7 1973 F.Black, M.Scholes, R.MertonaOs, B, ItoA:J,Tgno. jm, Dg,Ks, 9GIXA:Jwh,K+3j>s, Fi 1979 Cox, Ross, Rubinstein BO> <pXM, ÆHr Q, 4F1q>sA:JJin.
2. FOB: OBx arbitrage, TmMn. ^ M0k)., a!n?Ba. Sl5, s)anBaHn, Æ H X A:;J Yn i| Yn − Y0 ≥ 0 (,mM) F P(Yn − Y0 > 0) > 0 (,Kn). OBban*M7AnMF3iG.
3. -n``: 1anA:+, qNNa4^b``C+. jX~!Xv, 4&an``). -n``-n<, jana5~!v/4Vana5~!3v/a\. _)C+BaI, aI-3&+0(, )whUXENlm. )3``C+^, S``C+P# m. z\!X Ω, P 4san``C+, x) Ω^a8Az/. j Q Ω sVlan``, F1ba A, P(A) > 0 Fz Q(A) > 0, k5 P Q
+ -!. 37S.
4. A:A+K: >sS``C+3aaijI, ``C+1A:aAaa+K9/, S``aa7+K9/, HM3_)A:A, ^532 A:a)WY!A, ^32 Y!Aa)WA:. j5A:)K ξ 3``QS 1,−1, )SC+, HQS+ 1,−1, V3``_)M, )WM p, q. 5H32 A:a)WY!, Tp, q !K, sj p = 0, k3 1, )X)3, B_)MA:A; 5H32 Y!a)WA:, T p+ q = 1, sj p+ q < 1, kH#[ 4Hn33Q S)Q , B^_)MA:A.
5. SC+: janA:)KHO Xn, H)`` P 3, )n P S`` Q n, k5 XnSC+. z Yn 'DJ4A:HO, 3sz 0, kXn = Y1 + · · · + Yn . HmSC+? i , Yn QSE[ (BOBs)), H3SC+, P(Yn < 0) > 0 , Ha!SC+. ([ Xn E['DJ4A:HOY;;9r .) B5kSÆ? j Yn W*0$, k Yn E[5maq32$A, 4 P(Yn < 0) > 0 , 33s2z, Gs)$A.^5$A&s), SC+s), P/3s). 9Z, BLswh!Sa9!>d=. a9!>a> z`AJn 1980 8eDnq.%Z5.1 (a9!>) OB:232SC+xs.
6. ZwhM: 0ans~*B` r 8!Man)
+ -!. 38 n Sl Sn ~*M. qRz~*Sl'DJ4A:)KY;, CSn = S0ξ1ξ2 · · · ξn, (5.1)4^ ξn 'DJ4, S0 ~*8>Sl. j ξn QJnK,S
d, u, d < u ``J- q, p, p+ q = 1, ξn QS u ,~*) n :, P/,. kq5Ban7(<. Rzqipt>!\s*7A~*, 1~*^iAlfh. s Sn A:, 4y_~*#[s)zKa:>. a:A:+jA=.
7. OB: qM, RzMOB, /#d < 1 + r < u.s, j 1 + r ≥ u, ^5B`g, kaqih
(f!) 1 ~~*, 6tw S0, skt>, )aQ s*f1~*, 6B (1 + r)S0 − S1, s 1 + r ≥ u > d, FimA:)KE[, 4FiK``z 0, smMs)OB. \qiMjB` 1 + r ≤ d, kaqi~*Sf~*46Mn, M^OB:2.
8. 3T: 3TYMV, sHb~*B\MVTSS. T5, 3TaK, ifk3TC, Hpa)nr!Xir!Slzf~*TB. zr!X m,r!Sl K, j ~*Slgr!Sl, kaB, 6B (Sm −K), =P, GBe, VD;TB, Fi3TSSA:, Vm := (Sm −K)+. (5.2)
+ -!. 39ofk3TF3, HWa)r!XPbX&ibr!Sl K fk~*, FiSSVm := max
1≤i≤m(Si −K)+. (5.3)a, an n 3YMVan S1, · · · , Sn CA:)K Vn.
9. y: 3TaaTB, w[32vI, Hy^h2t?w[lmS, 41,!Ba3T>:y35, 1Gq35, w[mbM,!3Ti6aK>II. B\BaPs)? 3Tbaawhk7i , gh)R~M. ja!~*, smzK !~*i)Sfk46B, MBU~*pas:|3M, Gl#?faK:3T, B\iM)GXI-3YBJz. 3835, Hr2a:ArN+a:4WDzMM. Ewhw:i53TB;whk7u7.
10. : sB` r 8!Ms), n a?#n + 1 1 + r ?, Fie3?))3D, Cn + 1 K ?) n VS K(1 + r)−1 . Ean!, 3kI, jD, qiRz r = 0,B321wN ,Wb3. Vl, )z2,I-,
m = 1 ^|5SM.
11. !S: 3Tan<A:L,, *MA:L,nU;9. Fi& !S9z,!_, CSly^A:L,
+ -!. 403sS(1 + r)−mE[Vm]. (5.4))m4H9I-, BzaJ., Ba!S6&Z, & ÆzM, 9-e, HMAs\K (zfK) 1P4[t, B3T>:yb_3R. VlBn!S`` p , )LI-, Bn`` qYI?. kwvu3T!S6&1EN)L.
12. 6&Zx): 3T!S6&5\3T6Bi\1~*at6.
13. 0k: Rjbtw x0, 3\at~*n 3TSS Vm, 3Cp m = 1 . f x ~~*, #M xS0 ?, [ ? x0 − xS0 skt>, B\?anatBa, 4^x0, x !<,. B\ 1, ~*SS xS1 4skt>?SS (1 + r)(x0 − xS0), 73, atSS
xS1 + (1 + r)(x0 − xS0).Vn13TSSpzfA, Fi3s3)kI-, &φ(S1) = xS1 + (1 + r)(x0 − xS0), (5.5)4^ φ(S1) 3TSS, ):I=, φ(S1) = (S1 −K)+.
14. sZ: swZ (5.5) 3WDÆ? a3, sqL)FI-WD. )qBn7(<P-Rz<. S1 = S0ξ1 VQJnS S0d S0u,smφ(S0d) = xS0d+ (1 + r)(x0 − xS0),
+ -!. 41
φ(S0u) = xS0u+ (1 + r)(x0 − xS0).Ks JnOKx =
φ(S0u)− φ(S0d)
(u− d)S0
,
x0 = (1 + r)−1 (u− (1 + r))φ(S0d) + ((1 + r)− d)φ(S0u)
u− d. (5.6)^5, Va x0 ?, >ri\isat63T\SS?, FiBn3TYty^hBk2?. B\3Mj)., >r32bB. Fi x0 y^3TSl, (5.6) 3T!Sv.
15. nor: qy^ , Ijmnos7(<Rz, ENP#.
16. Cv:Rz~*)Sl 50, ni 3/4``: 20%,
1/4 `` 20%. n3faKa 3r!Sl52 :3T, RzB`ia3I, k3Tz,!_!S
(60− 52)× 3/4 + 0× 1/4 = 6, 4sw!Sv, 3TSl(60− 52) · 20%
40%= 4, swv+h>qBk31_M, St>u 16 ?, Psh3TF 4 ?f 2/5 ~~*Ds, an,j~* 40 , k3T3230, qh~* 16 ?+pt>; j~*: 60 , h~* 24 , +pt>
16 , (p3T\a 8 ?F.
+ -!. 42
17. 3Tfb: uswBnCv3sa3Tfb.jap?, G98~*2:, &umzna , jf~*Vf 1000 ~, ~*s:, , Gnap, Gt,3&u?, *a, f3T, if 12500 K3T, j~*:M, GnK3Tn 8 , xp, ^n1p. B3T12alm, -tM, jG?~*M, k3TWa8HW, pM; 4mf~*, +8p).Fi3Tfb, KDzMA)n^DzMM, yP there is
no free lunch.
18. H: uH3T!Sv (5.6), q8Bnv p m, Ban-, s p anWab`,, Y!, ,sM. Vlv^Jn 1 K,u− (1 + r)
u− d,1 + r − d
u− d,JA p′. BJn,5z, qib7!m`` P Æ
P(ξn = u) = p′, P(ξn = d) = 1− p′,k P P anS``C+ÆE(ξn) = d
u− (1 + r)
u− d+ u
1 + r − d
u− d= 1 + r,BZ~*Sl (1 + r)−nSn ) P Wan. Fi
P P )almanSC+, 4Fq^ , P Æ (1 + r)−nSn WzaSC+. Bwh!S> 79!>.%Z5.2 (79!>) 3TM!SFzSC+zas).
+ -!. 43
19. (!Sv: SC+, !Sv (5.6) i5Wx0 = (1 + r)−1[q′φ(S0d) + p′φ(S0u)] = (1 + r)−1E[V1],4^ V1 = φ(S1), H3Tn)7``3sS,z,!_!Sv (5.4) =#, V``3. ra,iX m r!XYMV Vm ) 0 !Sy^
x0 = (1 + r)−mE[Vm]. (5.7)wqs>M.
