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E[Wn] = 0, n = 0;1;2;:::; Var[Wn] = n. 2. x13x58m > n > 0, x16x7e Wm ?Wn x67Wk, k = 1;2;:::;n;x4fx50x45x31x5a 3. x13x58m > n, x16x7e Wm ?Wn x67Wm?n ?W0 xdx62x17x22x7dx23x51x7ex5a x42x52x40x41x48x2cx47x0x22x18x19x5cx62x8x10Brownx5x3a (Bt)t>0xax2bx74x40x41 x2ex5ax62x79xfxax40x20x59tx5dx5cx22x40x41x5dx5dx62x5cx6axdx33x0x10x68x2b 1. (x27x28x4e)x2b (Bt)x22x5ax5bx61xdx22x5axfxax18x19x22x61 2. (x54x6ex43x31x32x4e)x2b x13x58s < t, x16x7e Bt ?Bs x26x61xdu 6 sx22Bu x45 x31x61 3. (x54x6ex43x1cx55x4e)x2b x13x58s < t, x16x7e Bt ?Bs x67Bt?s ?B0 xdx62x17x22 x7dx23x51x7ex61 4. (x56x45x4e)x2b x26x14x12x14 t > 0, Bt xax65x66E[Bt] = 0x53Var[Bt] = tx22x75 x70x3bx36x51x7ex22x40x41x5dx5dx5a x1fx79x4ax27x18xdx1fx3ax70xaBrownx5x3ax22x3cx52x62x4ax23x3cx2bx70xax3cx5dx22x6dx3 x31x32x62x4ax41x7ex65x3bx1cx57x58x20x5x3ax5ax3cx4cx70x23x22x3bx36x10x63x4ax60x1fx3ax70x9x72x62 xex15x60x1ex3e8.1 (x23x24x6fx30x3cx3e)x63x50x72x62x42x52x44x45x45x31x22x1cx40x41x51x3ax20 x59x67x22x40x41x48x2cx47x0x62x73x2fx3bx7bx52x33x4ax75x70x3bx36x51x7ex22x40x41x5dx5dx52x6f x30x5a x32x76x67x20x59x55x3bx1ax6ex63x9x72x5a x76x33x22x5fxax8x10x5cx31x32x5a x66x2bx32x69x62 Brown x5x3a (Bt) x22x21x5dx2ex5a (V st )t>0 = (Bt+s ? Bt)t>0 x65x66E[V st ] = 0, E[V st+?V st ] = ‰ 0; ? > s; R(?); ? < s; 227 x1fx79R(?)xa? x22x6cx6dx62x16x4axaR(0) = Var[Bs] = s. x13x58x2bx74x40x41x2ex5a ft x65x66x2b E[ft] = ?x4ax4bE[ft+?ft] = r(?) x43x67tx2fx39x62 x16x7exfx49x52(x5a x5b)x1cx55x70x4bx5a Brownx5x3ax22x21x5dx2ex5axexax2bx74x74xax2ex5ax5ax74xax2ex5ax63 x4ax78x4ex69xexax2bx3bx12x44x5ex10x6dx40x20x59x5dx30x22x40x41x2ex5ax5ax1fx3bx40x41x2ex5a x22x76x2bx5dx22x5ex10xaxfx2bx3cxax2dx37x1cx40x41x5cx5dx20x67x2bx74x3fx6dx22x59x67x5ax61x13 x1cx40x41x5cx5dx20xexax5ax5bx52gt = asin!t + bcos!tx22x40x41x2ex5ax62x4ax23! (x5e x23)xax3fx6dx62 a;bxax6dx3x41x6bx52x6cx22x40x41x5dx5dx62x72ax53bx22x32x76x38x53x5f x1dx5fx1fx2bx1cx40x41x5cx5dx20x22x73x78x5ax2bx74x74xax40x41x2ex5ax2bx3cxaxdx30x74x2bx72x2f x30x74(x5ex53x2bx72x5ex6dx51)x40x41x5cx5dx22x59x67x5a x13x58xfx22x1cx40x41x5cx5dx55x51x20x23 xdx2bx74x1cx55x51x20x22x1cx73x78x20x5ex20x73x62 x16x7ex1fx74x74xax2ex5axex14x60x72x2bx3cx22 x2ax41x10x5ax1fx3bx32x11xdx20x6ex3x13x1ax51x3dx47x3d(x2bx47x48x21x22)x22x2ax41x10x62x6c xax1fx17x21x4ax1fx62x38x36x5dx70x20x47x3dx47x0(x2bx47x48x21x22x47x0)xax51xax74xax22x62 x2bx72x5dx32x11x33x4ax1cx74xax5cx20(x5cx13x58x55x4ax23x22x21x6bx7cx35)x5a Brown x5x3ax22 x21x5dx2ex5axax2bx74x1cx5ex5fx20x35x4x22x74xax2ex5ax62x79xfx22x61xdx1cx40x41x5cx5dx55x51x20 x22x1cx73x78x20x43x76x6dx37x5ax61xdx5ex23x22x40x41x5cx5dx55x51x22x73x78x43x2bx17x22x74xax2e x5ax49x52x1cx16x3ex3fx20(x49xfxax1cx16x20x22xax60x13x1cx16x6cx7ax20x52x57x75x65x3bx61x6c(x5e x23)x22x7ax22x75x52)x5a Brownx5x3ax22x21x5dx2ex5ax35xfx36x14x1cx16x3ex3fx20x5ax2bx3bx6d x3x68x6dx53x54x52x22xex11x39x52x62 Brownx5x3ax26x20x59x22x1cx9x6dx20xax1cx16x3ex3fx20x5a x67x40x41x48x2cx47x0x2bx17x62 Brownx5x3ax6ex63x4ax67x6ex22x7dx3fxcx3ax73x2bx50x5a x1fx79x40x1x6ex54x5d -x77xbx3fx58x12x2ex5ax38x6ax22x7dx3fx5ax4ax34x4ex68x6fx79x73x33x69 x67x72x5ax1fx79x68x6fx6bx6ax36x78x4ex69x1ax3ex35xfx5ax38x36x62 x13x62x40x41x48x2cx47x0x2b x17x62 Brownx5x3axax2bx74x6e(x40x41x2ex5a)x62x79x26x14s < tx1axex62 Btx39x14x20x62 sx22 -x77 Fs x22x70x71x41x6bxexaBs. x2bx72x13x8x3ex41x1fxex62x73x20x62 s x26x4dx1a x20x62 tx22Brownx5x3ax1ax50x44x62x67x31xexax20x62 sx22Brownx5x3ax5a x60x8x10Brownx5x3ax22x63x2ax55x4ax7cx22x6ex5ax76x1x74x22x7cx5axa(aBt +b), x1fx79 a;b x52x3fx6dx5a xfx14x1x6exax6ex5a x75x3dx51x22x7cx5axax6ex7x74x32x5x6x5ax6c xa (B2t) x6dxax6ex62x44x52 B20 = 0, x6c E[B2t] = t 6= 0. x1x72x62x63x4ax56x1dx62 (B2t ?t)xax6ex5ax1fx3bx6ex49x52x1cxdx6dx5a x66x2bx3bx5x6xax50x6dx5x6x5a x26 Brown x5x3ax1x74x22x50x6dx5x6x6ex6dx76x2ax55x6ex5ax5cx13x62 (expBt)xex6dxax6ex5a x1x72x62 (exp( Bt ?( 2=2)t))xax6ex62x1fx79 xax3fx6dx5ax1fx3bx6ex49x52x62x74x6dx5a x26xfx6f x22x3fx1dx79x73x33x69x67x72x5a x73x7fx0x1x2x3x23x62x1x3fx1x74x69x31x3cx47x48x3dx52 St x22x5dx5cx65x66x33x0x40 x41xax51x32x5ax2b dSt St = ?dt+ dBt; (1) x1fx79? x49x52x63x64(drift)x62 x49x52x68x69x6a(volatility)x5ax1fx2bx32x5ax22x54x52x6d x3x75x52xdx32x73x6ex69x68x7bx62x6cxax68x6fx73x1fx79x63x4ax78x4ex69x38xfx3ex35x52x2bx48 x37xax40x41x5dx5dx1dx1ex22x1cx67x20x21x22x23x20x62 xfx22xax17xa?dt, xfx22x32x76xa 2dt 228 (x44x52x3cx6dx53x54x52x22xex11x62 dBt = Bt+dt?Btx22x32x76xadt)x5ax6exexaxex62x1f x79x25x65x6aMarkowitzx13xax17x4x32x76x1ax62x63x21x22x23x22x11x12x21x11x5ax62x20x62x60 x14Brownx5x3ax22x5ex10x62 x1cx67x20x21x22x23x20x22xax2bx74x3bx36x51x7ex22x40x41x5dx5dx5a x66x2bx32x69x62 x38x68x1cx32x5ax73x5ax5bx68x5bx55 dSt dt = St ?