20. at: BatBl6!) k 2t?f~*. Rz)F Xk 9\, ?f (h) Hk K~*, sk (~)t>. k)a9\ySXk+1 = HkSk+1 + (1 + r)(Xk −HkSk), (5.8)anatHO Hk 1ysan9\HO Xk. es'Z(1 + r)−k−1Xk+1 − (1 + r)−kXk
= Hk
((1 + r)−k−1Sk+1 − (1 + r)−kSk
).qOÆ (1 + r)−kXk (1 + r)−kSk J-9\~*SlHO, A Hk AA:;J. sA`` P F Hk S1, · · · , Sk C, Fiq Doob 9!>O (1 + r)−kXk `` P ^, C9\)7C+^an.
21. wht: Bnb`4(vua, whtl6I32w, T)atZ^eMatwl3(Pk7AQ t
+ -!. 44w. Z (5.8), W Bk := Xk − HkSk, Bat~* 3Dkt>, )a, Bk+1 = (1 + r)Bk, smXk+1 −Xk = Hk(Sk+1 − Sk) + (Bk+1 −Bk), (5.9)^59\1P3w~*1St>B, m4G3,Bwht.
22. !S: jqbtw x0, &\at Hk : 0 ≤ k < m )r!X m w 3T Vm a\9\, C Xm = Vm, k3s3)Ax0 = (1 + r)−mE[Xm] = (1 + r)−mE[Vm].B\MMq!Sv.
23. X\!S: u(A , swvanWa, ^s) Hk : 0 ≤ k < m Æ Xm = Vm. MZ, q) r = 0RzsBnS. j Hks), kqanCs)at9)~sn Vm, FiB\!S^VX\!S, s3Tn?+i\~*at3. k Hk s)lskÆ? BKwF5,S. Sn)7C+ P , HYJ Fn. s Vm OÆ3T) m SS, WVk = E(Vm|Fk),k (Vk : 1 ≤ k ≤ m) Y, Vk 3T) k MSl, H Vm zaY!. jYi,, ks) Hk Æ#yA:;J Xk = Vk 1F 1 ≤ k ≤ m WD, SsM.
+ -!. 45
24. T?: +Rz r = 0, P/ Sn Sn i>iM. p m ,
Vm − Vm−1 = Hm−1(Sm − Sm−1),4^ Vm S1, · · · , Sm Æ,, Hm−1 Sm−1 S1, · · · , Sm−1 Æ,. Rzq) m − 1 , B S1, · · · , Sm−1 gO, qs Vm−1 Hm−1. s Sm #1 Sm−1 JaA Sm−1u Sm−1d, Fiq JnZVm(S1, · · · , Sm−1, uSm−1)− Vm−1 = Hm−1Sm−1(u− 1);
Vm(S1, · · · , Sm−1, dSm−1)− Vm−1 = Hm−1Sm−1(d− 1).<JnZJnOK, sHm−1 =
Vm|u − Vm|dSm−1(u− d)
;
Vm−1 =(u− 1)Vm|d + (1− d)Vm|u
u− d= E[Vm|S1, · · · , Sm−1].
25. !SatBa: a, 1 0 < k ≤ m Hk−1 =
Vk|u − Vk|dSk−1(u− d)
;
Vk−1 =(u− 1)Vk|d + (1− d)Vk|u
u− d= E[Vm|S1, · · · , Sk−1].4^ Vk|u = Vk|ξk=u, Vk|d = Vk|ξk=d.
26. S: z ξ E[A:)K, M: P(ξ > 0) > 0 Fz Eξ > 0.
27. S: za~*) 0 Sl 50 , )nnX.&i3/4 ``s: 20%, i 1/4 `` 20%, RzB`ia,
+ -!. 46n)MV>v7zf) 2 3r!Sl 52 3T, ^5G) 2 i 52 Slhp>v7a~~*TB, BK3Ty^h2t?? >v7y^0kat31_M, j 1 ~*, BK3T+S2t??>v7y^0k??
28. S: RzsS^s~*B` r = 5%, #ySy^0k?29. S: j (5.1) ^ ξn QS 0 < u1 < u2 < u3 ``J-
p1, p2, p3, 4^ p1, p2, p3 KF 1, s)``C+ÆSlZ (1 + r)−nSn WWak? zae?
# e J "_&GE&[1. wqJaP#I-N M3T!SS: 7(3T<
Black-Scholes <. BJaI-, qi 3T!S,w. )a3RztLSl)'i|aP#WaI-, w[33s k,w, a29>d[1, BÆqi>swh^2r_. Bgs, BnZ~5,K> i3swh^2X'+. i , dDAo,KO J.L.Doob,Gh ‘Stochastic Processes’ .
2. !m: z``!X (Ω,F ,P) 9Y Ft, C Ft >sMtLSl t UF9. anA:Z XtV Ft y, j1b t, Xt Fn C, BZ5, t, Xt nUO.M. Canb`, Bn~lgsMBn`. ~*SlZyA:Z. anyA:ZV, j1bX t > s,
E[Xt|Fs] = Xs.
3. s: Xt − Xs VA:Z1K, S51K s 9Wa3sT, 7A59S1e3nCT. Wa3s3s`i, 3s9C, Wa3s96J9C. kl,Ks1Wa3sA:)KL;, q)B32Kp. v/~l, \N*0^9\;:Z, ^5*0^A:A3(*ba.
47
* 48
4. s: !m^_zE[Xt|Fs] ≥ Xs,B Xt V. 1K s Wa3sE[. j Xt wF9\;:Z, kyT351wF(, 7A5B. =PjE[Xt|Fs] ≤ Xs,k Xt Vs, [, s1wF3B.
5. Xan1Fd (ao7o,&E[) Æ,d[, Cj Xt , g 1FdÆ,, k g(Xt) ,sg(Xs) ≤ g(E(Xt|Fs)) ≤ E[g(Xt)|Fs],4^Jn3J-B1AdÆ, Jensen 3. smj Xt , / X+
t ^.6. ;9, XandÆ,. Fi Xt , |Xt| X2
t .7. SC+: z P ^``C+, j1ba A, P(A) > 0 S P(A) > 0, 7A5 P(A) = 0 S P(A) = 0, kV`` P P S. p!A:HO Xn, js)SC+ P Æ Xn`` P , k5s) Xn SC+.
8. ZI=: z X anA:)K, 3s)SC+ P,ÆX 3sT, C E[X ] = 0? j X P E[, C P(X < 0) =
0, k X P ^E[, sm E[X ] = 0 S P(X = 0) = 1, i
* 49 P(X = 0) = 1, ^5, eE X = 0 P/3s)B\SC+. 1 P(X > 0) = 0 A:)K\r . eMBJaI-, Y2s)B\SC+. XS. FiSC+s)x)B\I=: j A:)KW3n(, n"A&Cs).
9. C: z X J4P(X = u) = p = 1− P(X = d),4^ d < u, p ∈ (0, 1). k d < x < u , s)zaSC+
P Æ E[X ] = x. )S`` P P, X ^VQ d, u JnS, kE[X ] = qu+ (1− q)d,4^ q = P(X = u). sm E[X ] = x & szas
q =x− d
u− d, 0 < q < 1, 4 d < x < u.
10. at: z Sn MtLSl, nn Sn iA:*Ki,w2aMtL. !VlanA:HO Hn, !mA:HO Xn Xn −Xn−1 = Hn−1(Sn − Sn−1), X0 = x0, (6.1)7A5
Xn = x0 +H0(S1 − S0) +H1(S2 − S1) + · · ·+Hn−1(Sn − Sn−1).^5 X 1K3w S 1K H Y;. B\!mQ m, an*, j Sn )*rsa9\
* 50Z, ^'MtL, 4 n n Sn − Sn−1, k Xn%a9\Z, H G ‘at’ )*rsmaBa, 4 n nHn−1(Sn − Sn−1).(~*at, j Sn n ~*Sl, ata) n− 1fk Hn−1 ~, kGtLy, Xn−1 = Hn−1Sn−1, n~*Sl) Sn, atatL)WXn = Hn−1Sn.sm
Xn −Xn−1 = Hn−1(Sn − Sn−1),^59\_)3w~*Sl_). B\atBaVwhtatBa.
11. A:;J: S H 3!m X (6.1) ENa, aaat=9\Z, BanEN_ HyA, ^5, ata! n− 1 atBa, G3OÆ n tL)'I-, C Hn−1 S1, · · · , Sn−1C. BRL, ataa35m2OE.
H i|BnCAWa, q5 (6.1) !m X H S A:;J.
12. Doob u>: (1) j S , k H S A:;J^.s, Wa3sA, s Hn−1 Fn−1 C, E[Xn −Xn−1|Fn−1] = Hn−1E[Sn − Sn−1|Fn−1] = 0.
* 51;9, (2) j S , H E[, k H S A:;J. BJnr&Q, an51anv/*0, ÆkBaS*, V32O, kr321rB. Van51an1FB*0, Vno&44, krd[1B. Doob Bn!>ZQ, VP, A:J9, JnA:JYwh,Kzd4s.
13. Z: A:X3, jq5, k8naOAv?? kf~*2:? y? ~*)g , . BwX&3Y! (4Fw3 ),FiA:X. qvua;lP#ENbA:X. anA:X T VZ, jHP8) n i n93/. Z`ENQ, sw5naOAv?~*:BJnXZ, s k, &iq 99/BJn+Pgw , j?, AvSl. ~*gBnX3Z, s/)~S3)g, Vi)93, +OÆ3, sgJT`. jVan7g, kS)9/. FiZanq S)9/PwWA:X. ,KV,w, 1b n,
T = n ∈ Fn.[!XZ.
14. ZUZ: 1Jn, a, b, a ∧ b ,4^/n,. j
* 52
Xn A:Z, T A:X, !mXT
n = XT∧n, ∀n,k XT = XTn ^A:Z, H) n ≤ T Xn a\, )
n > T , HZU) XT i, FiVZUZ. i*0CQ, ZU*f67<3mM, ^aaBa, 4Fs92O, BaBa^3_)K. q0k3> Hn−1 ÆXT∧n −XT∧(n−1) = Hn−1(Xn −Xn−1)? n−1 ≥ T , T ∧n = T = T ∧(n−1), XT∧n−XT∧(n−1) =
0; n ≤ T , T ∧ n = n F T ∧ (n− 1) = n− 1. smXT∧n −XT∧(n−1) = 1n≤T(Xn −Xn−1).^5
Hn−1 = 1n≤T = 1− 1n>T.s1n>T = 1T=1 + · · ·+ 1T=n−1,F T Z, Fi 1n>T Fn−1 C, i Hn−1 Fn−1 C4F[E[. sm Doob u>i anZUZ+, anZUZ+.
15. OB: OBTMn?:2. i,KV35, z Sn MtLSlZ, 5HOB, Ts)anatBa Hnn m, j Xn , H S A:;J, kX0 = 0, Xm ≥ 0, P(Xm > 0) > 0.
* 53sw X0 = 0 Tat3D?, Xm ≥ 0 T) m 3"?, FiatM, anWa5n?``K. kl)Xm ≥ 0 R, P(Xm > 0) > 0 E[Xm] > 0 S. )^, 8!yXSS, ^5B`, Fie3tLSS))73#F, zB` r. anOBM\NV1M.%Z6.1 ("KD%Z: T:) 2aMtL Sn MOBFzs)C+ P Æ4^btL)
(1 + r)−nSn)BnC+. P VSZ+.
16. YMV!S: m 3YMV Vm S1, · · · , SmCA:)K. P-, MtL|^YMV. 0kpanA:L,!S? Ban 1xS. \N35, ?z,!_, y^iA:L,SS,K3s3!m. YL,3aA:L,, HSSbVlanVMtLA:L,. Cj, jQhJnaSman9 :, Jnamaon&A:)K, 9l. jQhJna&faK\~*, an!, [~*SS9C, i/!GqJnna\. B9MtLA:L,A:L,3, Fi!S)Blm^3. jMOB, kYMVSly^YMV)C+3sS, VOB!S. BJnS, a, Hz,!_!Sa\s)M; 7,ÆMtLWC+3za? j3za, !S3z
* 54a. pswnCv, whMs!S3zawhL,#[2LOB:2. YMV!STM!S, Hl65=\G b'iPUN%;'i M-2iYA, 5Mi1_, B\ !SVM!S, 7AX\!S. 1_Bas)whMnM, w79!>p nM,Ks.%Z6.2 ("*D%Z: () a9!>^SC+zaFzbanYL,SS Vm &i\1MtLat36, Cs)yZ Hn ÆVm = V0 +H0(S1 − S0) +H1(S2 − S1)
+ · · ·+Hm−1(Sm − Sm−1),4^ Sn = (1 + r)−nSn tL. i|BnAMWnM, 7(3T< Black-Scholes <&nM. B[ mR>YMV Vm Sly^4S (1+r)−mVmSC+3s, CV0 = (1 + r)−mE[Vm].
17. 9!>y: Jnwh!S9!>) 1979 1981 oX M. Harrison, D. Kreps, S.Pliska oao' nW, EN. HSlM\IMan8CKnW. qvuHJny, q^3SlY!, z Sn MtL,H 3e T 3Sl F ,k 3e3SS S(T )−F .QSC+ P, Bs3)bmSS, FiE[ST − F ] = 0,
* 55^sv, i 3SlF = E[ST ] = (1 + r)T E[(1 + r)−TST ] = (1 + r)TS0.
18. o3TSS (B6J 6q|KKvÆ): VlanCvo:3TSS, o:3T:3TI-oi) 3ebanXR>, BnXiA:X, Ha!Z, satA)/3)p!XR>3TF99BnXi9. J.anZσ ≤ T , j3T) σ R>, ksmJ.) σ R>o3TSly^R>SS3s, C
V0(σ) = E[(1 + r)−σ(Sσ −K)+]ko3TSly^V0 = max
σV0(σ),4^ σ !*FQS) [0, T ] PXZ. 73o:SSQ:, Ls32. qM
V0 = (1 + r)−mE[(ST −K)+]. ≥ [. qM1bZ σ,
E[(1 + r)−σ(Sσ −K)+] ≤ E[(1 + r)−T (ST −K)+]. (6.2)s (1 + r)−nSn P F r > 0, (1 + r)−n(Sn −K) : n ≥ 1
* 56E(Sn|Fn−1) = (1 + r)Sn−1 ≥ Sn−1,(s x 7→ x+ 1dÆ,, Fi
(1 + r)−n(Sn −K)+. y Doob !>, s σ ≤ T , Fi (6.2) WD.
# J y&G31. sa7g<pX3T!Sv, EFa&2 RzXGI)'rRLI-, FiBa7qe[tGIX3T!S> . zX t )NX [0, T ] ^)', 4^ T i>s 3e, 0 h 3TX, VbX. K r!Sl. S = S(t) ~*)X t Sl, nA:)K, HqWanA:Z, x)M~*. Black-Scholes 3T!Sv92, i<pX|, i,C+)-9, 3:k9, (A&D1 Ito A:J> #>s. qvuvJZ9, H^ Black Scholes )Gq ^9.
2. V (S, t) ,3T) t Sl, 4^ S = S(t). B,3TSlX t t ~*Sl S(t) Æ,. ) 3e t = T, V (S, T ) gO, b43T,a, j:3T,
V (S, T ) = (S(T )−K)+;3TV (S, T ) = (K − S(T ))+.1aA:Z S = S(t), V =?3 3, )P#Rz, Black Scholes 5< M,w.
3. Rz: S = S(t) E Brown %$, ~*Sly^E[SA:$. z B = B(t) +q Brown %$, z S i|A:vJZdS = µSdt+ σSdB(t), (7.1)
57
! ' 58 Ito vi S s 3S = S(0) exp
((µ− 1
2σ2)t+ σB(t)
), (7.2)4^ µ bl,, σ K,, &<<,, S(0) bSl. ^5Bn<Jn<,, BBn<w+. FiSl53TSl+ K, µ σ . A:J^ba
S Jo)I3T, wvI?(dS)2 = σ2S2(dB)2 = σ2S2dt.
4. Delta 1_: Black & Scholes i V (S, t) Sl) t h aK3Ty^fk∂V
∂SK~*31_M. BnKV ∆, Bn6&V ∆ O3S7Delta 1_. kÆ? zfk x K~*_Wanat
Π = xS − V,k x = ∆ , Π S )'`T, C∂Π
∂S= 0,B56 (/X) , Π SS3b4 S, 4 S 4^zaM3, ^5, at Π 6mM. jGI3/bmJat, ki $vM.
5. Vlan6&5, jmMM, katFy^HatMtLF, CdΠ = rΠdt, (7.3)P/2LOB, BW&M3iG8.
! ' 59
6. pXan1K dt, at Π 2an)'dΠ = xdS − dV, Ito v
dV =∂V
∂SdS +
∂V
∂tdt+
1
2
∂2V
∂S2(dS)2
=∂V
∂SdS +
(∂V
∂t+
1
2σ2S2 · ∂
2V
∂S2
)dt.sm
dΠ = −(∂V
∂t+
1
2σ2S2 · ∂
2V
∂S2
)dt,k (7.3), vJZ
∂V
∂t+
1
2σ2S2 · ∂
2V
∂S2= −r
(∂V
∂SS − V
),'Z
∂V
∂t= rV − rS
∂V
∂S− 1
2σ2S2∂
2V
∂S2, (7.4)mZV Black-Scholes Z, .F V i| 3eWa (`,Wa, bSWa1y), j1:
V (S, T ) = (S(T )−K)+.