+ dBtdt ? ; x68x6fx22x63x4axex62x47x3dx22x21x6bx23?x66x78x73x78x52 x22x1cx16x3ex3fx20x22x45x4bx5ax1f x17x3cx52x22(St)x6ex63xdxfx5fx60x52(exp( Bt +(?? 2=2)t)). x2bx9x1axex62 xf x6dxax6ex5ax6cxax13x58? = 0, (St)xex5dx55x50x6dx6e(exp( Bt ?( 2=2)t)). x60x71 x22x63x4ax21x72x62 x26x14x12x14 tx1axex62x40x41x5dx5d St x75x70x26x6dx3bx36x51x7ex5a x26x47x48x3dx52x42x1fx17x22x31x32x5cx6dxax66x58x72x1ax22x62x72xax2dx37x7fx0x44x5d x1x2x3x66x6fx6bx41x51x3dx6dx45x61x64x78x22x59x58x5ax68x6fx38x68x1cx40x41xax51x32x5ax33 x1ex5c (x1fx63x6dx53x54x52x69x39x52xax7d x1cdt = 1x20x72x64x78x22)x62x5fxd Sn+1 ?Sn Sn = ?+ "n+1; (2) x1fx79"nxax40x41x51x3ax47x0x5ax1fxexaxex62 x32x5a(1)x27x28x68x72x40x1cx67x68x1dx4bx43 x8x9x6ax49x4a (x69x6ax6bx3x8x9x6c) x6dx3bx3dx6ex3dx3ex6fx69x49x4ax20x1fx2bx31x32x5ax6 x7x68x62x41x1xdx75x39x52x7ex7fx56xdx24x10x31x32x51x3dx14x47x48x3dx52x47x0x5ax55x2bx74 x40x41x48x2cx47x0x62x4ax38x36x72x40x1x6exa Bachelierx5ax6cxax1fx3bxex11x48x1x73x3e x3fx53x27x1dx3bx32x69x43x3x51x3cx5ax27x28x68x62x65x5dx27x1dx51x3dx50x72x62x47x3dx76x51x47 x0x3ex3fx6dxax74xax47x0x62x72x47x3dx21x22x47x0x46xax74xax47x0x62x44x72x46x63x76x5a x55x40x41x51x3ax47x0x5a x66x2bx32x69x62 (1)x22x63x3ax42 dlogSt = ?dt+ dBt: (3) xfx22x33x1ex5cxa logSn+1 ?logSn = ?+ "n+1: (4) xfx7bx29x2ax67x68x1dx4bx43x27x74x70x57x49x4ax6dx3bx3dx6ex3dx3ex6fx69x49x4ax5a 1 (3)x67(4) x5cx6dx2bx5dx62x6cxaxfx6fx73x1cx2bx46x36x6bx20x22x7bx52x33xax2bx5dx22x62x79x68x6fxdx36x6b x51x5bx2b log S n+1 Sn ? … Sn+1 ?SnS n : x1fx2bx36x6bx51x5bx1ax40x2b x1b jxjx35x59x20x62 log(1 + x) … x. x44x71x62 x1b Sn+1 x35xf x36Sn x20x62x68x6fxex63x38 log ? Sn+1 Sn · x1bx42x47x48x21x22x22x36x6bx17x5ax27x28x68x62x60 1x40x15x54 1988x56Lox31MacKinlay [13]xfx18x51x2bx2excx15x2x50x20xcx1dx17x7fx6x24x25x16x23 x7dx7ex31x54x8x39xcBlack–Scholesx36x37x64xcx11x28x7ex7fx2bx2exbx31x7ex2ax70x22xcx16 229 x14x73x6dx3x68x62 log ? Sn+1 Sn · x1ax1bx26x45x37x3ex62x73x27x1dx51x3dx20x62x1x3fxex38 (x1b) x21x22x23x1fx17x3cx52x5a x43x73x68x6fx1ax53x6dx2bx9x22x40x41xax51x32x5ax2b dXt = Ktdt+HtdBt: (5) x1fx79 (Kt) x53 (Ht) xax3bx74x40x41x2ex5a (x73x68x1cx22x47x3dx32x5ax23x62 Kt = St?, Ht = St )x61 (Xt)x5fxax60x71x3cx52x22x40x41x2ex5ax5ax68x6fx43x73x5dx53x6dx62 x26x14Xt x22x6cx6df(Xt)x12x7dx65x66x6cx17x22x40x41xax51x32x5ax5a x13x58x68x1cx32x5ax23Ht = 0, Kt xatx22x8x3ex6cx6dx62 x16x7ex1fxexax3dx4cx6cx6dx5ex9x6dx22x69x54x62x79x68x6fxd