7. Black-ScholesZan <"A(vJZ, <s,sH)an)K-)Wan+q`jZ. T0k Black-Scholes Z'W`jZ1E,Kl_a35.3b, ql)B55ii|2K58.
(a) &9-S V (, BBT,Æ,L3)A∂(er(T−t)V )
∂t= er(T−t)
(∂V
∂t− rV
),
! ' 60i V ′(S, t) = er(T−t)V (S, t) i|wZ∂V ′
∂t+
1
2σ2S2∂
2V ′
∂S2+ rS
∂V ′
∂S= 0. (7.5)
(b) X3, W τ = T − t, t (, τ (,)∂W
∂τ=
1
2σ2S2∂
2W
∂S2+ rS
∂W
∂S= 0, (7.6)4^ W (S, τ) = V ′(S, T − τ), `,SS)WbSS.
(c) U- S = eξ, )K S -W ξ, ξ )->y,. ijZM∂W
∂ξ= S
∂W
∂S,∂2W
∂ξ= S
∂W
∂S+ S2 ∂2
∂ξ2.k (7.6) anN,vJZ
∂W
∂τ=
1
2σ2∂
2W
∂ξ2+
(r − 1
2σ2
)∂W
∂τ. (7.7)
(d) an)- ξ = σx − (r − 12σ2)τ , W U(x, τ) = W (eξ, τ), i U i|`jZ
∂U
∂τ=
1
2
∂2U
∂x2, (7.8)
(e) ( V s 3V (S, t) = e−r(T−t)V ′(S, t)
= e−r(T−t)W (elogS, T − t)
= e−r(T−t)U(log S/σ +1
σ(r − 1
2σ2)(T − t), T − t).
8. bSWa U(x, 0) = f(x) `jZsza, `,jU(x, τ) =
1√2πτ
∫ +∞
−∞exp
(−(x− y)2
2τ
)f(y)dy.
! ' 61z`,Wa V (S, T ) = v(S), BS U bWaU(x, 0) = V (eσx, T ) = v(eσx).sm (MZq, 5 T − t τ)
V (S, t) =e−rτ
σ√2πτ
∫exp
(−(log S + (r − 1
2σ2)τ − y)2
2σ2τ
)v(ey)dy
=e−rτ
σ√2πτ
∫ ∞
0
exp
(−(log S
u+ (r − 1
2σ2)τ
)2
2σ2τ
)v(u)
du
u1:I-, v(u) = (u −K)+, Φ ,+qKKJ4, kV (S, t) = S · Φ(d)−Ke−rτ · Φ(d− σ
√τ ), (7.9)4^
d =log(S/K) + (r + 1
2σ2)τ
σ√τ
.1I-, v(u) = (K − u)+, kV (S, t) = −S · (1− Φ(d)) +Ke−rτ · (1− Φ(d− σ
√τ)). (7.10)
9. A:J9irQsBnS, .FvSMswsBkan(vJZ;DZ, B^i5A:J)Ly, A:;JQlm)Bi^J3. 3GIX+<pX, 3T!S9> ;9, 3<!Sv. z S = (St) ~*) t Sl, r > 0 B`, GIIB. an T 3Fr!Sl K :3T) T SS[VT = (ST −K)+ = max(0, ST −K).
! ' 62aYMV) T SSan FT CE[A:)K VT .
10. wht: anatBa H = (Ht) ata) t 9\Xt = HtSt + bt,4^ HtSt at)~*s?, bt D)t>?, t>B`
r, GIXB dbt = rbtdt. <p< (5.9) , whtl69\1Pz3w~*1St>BdXt = HtdSt + dbt, (7.11)j<pMa\, ^, J6;Jvi
d(e−rtXt) = e−rt(−rXtdt+ dXt)
= e−rt[−r(HtSt + bt)dt +HtdSt + dbt]
= Htd(e−rtSt),C9\1P3w~*1S, smbat
X0 = x0 whtk,9\e−rtXt = X0 +
∫ t
0
Hud(e−ruSu),9\ H ~*SlA:;J. BA:;JQA MS=ZT.
11. !S6&: jq> an``C+ P, Æ~*Sl)m``, kYMV) 0 Sly^V0 = e−rT E[VT ].
! ' 63ji,, k VT Y!e−rtXt := e−rT E[VT |Ft]i H (e−rtSt) 3,, H i31_sh aK3TF|3MatBa, BswSlVX\Sl. 3X\SlLsmlm.
12. C: anPC, q3N Black-Scholes 3T!S<,
Black Scholes <Rz~*Sli|wA:vJdSt
St= σdBt + µdt, (7.12)4^ σ > 0 V/$`, µ ∈ R Vn`. A:vJ (7.14) is, Ito v,
d log St =dSt
St− 1
2S2t
d〈S〉t
= σdBt + µdt− 1
2S2t
σ2S2t dt
= σdBt + (µ− σ2/2)dt,smSt = S0 exp
(σBt + (µ− σ2/2)t
)= S0 exp
(σBt + (r − σ2/2)t
).
13. Markov A: Markov ATjOÆ~*)pK, k1e3CS. ^5, Fw5 BS93C329&m> b . ,KVx)Markov A
E[a < Su < b|Ft] = E[a < Su < b|St], (7.13)
! ' 644^ u > t > 0, b > a > 0. Mj, Brown %$ (Bt) 'D1KA, FiSu/St = eσ(Bu−Bt)+(µ−σ2/2)(u−t) Ft 'D, ^ St 'D.
14. SZ+Y!: J6;Jvd(e−rtSt) = e−rtdSt − re−rtStdt,sm~*SlZ (e−rtSt) i|A:vJd(e−rtSt)
e−rtSt=
dSt
St− rdt
= σdBt + (µ− r)dt
= σ
(dBt +
µ− r
σdt
).W
Bt = Bt − (r − µ)σ−1t.jq>an`` P Æ B = (Bt) )m``, k~*Sl)7``. C+)-!>, BV!m7``C+P(A) := E
[exp(aBT − 1
2a2T );A
],i M, 4^ a = (r − µ)σ−1. C)`` P , B Brown%$.
15. 3T!Sv: )e−rtVt = e−rT E[VT ],
! ' 65j^: VT = (ST −K)+, kVt = V (S, t) = e−r(T−t)E
((ST −K)+
∣∣Ft
)
= e−rτ E
[(S0e
σBT+(µ−σ2/2)T −K)+ ∣∣Ft
]
= e−rτ E
[(S0e
σBT+(r−σ2/2)T −K)+ ∣∣Ft
]
= e−rτ E
[(S0e
σ(BT−Bt)+σBt+(r−σ2/2)T −K)+ ∣∣Ft
]
= e−rτ E
[(Ste
σ(BT−Bt)+(r−σ2/2)τ −K)+ ∣∣Ft
],4^A:)K BT − Bt )`` P Ft 'DFTpKKJ4
N(0, T ), A:)K St Ft C, sm3TSl,wV0 = e−rτ
∫
R
(Ste
σx√τ+(r−σ2/2)τ −K
)+φ(x)dx,4^ φ(x) +qKKJ4u+Æ,. 5rY, +qKKJ4Æ, Φ(x) ,, Black-Scholes v
V (S, t) = StΦ(d)−Ke−rτΦ(d− σ√τ ), (7.14)4^
d =log(St/K) + (r + σ2/2)τ
σ√τ
.
# J exotic options
1. o3T+q3T, V$A<3T. Ms+s)G2E+q3TL,. Hqs)>2a2\, s1_DL, ^s4Hj2, 2I, 9_7A\ÆHqua.
2. 5r3Twh,KUJban6J, B23Tq P!DL4k, 3, +tL<g+#, qY>!S, 3TM /X, ijZQ 3B.15r3T>J; , iq YL,N^a2A315r3T>zZ&J, .15r3T>yTvu.
3. wY: TeM 3wYPl, )13XP4HwY. )<pwYI-, ^iman3T4W. RzaK!) t0 NZ\Aw q, /)MOBRz, SlV (t−0 ) = V (t+0 ) + qB q X+tLY!Æ,. j q A:)K, Cj q SQ ~!K=wS, q = 1 ~!Kw; q = 0 ~!=w, ks)SlwY3a!WD, swY3Y!AZr\Ana(. )GIwYI-, VCKa Black-Scholes ZC
∂V
∂t+
1
2σ2S2∂
2V
∂S2+ (r − q)S
∂V
∂S− rV = 0B q BNZ`.
66
exotic options 67
4. Xb4A: T)PH^PkXb4PA. CjÆz3T, V)p!e37Ap!X.RR>3T; =3T, =[iAsX43, nn!&iz!ansn!rg1/, B23T&iVXsE$.
5. \b4A: T313z 3+tLSl, +tLSl\^. C\b4: 3TSl)+tLSlX9gsYt)2an'D)KÆ,, CjQ3T, Q+tL)3T13X/Sl, B;3T!SDJ+tLJnZSl, C1Pa|)K; n\b4: 3TSl+tLSl\, )3T!S(vJZ^.3DP;93T1P7'D\b4)K, Cj=3T, +tLSl)Y!XhBnY!=1/, =3TD (H) 7Dk (1), B;3T\b4, Ls(vJZ3TLs(vJZV)%tWasI-, .3xB1Pa|)K.
6. |+: T3T!S^'D)Kn,. 3T t St Jn)K, J|. 2|=, aa3T^2n+tL, Cjaa3T, 31QJa+tLSlzS, BJa+tL&A:, na&wF/$`, HqPX+#, )Black-Scholes Z^2 1natLSl7o(, B Mo|S; VlaaPR C\b4, CjQ3T^Sl/S, 3TSSb4BnK, 3TSlZ^D(1P7)K, B3TSl1Bn7)K,Vao, Bn7)K73r)an+Xa\)K, B2+tLI-3.
exotic options 68
7. o,: T3T31SSQ23TSS. N3Tao, 31QmQ+tLSl, 4Gj\b43T, j\)KQm'3TSl(, ^ao. <7o3TCvX3T, jan:3Tp\Azfan3TTB, X3T) t1 3, 44w)Kn+3T/)ran t2 3. pLj+3, go3Ts)R MbdS: X3T31Q+3TMSS4E> SS, 1Jo3T&Æ> <,Bgo3T1<KYPEN| , D/8i>.
8. >TBa: T^BaR>J.. Cjo3T, iR>TanbPH, Vlm-2V^;9PH, R>T_)0k6zn, B4^& M>TBaJ.S.
9. o3T: z6JN^5r3T!SS&irLs(vJZ, o3TSÆ,V (t, St) = sup
τ∈Tt,T
E(e−r(τ−t)h(Sτ )|Ft)4^ Tt,T QS [t, T ) A:A, h(·) p!Æ,.Ls, (t, x) 7→ ν(t, x) i|
∂ν
∂t+
1
2σ2x2∂
2ν
∂x2+ r(x
∂ν
∂x− ν) ≤ 0,
ν ≥ h; ν(T ) = h,(∂ν
∂t+
1
2σ2x2∂
2ν
∂x2+ r(x
∂ν
∂x− ν)
)(h− ν) = 0.
exotic options 69
10. Q3T: C\b43T, D1Pa|)KLs. HSÆ,V (t, St) = E
(e−r(t−t)(
1
T
∫ T
0
Stdt−K)+|Ft
)zSt =
1
t
∫ t
0
Ssds,/Vt = V (t, St, St),`,Wa
ν(T, x, y) = (y −K)+.d(tSt) = Stdt = Stdt+ tdSt
dSt =St − St
tdtmCi
∂ν
∂t+ r(s
∂ν
∂x− ν) +
σ2x2
2
∂2ν
∂x2+
x− y
t
∂ν
∂y= 0
11. Basket Option: +tL d n, nn+tLSl$KZdSi = µiSidt+ σiSidWi4^ E[dWidWj ] = ρijdt. B Ito v3TSl$KZ
dV =
(∂V
∂t+
1
2
d∑
i−1
d∑
i−1
σiσjρijSiSj∂2V
∂Si∂Sj
)dt+
d∑
i−1
∂V
∂SidSi
exotic options 70RzatΠ = V (Si, · · · , Sd, t)−
d∑
i=1
∆iSi4^ ∆i ,#y+tLK,, /dΠ =
(∂V
∂t+
1
2
d∑
i−1
d∑
i−1
σiσjρijSiSj∂2V
∂Si∂Sj
)dt+
d∑
i−1
(∂V
∂Si−∆i
)dSiQ
∆i =∂V
∂Si,/ii13 2n+tL Black-Scholes Z
∂V
∂t+
1
2
d∑
i−1
d∑
i−1
σiσjρijSiSj∂2V
∂Si∂Sj+ r
d∑
i−1
Si∂V
∂Si− rV = 0.jGI8DB, CGIwY, /
∂V
∂t+
1
2
d∑
i−1
d∑
i−1
σiσjρijSiSj∂2V
∂Si∂Sj
+d∑
i−1
(r − qi)Si∂V
∂Si
− rV = 0,4^ qi ,B`.
12. i-3T, W\Ai-+tLTB, 4i-` D1, D2 p!, 31 max(D1S1 −D2S2, 0). )GI8DBI-, 3TSli|(vJZ∂V
∂t+
1
2
d∑
i−1
d∑
i−1
σiσjρijSiSj∂2V
∂Si∂Sj
+d∑
i−1
(r − qi)Si∂V
∂Si
− rV = 0.R!V (S1, S2, t) = D1S2H(ξ, t),
exotic options 714^, ξ = S1
S2, )LsZ^hM|,. )K-
∂
∂S1=
1
S2
∂
∂ξ,
∂
∂S2
= − ξ
S2
∂
∂ξ,
∂2
∂S21
=1
S22
∂2
∂ξ2,
∂2
∂S22
=ξ2
S22
∂2
∂ξ2+
2ξ
S22
∂
∂ξ,
∂2
∂S1∂S2
= − ξ
S22
∂2
∂ξ2− 1
S22
∂
∂ξ,ks)(vJZ
∂H
∂t+
1
2σ′2ξ2
∂2H
∂ξ+ (q2 − q1)ξ
∂H
∂ξ− q2H = 0,4^ σ′ =
√σ21 − 2ρ12σ1σ2 + σ2
2.
13. =3T, yEN?, awi31_;9Aa2wY, Vaw, jOÆ+tLSl, kzf=3TN3T3(f2. )=& P, =3TN3Ti|(vJZaZ∂V
∂t+
1
2σ2S2∂
2H
∂S+ rS
∂V
∂S− rV = 0,=Wa/=)#y%tWas. D 3T+tLSlw D =1/, 3TH, sm%tWa t < T ,
V (H, t) = 0, 4^ H *s7*=1/, 1an*sD =3T35, ) 0 ≤ St < H Was Black-Sholes(vJZ, ^tLSlw H %tWa V (tH , H) =
0, j=1/mw , jan:3T, %tWa
exotic options 72
max(ST −K, 0). Dk=V)=1/hB8SS,sm, jm w=1/, /31 0, 1Dk3T35,4SS) w=A, jan*sDk3T, k)tLSl ws!, 3TSSan#yN3TSS, =hB, Ys|, smanDk3TLsan7o, sm#sD 3TJX, 8 Dk3TSS. )3^XI-, #R>Sl 3X=1/Dk3TD 3Tj: Dk3T + D 3T = R>SlX#N3T, Bs tLSlPhB=1/, Dk3TD 3Ty N3T#1. Bn)=3T!S^lm,VDL 4^an=3T, C VanSS.
# R J .37[1. 1_: M3wA:A, b[tA:A7AM>$a&V1_7AO3S. anh 3Twh:ywRsM, [D&93>M, ÆM). jMtLnU>&'7(<7AE Brown %$, kp> sgW>M9. 3>&, FiOLZ>9#.
2. /<1_9: naked covered. Rjryh aK:3T, ijIJ6JW:3TSl +3TijII.
naked Tk&3, km~*) 3e D!SlP,k:ynMijI; j~*s: D!SlPs, "eR>3T, :y#FpMsf3~*hp", M4j^, ~*:g, M^z. covered T)h:3TDpMsf3~*qs, k naked I-#=, m~*:D!Sl, "R>3T, :y ~*hp", :ymB, m~*, "D;R>, :yDifkSrSl ~*, B:y2[M. FiBJa1_9[Z, &zM.
3. UBBa: +aaBaB\zI, :y Hij~*Sl, ~*SlD!Sl K , Hk&3, ~*Slag K , Dfk~*. jBnBa>, :ymM. BnBaJnS (1) ~*fhijII; (2) :73
,Æ 74y3 YjBnSl"? [anS^G733S, ijIIa, 2, H)ij+;2WS, Sij32+;? 7nS, jijBWSl".FRz~*A:$, k Brown%$Ah>q, H)-X2J2o'BW", ^5ijs4+;. Ls, janSl"^3, y2anU, BnUzMZM, U/MZijIÆz, ^5, 4^3-e_M.
4. Delta 1_: Ls, ) Black-Scholes vi^gh>qatMan&$, B3T Delta, 3T Delta K^3T7Aat14MtLSl| Z+, C3TtLSl)'`∆ =
∂V
∂S,4^ V 3TSl. O3SaabG(. \JJa3MtL4nU-eM9aV Delta 1_.
5. T5, :3T3T Delta J-∆(:) = Φ(d), ∆() = −Φ(−d).BJnv;9, VManM, MBnvanLHS. qa5, whMsmOJa:AO3SA. a:A=ÆIBOB36B, Gq[32S1_ij, O3SAJJa, aaatM~*D1_4^6JM, Vaafh3TD1_4^M. aaO3SA8Æ Delta O3S, 1_Mi6Qfh3TIS.
,Æ 75Vlq ∆Sl S X tÆ,, BlstL#FGI)' (3/fk7A !~*) i\an ∆, V Delta^A, BanZV$KO3S. BLMgY$AijI.
6. Gamma: 3T7AtL Gamma 7o,Γ =
∂2V
∂S2,C∆tLSl)'`. Hp MantLM\ Delta^A4#F>$KO3S,K+;Z+. Gamma 1S/, Delta )',j, BM Delta ^AFij3J+;, Gamma z, Delta 19g+tLSl)| . :3T Gamma a\, &
Γ =Φ′(d)
σSτ.
7. Vega: 3T7AtL Vega Sl/$`ao,mathcalV =
∂V
∂σ,jan3T7ij Vega 1Sz, ijSl1/$`)'EN| .
8. Theta: Theta 3TSlAX)'`Θ =
∂V
∂t.
9. j Π = xS−V an Black-Scholes Rzat, k;9 Black-Scholes Zi∂Π
∂t+ rS
∂Π
∂S+
1
2σ2S2∂
2Π
∂S2= rΠ.
,Æ 76smΘ+ rS∆+
1
2σ2S2Γ = rΠ,1 Delta ^Aat, ∆ = 0, 4
Θ+1
2σ2S2Γ = rΠ,m5 Θ Γ PXa!1y: Θ K4Fz, Γ #[4Fz, =Pk[.
10. /$`3T!Sv^za3QC<,. /$`!m: n)K/$` σ !mBa)K)XGIXBn`+qI. /$`3T!S, X\N!ma, sm/$` 1 GIXBn`+qI: /$`M#\, X\N 1 V, m/$`1ynVGIXBn`+qI. a3g, σ√T )K ln(ST
S0)+qI, ST M)K)X T Sl, S0 mM)KSl. aa/$`w[Rz, C/$`dd wM9Fu7. B29rÆata_)1~*Sl, ~*Sl)'^rWM/$`)', Ba/$`)'qm TrZM, P#1Var /$`)aZ+sij|,W.
11. A /$`: q A , |?)K/$`. |?, HXXm\Na!XNXj 1 V1 n!, !mn+ 1: Co,Si: i nX.r+)KSl, i = 0, 1, · · ·, nτ : XXmO+ui = ln( Si
Si−1): i nNX1, i = 1, 2, · · ·, n
,Æ 77
ui +qI\N|I s √1
n−1
∑ni=1(ui − u)2, 4^ u uS.
ui +qI σ√τ , )K s σ
√τ |IS, σ |9 σ , 4^
σ = s√τ. τ iIK, I? /$`1y)'`,iVIK, I? /$`1ye)'`.
12. v /$`q 3TMSl=iI? /$`. ) Black-
Scholes <^, R!n`TpKKJ4, )^, 3`B4HM)KFTpJ46KKJ46Gz. 20 M 80 ^3, a2ij 3`J4^GAK, 44Gijd[ 1,KKJ4>13`J4KY ij:Mfk+ES:B3T, .F6MÆz|B. 803, EFa& ES3TF1yv /$`g.v /$`iq 3TMSlB,S9I?4. \JOlimσ↓0
CBS = (St −Ke−r(T−t))+,
limσ↑∞
CBS = St.4^ CBS q Black-Scholes < 3TSl.
∂CBS
∂σ= SN ′(d1)
√T − t,
∂2CBS
∂σ2=
SN ′(d1)√T − t
σ
(ln2( St
Ke−r(T−t) )
σ2(T − t)− σ2(T − t)
4
).4^, N ′(x) = 1√
(2π)e−
x2
2 , d1 =ln(
St
Ke−r(T−t))+ 1
2σ2(T−t)
σ√
(T−t).k, σ 7→ CBS(σ)) (0,
√2| ln( St
Ke−r(T−t))|
T−t)sdÆ,, ) (
√2| ln( St
Ke−r(T−t))|
T−t,∞)
,Æ 78sÆ,, sm CBS(σ) = CMarket i Newton 9,S9Ls σ.
13. 6/$`: M)<^Tv /$`A, \<d6QA, w[d9R!/$`X+tLÆ,, C)M^AC++tLSlj$KZ:
dSt
St
= rdt+ σ(t, St)dWt)6/$`<^, Rz31 h(ST ), /3TSli|Z
∂C
∂t+ rS
∂C
∂S+
1
2σ(t, S)2S2∂
2C
∂S2− rC(t, S) = 0,
C(T, S) = h(ST )./σ2(t, S) =
rC − ∂C∂t
− rS ∂C∂S
12S2 ∂2C
∂S2kl) t , St !, FiL3 (,. Bruno Dupire (1994)B3R>Sl 3X3TSl3Ls6/$` σ(t, St),GMM:3TSl C(t, St, T,K) i|j(vJZ
∂C
∂T=
1
2σ(T,K)2K2 ∂
2C
∂K2− rK
∂C
∂K,
C(t, St, t, K) = (St −K)+.BMsg3T, \ES9, iLs6/$`:
σ(T,K) =
√2
∂C∂T
+ rK ∂C∂K
K2 ∂C
∂K2
.[, )Ly^, 3FR>Sl 3e3TSl&H , Fi3ES91r'z; Vl, )7o(
,Æ 79I?^, , (IZlzI?ÆI. a5T,6/$`/wAX)$, C: σ(T,K) = σt(T,K), ^5)a5d^DukA:/$`.
14. 6/$`v /$`PXF: R! C(T,K) = CBS(T,K, I(T,K)),
I(T,K) K T v /$`. k Dupire v, σ2(T,K) = 2
∂CBS
∂T+ ∂CBS
∂σ∂I∂T
+ rk(∂CBS
∂K+ ∂CBS
∂σ∂I∂K
)
K2(∂2CBS
∂K2 + 2∂2CBS
∂K∂σ∂I∂K
+ ∂2CBS
∂σ2 ( ∂I∂K
)2 + ∂CBS
∂σ∂2I∂K2 )
=IT+ 2 ∂I
∂T+ 2rK ∂I
∂K
K2( 1K2IT
+ 2 d1KI
√T
∂I∂K
+ d1d2I( ∂I∂K
)2 + ( ∂2I∂K2 ))4^ d1,2 =
ln S
Ke−rT± 1
2I2T
I√T
. j ∂I∂K
= 0, / σ2(T ) = I2(T ) +
2I(T )T ∂I∂T
, CI2(T ) =
∫ T
0σ2(s)ds
T.
# J _h1. B`TXFNZBKFuk8!w`,B`gMs8!uL. B`#whL,2, CjT2V2VU$B`2VN^2VL,, +3B`3pB`. N B`X3!SVB`3!ry. B`3!\nO"3x), Cj2nO", C 3 n!6 n!1 QY 30 2B`S, HqNoawhL,!S9q.
2. T2V: e3 1 o[tL, Z(t, T ) ,^2V) t Sl, / 3SS Z(T, T ) = 1, Z(0, T ) ^2VSl.
3. n: lx)M2VFv /B`, jB`N, r, /T2VSl Z(t, T ) = e−r(T−t), 7A5r = − lnZ(t, T )
T − t.m2V/
P (t, T ) =N∑
i=1
Cie−r(ti−t) + Pe−r(T−t)BRz [t, T ] XXmJe, Ci ti NZB, P w. mB`3N,, /!m/n R(t, T )
R(t, T ) = − lnZ(t, T )
T − t.
80
% 81
4. 3B`: !mrt = lim
∆t→0R(t, t+∆t) = lim
∆t→0
[− lnZ(t, t+∆t)
∆t
].[
rt = R(t, t)
= lim∆t→0
[− lnZ(t, t+∆t)− lnZ(t, t)
∆t
]
= − ∂
∂T(lnZ(t, T ))|T=t
= − ∂
∂T(lnZ(t, t)).
5. 3B`: T6)yVj+C t3B`, a f(0, t),. 3B`T2VjZ(t, T ) = e−
∫ T
tf(t,s)ds,7
f(t, T ) = −∂ lnZ(t, T )
∂T.s/n
R(t, T ) =1
T − t
∫ T
t
f(t, s)ds,/f(t, T ) = R(t, T ) + (T − t)
∂R(t, T )
∂T.Fi, 12VSl3B`noA^blanZdD<, C VlJA<.
6. 3: gB3!m− 1
P (t, T )
∂P (t, T )
∂r=
1
P (t, T )
((T − t)Pe−r(T−t) +
N∑
i=1
Ci(ti − t)e−r(ti−t)
),
% 823i>swY4SbYiwY8X, anPT/X`. Rz P (t, T ) = Xe−r(T−t), /∂P (t, T )
∂r= −(T − t)Pe−r(T−t) −
N∑
i=1
Ci(ti − t)e−r(ti−t)
= −X(T − t)e−r(T−t).− 1
P (t, T )
∂P (t, T )
∂r= T − t.3il3m|3L,n`, kl3!A, Cj 30 32V 10 3T2V3#, 30 32VMrz.
7. B`!-, !-N^anL,. !-3Ta\, aabwhYs, HSa:OB9t. !-;7 20M^3, aK!- ) 20 M 80 b, wi,!-MMF=83, )gWLwhManbW6J, N^!-8!!-B`!-8!B`!-9qB`!-tL!-r,!-B~T!-. N^!-B`!-, B`!-ij/r!/, ia8!#w3~mwI?9g, )#3!, i-!B`BU$B`BNZij. B`!-gh.E!htW, ._)2 7tLA7a;. )whU, u*aPXs)st9DIrBwhMPXs)s#!JjW , Æ3u*aPXhtWo3#. p4B`!-RuM. Cj, Qv79Dg, au!B`twW 7%, ukU$B`twW LIBOR + 0.3%; hv79D, au!B`twW 8.2%, ukU$B`
% 83twW LIBOR+1%. ^3au!B`tw+U$B`tw, Qv7&1, hv7)ukU$B`tw#1. R!Qv7DU$B`tw, 4hv7D!B`tw, )BaI-, /80ow)2VMt>9~Ms#1, .\B`!-ij3eBam'LBn. /wWj!-3p: Qv7 7% !B`autw, hv7 LIBOR + 1% U$B`autw, [>B`!-, !-ijWa: hv7i 7.95% !B`2 Qv7i LIBOR + 0.05% U$B`2 #i-, /opBW 0.25%.
8. B`!-2VPX, i\T2VY!B`!-^!B`S, ^V!-B`Sl. RzB`!-yX3! [t, T ], !-) Ti >, 4^, t = T0 < T1 < · · · < TN =
T , τi = Ti+1 − Ti, NX [Ti, Ti+1] sw Bi, !B` rs,/!B`NZNZy3 rs∑N−1
i=0 τiBiZ(t, Ti+1),1G35, U$B`B ∑N−1i=0 Bi(Z(t, Tk) − Z(t, Tk+1)), q OB>, )wW3p,
rs =
∑N−1i=0 Bi(Z(t, Tk)− Z(t, Tk+1))∑N−1
i=0 τiBiZ(t, Ti+1).B`i-ijEN?, smiaa+tL, 4F\B`!-, iY!n`O", BnZ^mxB B`<y, s, jwYPXnU1_(i$vÆ<, s<v/(IZÆzB.
9. B`<, 1-335, B`iY!, jN,7AY!XÆ,, 1O335, Dyd<, Bgvu
% 84s<B`<, d1+3B`, )L^iS 3ean!2V. Rz3B`i|$KZdr = u(r, t)dt+ ω(r, t)dW,4^ u, ω r, t Æ,, W p!C+!Xa|47%$.B`3fh, 33X\tL, izf3 3ea2Vy,tL31_M, RzT2VSl V1(r, t;T1)
V2(r, t;T2) V" r, t, T ', y,tL Π = V1 −∆V2, ItovdΠ =
∂V1
∂tdt+
∂V1
∂rdr +
1
2ω2∂
2V1
∂r2dt−∆
(∂V2
∂tdt+
∂V2
∂rdr +
1
2ω2∂
2V2
∂r2dt
).Q
∆ =∂V1
∂r∂V2
∂r
,)whtOB>Rz,
dΠ =
(∂V1
∂tdt+
1
2ω2∂
2V1
∂r2−
∂V1
∂r∂V2
∂r
(∂V2
∂tdt+
1
2ω2∂
2V2
∂r2dt
))dt
= rΠdt
= r
(V1 −
∂V1
∂r∂V2
∂r
V2
)dt,J>
∂V1
∂tdt+ 1
2ω2 ∂2V1
∂r2− rV1
∂V1
∂r
=∂V2
∂tdt+ 1
2ω2 ∂2V2
∂r2− rV2
∂V2
∂r
,s% T1 Æ,, % T2 Æ,, `+ 3∂V∂tdt+ 1
2ω2 ∂2V
∂r2− rV
∂V∂r
= a(r, t),
% 854^ a(r, t) ,a;3lMM, HiMSl λ(r, t) 3x):
a(r, t) = ωλ(r, t)− u(r, t),/ 3e T 2V!SZ∂V
∂tdt+
1
2ω2∂
2V
∂r2+ (u− λω)
∂V
∂r− rV = 0,p 3%tWaY! u, ω, λ, i s)Zss7,Ss. d Y! ω u− λω Æ,=.
10. λ(r, t) m: sdV (r, t, T ) =
(∂V
∂t+ u
∂V
∂r+
1
2ω2∂
2V
∂r2
)dt+ ω
∂V
∂rdW,4
∂V
∂tdt+ u
∂V
∂r+
1
2ω2∂
2V
∂r2= λω
∂V
∂r+ rV,Fi
dV (r, t, T ) =
(λω
∂V
∂r+ rP
)dt+ ω
∂V
∂rdW,7
dV (r, t, T )− rPdt = ω∂V
∂r(λdt+ dW ).s5T2V.EM2V, sM2V,y^T, ,is[MQ3n, 4Æ, λ(r, t) \NVMMSl.
11. Vasicek <V” S=m” <, B`lg, H=52Dj, u*a1twDL2[t, r2ZB`h. B`
% 86l, u*a1twDL21P, rZB`s. ^<Rz3B` r(t) i|A:vJZdrt = α(m− rt)dt+ σdWt4^, m S, α i12E+, JAD\|IWp!N,, Wt M^AC+47%$. s^A:vJZ
rt = m+ (r0 −m)e−αt + σ
∫ t
0
e−α(t−u)dWu,EP0 [rt] = m+ (r0 −m)e−αt,B
CovP0[rt, rs] = EP0
[∫ t
0
e−α(t−u)dWu
∫ s
0
e−α(s−u)dWu
]
=σ2
2α
(e−α|t−s| − e−α(t+s)
).m t → ∞, rt Tp N(m, σ2
2α). T2V3B`
Z(0, T ) = EP0
[e−
∫ T
0rtdt],4
∫ T
0
rtdt = mT + (r0 −m)Λ(T ) + σ
∫ T
0
Λ(T − t)dWt,4^ Λ(u) = 1−e−αu
α. Fiii
Z(0, T ) = e
[−(mT+(r0−m)Λ(T ))+σ2
2
∫ T
0 Λ(u)2du]
,iBZ(t, T ) = e
[−(m(T−t)+(rt−m)Λ(T−t))+σ2
2
∫ T
tΛ(u)2du
]
,
% 87T2VSli|$KZdZ(t, T )
Z(t, T )= rtdt− σΛ(T − t)dWt./nÆ,=
R(t, T ) =− lnZ(t, T )
T − t
= m+ (rt −m)Λ(T − t)
T − t− σ2
2(T − t)
∫ T
t
Λ(u)2du.
12. Heath-Jarrow-Morton (HJM) <\Y!2V<(3I?B`, iT13B`dW, s3B`3C, #FI? /n`7A2VSl, \3/J<,8Æ/n7A2VSlMCSaZ, BaZ2P<LÆI. HJM <RzT2VSli|$KZdZ(t, T )
Z(t, T )= rtdt+ σ(t, T )dWt,B Ito v
d(lnZ(t, T )) =
(r(t)− σ2(t, T )
2
)dt+ σ(t, T )dWt,J%;J
lnZ(t, T )−lnZ(0, T ) =
∫ t
0
r(s)ds−1
2
∫ t
0
σ2(s, T )ds+
∫ T
0
σ(s, T )dWs.)s^Q T = t, kl Z(t, t) = 1, ∫ t
0
r(s)ds = − lnZ(0, t) +1
2
∫ t
0
σ2(s, T )ds−∫ T
0
σ(s, T )dWs,FiiiZ(t, T ) =
Z(0, T )
Z(0, t)e(−
12
∫ t
0 (σ2(s,T )−σ2(s,t))ds+
∫ T
0 (σ(s,T )−σ(s,t))dWs).
% 88/ni|R(t, T ) =
− lnZ(t, T )
T − t
= − 1
T − tln
(Z(0, T )
Z(0, t)
)+
1
2
∫ t
0
(σ2(s, T )− σ2(s, t))
T − tds−
∫ T
0
(σ(s, T )− σ(s, t))
T − tdWs.q 3B`T2V
d(f(t, T )) = −d
(∂ lnZ(t, T )
∂T
)
= − ∂
∂Td(lnZ(t, T ))
= − ∂
∂T
((r(t)− 1
2σ2(t, T )
)dt+ σ(t, T )dWt
)
= σ(t, T )∂σ(t, T )
∂Tdt− ∂σ(t, T )
∂TdWt,kl σ(t, t) = 0,
σ(t, T )∂σ(t, T )
∂T=
∂σ(t, T )
∂T
∫ T
t
∂σ(t, s)
∂sds,C)Bn3B`<^, /$`!)e`, BV HJMPs.
# L J .3(b: VaR
1. A:A, MsM is), anatM0kKanENb4/ S, I7A5/$`\NMaa+K, /$`z/LsVh>qatSS!Z+. wh:yDrZQmaaM+K.
2. VaR=Value at Risk: ^at (7Aanwh:ytL) iÆJP X ``M)m3 N VtL.B32gan,S R. Bn R SBnat VaR, VN ,tL) N V3X)', HanA:)K,
P(VN ≤ R) ≤ 1−X%,[ X N 4^Jn<,. ) 1996 Basel 2!nnt>#FI?.v4) X = 99 N = 10 VaR iv OÆt>M. Fi VaR SLsanJS. \;5, VaR 1xB\anS: ‘I)2 kZ+?’
3. VaR Jn<,: X N [9+ X%. VaR NNU:y3Y!tw,K, 1MM, U:yrrLtw)3 10V 99% VaRSma,; 19M?M,U:yrrL)twI?^, : 1 \3 99.9%[9+. )^, q\NQ N = 1, [NV VaR = 1 V VaR×
√N.B[|9, Rz 1 VtLSS)'STKKA:)K, nVSS)''DJ4, BnYWD, ijM.
89
%1 ,: VaR 90
4. 12ntL35,
V aR = −N−1(1−X%)
√√√√TM∑
j=1
M∑
i=1
∆i∆jσiσjρijSiSjBRz M atL, ,K ∆i, Hq/$`J- σi, JatLPX#, ρij .
5. _II?: A , 3I?N9. j5q500 V, , JMnV"atS. n V, [SaV, n n)', , B\q 499 n)', , Hqp/ z73, 5 n, 99% VaR. \NqSXs<jB`', jz i S vi,k nV)'`&6 aV, W i VCKS
v′i = v500 ·vivi−1
.
6. VaR |I, )gOA:)KJ4I-, iQm|I VaR S. CjRz X1, · · · , Xn !#'D, F&TpT,J4, u+Æ,f(x) =
λe−λx x > 0, 0 x < 0. P(X1 ≤ R) = 1−X%
∫ R
0
λe−λxdx = 1− e−λR = 1−X%,CR = − ln(X%)
λq z,!_, )a![9+, n → ∞, Xn =
1
n
n∑
k=1
→ E[X1] =1
λ,
%1 ,: VaR 91Fi VaR |ISR = − ln(X%)Xn.)OJ4<,I-, D|IZJ4C#y<,,[(>s)Z|I VaR S. Rz1 ∀x ∈ R, Fn(x) =
1n
∑nk=11Xk≤x, n → ∞ ,
Fn(x) → E[1X1≤x] = F (x).
7. /$` VaR: [I3 VaR, JA+uE#, zV 3sT+qI σ KKA:)K, k V
σTp+qKKJ4, Fi
P(V/σ < uα) = α,4^ uα +qKKJ4 α- J. sm VaR σ · u1−X%./$`tLn`+qI, jtLp V0 ) V1, k/$`
σ =
[E
(V1 − V0
V0
)2] 1
2
,/tL)'+qI σ ·V0. n V/$` σn aV/$` σ1 iσn = σ1 ·
√n.Cjj Microsoft ~*a/$` 32%, kaV/$` 32%/
√252 = 2%, 4^n 252 nser?. Rj:y\ 1000 po Microsoft ~*, /mate)'+qI
2%× $10, 000, 000 = $200, 000.
%1 ,: VaR 92B\BnataV3 VaR iI?M, H200, 000× u0.01 = 200, 000× (−2.33) = −466, 000.
8. w#A', tLSlnV)'.EnU'D, Rz i VSl)' δSi, Rz δSi δSi−1 #, ρ, 1bl i,
δSi I σ2, / δSi + δSi−1 +qI √σ2 + σ2 + 2ρσ2, /
T VSl)'+qIσ√T + 2(T − 1)ρ+ · · ·+ 2ρT−1,js)w#A, / V aRT = V aR1
√T |L VaR S.
9. MC+aZAWa Artzner et al. (1997) R :oPA: ρ(X + Y ) ≤ ρ(X) + ρ(Y ), JatLyWatMC+Sy/JatLowMC+SP. BnWa” 3 >D)an6v” M>6&aZ, CJp'atMa!/A^'atM.A: m X ≤ Y , / ρ(X) ≥ ρ(Y ), j)bWa, X n Y n, k X MC+Sa!z Y MC+S, jan1yVanI, /anMg, 4FDtw,Krz./e3)A: ∀ N, c, ρ(X + c) = ρ(X − c), j)ij^Pk c ,Kw, /MC+S#F[t c, s^6JwiBRu1_, #yqwL^[t.A: λ > 0, ρ(λX) = λρ(X), jantLF tL,a#1C3), tL,K1Y3,K λ , /7MC+Sy^MC+S λ .
%1 ,: VaR 93
10. VaR 3i|anWa, 4 oni|. Cj, R!Jn'D~*() 1 0.02 ``B 1000p, 0.98 ``B100 p, blan ~*)\3 1 , 97.5% [91/ VaR 100 p, / VaR 200 p; eJn~*, 0.02× 0.02 = 0.0004 ``B 2000 p, 2× 0.02× 0.98 = 0.0392``B 1100 p, 0.09× 0.98 = 0.9604 ``B 200 p, k\3 1 , 97.5% [91/, VaR 1100 p, P 200 pg 900 p, xManWa.
11. C-VaR= Conditional VaR : [ VaR h>qI)2 kZ+, C-VaR 2h>qj3I8M, k.B2 kZ+. Bp,KsTn3 (100−X)% `` A^)k? A^)| VaR +J4)< VaR . 7V3.B, KMWa)*4h8B, B3sS2z. C-VaR i|aZAWa. msC, I? C-VaR,) 2.5% 6J4^, 2% ``B 1000 p, 0.5% ``B100 p, C-VaR 0.8 × 1000 + 0.2 × 100 = 820 p, Jn~*, ) 2.5% 6J4^, 0.04% ``B 2000 p, 2.46% ``B 1100 p, C-VaR 0.04
2.5×2000+ 2.46
2.5×1100 = 1114.4 p. / C-VaR P 1640 p.
12. ZJ: e VaR LB>1?Z. QYZ, jI?M\3 1 V, [9+ 99% VaR, /i> nVB^2toQMBa VaR S. jQV,4JTV, 1% , /5 VaR I?<lI, 1 1 250VRz, a 4 oi,I. jQV,Clz, 5 VaR S(, 4Betw,K(, a 12 o
%1 ,: VaR 94isDllk, jDyJ VaR S. a, jVaR \3 1 V, [9+ X%, j VaR <KY, /Q VaR `` 1 − X%, R! n nHe, 4^ m VBQ VaR , m m
n> 1−X%, 5 VaR |I(.
13. K_IYZ:
H0: 1blaV, Q VaR `` p;
H1: 1blaV, Q VaR ``z p.BQ VaR V,z m ``n∑
k=m
Cknp
k(1− p)n−k,Rz^YZFJ!YZ1/ 5%, /sI?Sz 5% ,m" H0, s/ 5% , H0, p4 VaR I?<.
14. Kupiec R /YZ:H_IK−2 ln[(1− p)n−mpm] + 2 ln[(1− m
n)n−m(
m
n)m],9L, sSzp![9+ χ2(1) J4> S, i^ VaR <.
15. 'DAYZ: atnVSl)''D, kQ VaR I=8y^l$J4)\NX, ^, Q VaR 8rrA)a7, Chistofferson R MYZ VaR 'DA9.pK 0 : aVmQ VaR I=8;pK 1 : aVQ VaR I=8.
uij VipK i F)7VipK j o,. !m_IK−2 ln[(1−Π)u00+u10Πu01+u11 ]+2 ln[(1−Π01)
u00Πu0101 (1−Π11)
u10Πu1111 ],
%1 ,: VaR 954^Π =
u01 + u11
u00 + u01 + u10 + u11,
Π01 =u01
u00 + u01,
Π11 =u11
u10 + u11.SlnV)'&'D, s)_IKTp χ2(1) J4, 9L, sSzp!YZ1/ χ2(1) J4> S, iSl)''DRz.
MYBlack-Scholes vB| 30 8373wh!S> A:J)Ly, ;9, A:;J, Ito v, ,!>, SC+B2snMs`4`+,K`!>[)BnU)ENQ, i5jÆ1, <j Riemann vJE& Einstein m#1 , BaI=1`+,K354 . KsBns, QHG Qv~[wht, ^sBns, Merton Scholes (6f BlackgS) 6M NobelHKf. B3nU&, whtBntC, aAL5rWDz, ,owhw:8&3Twhfbk3Ma, B2Fk7whL,whtIxq32\ , GqF>9yll 9\> wF$,Fi3OÆA:J)B\Ujm)LyR+3, Ba8IKjKO va\.
96
T.6[1] J. Hull, Options, Futures and Other Derivatives, H$zK y, 2008
[2] Paul Wilmott, Paul Wilmott Introduces Quantative Finance,
John Wiley & Son, 2001
[3] M. Capinski, T. Zastawniak, wh,K, ^azK y, 2009
(^)
[4] F. Delbaen, W. Schachermayer, The Mathematics of Arbitrage,
Srpinger & te% v7, 2006
[5] *^, yq, gk y, 2006
[6] *^, yzw~x, sa y, 2004
[7] yVd, vrnx, w&gm[8] yVd, ., s|, XzK y, s, 2005
[9] yVd, wr~, vtoup, XzK y, s, 2005
97