df(Xt) = f0(Xt)dXt = f0(Xt)Ktdt: (6) x6cxax73Ht x6dx51x14x6cx20x62 x7cx4bx6dx2x13x71x5a x5dxex19x44x22x5dx70xax51x22x3cx52xex50x5axax51x5fx8x5b df(x) = f0(x)dx x22 x27x28x75x52x73x14x2b x1b x x22x21x5d ?x x5dx5cx35x59x20x62 f(x) x22x21x5d ?f(x) = f(x + ?x) ? f(x) x63x5fx60x52 f0(x)?x + o(?x), x4ax23 o(?x) x5fx60 ?x x22 x4cx46x2fx30x59x62x79 lim?x!0 o(?x)=?x = 0: f0(x)?x x42x52 ?f(x) x22x1cxfx10 x21x50x20 x49x52 f x73x23 x x37x22xax51 df(x). x2x60x14 dx xex51x14 ?x, xex64x78 x68x1cx22xax51x5fx8x5bx5a x13x58 x x22xa t x22x6cx6d x = x(t), x16x7ex68x6fx22xdx2b df(x(t)) = f0(x(t))x0(t)dt. x1fxax44x52?f(x(t)) x26x14?tx22x1cxfx10x21x50x20x63 x4ax60xfx39x14?xx22x1cxfx10x21x50x20x53?xx39x14?tx22xfx10x21x50x1ax4cx55x5a x1x72x62 x26x14x40x41xax51x32x5ax1axex62 x7cx4bx6dx2x13x71x5a x2cx27x68x62x5cx13x62x73 Kt x53 Ht x43xa t x22x3cx3fx6cx6dx22x7cx5ax33x62x3cx6dx53x54x52x22x7bx52x62x36x6bx51x5b ?Xt … Kt?t+Ht?Bt x22x3ax37x7bx29x2ax2bx74xax17x52Kt?t,x8x10x76x52Htp?t x22x3bx36x51x7ex22x40x41x5dx5dx5ax14xax40x41x5dx5d ?X2t x22xax17x23x22x57x75?tx22x39x5a x1fx17x62 (6) x6dx2x55x31x5a x26x14x7ax9x6cx6d f = f(x), x1dx13xfx22 Taylor x5x69 x5bx2b ?f(x) = f(x+?x)?f(x) = f0(x)?x+ f 00(x) 2 ?x 2 +o(?x2); x68x6fxex63x9x64x33x0It?o (x71x72)x15x5bx2b df(Xt) = f0(Xt)dXt + 12f00(Xt)H2t dt: x1fx6exexaxex62 df(Xt)x42x52x40x41x5dx5dx62 xfx22xax17x1af0(Xt)dXt x22xax17x22x5d x37x72x2bx74x57x75f x22x56x46x9x6dx22x39x5ax1fxax40x41xax51x3x67x8x3exax51x3x22x44 x28x6dx62x2bx3dx4cx6cx6dx22xax51x15x5bx23x57x75x56x46x9x6dx39x5a x68x6fx13It?ox15x5bx1ax44x6x2bx40x1x74x22x6cx6dx2b 230 1. f(x) = x2, dXt = dBt, X0 = 0. x16x7e Xt = Bt, df(Xt) = 2BtdBt + 12 ￠2dt: x1fxexaxex62 B2t ?t = 2 Z t 0 BsdBs: x63x4ax50x72x62x3ax37xax74x6ex5ax35x68x14x62x1fx74x15x5bx67x2x40x41x7cx5ax22x5cx20x73x14 x37x6ax2bx39?t. 2. f(x) = logx, dSt = St(?dt+ dBt), S0 = x0:x14xa St = x0 + Z t 0 Ss(?ds+ dBs); x1dx13It?ox15x5bx62x63x64 log(St) = log(x0)+ Z t 0 dSs Ss ? 1 2 Z t 0 1 S2s 2S2 sds; x4ax24 log(St) = log(x0)+ ?t? 2 2 t+ Bt ? ; x2b St = x0 exp ?? 2 2 ? t+ Bt ? : x1fxax68x6fx68x69x48x1x46x78x2ex22x59x58x5a x4ax68xax2bx40x4ax6dx53x54x52x22x5ax5bx61x1bx1cx22x6dx3x59x58x5a x13x58x26x1fx40x59 x58xdx66x30x22x3ex35x62 x16x7ex26x2bx9x22x18x19x20x59x7fx0x3x22x6dx3x5fx8xex5ex4fx78 x5cx6dx53x36x3ex35x5a 10.2 x33x69x68x6fx36x38x68x69x46x78x22x2bx40x6dx3x7dx3fx13x1bx54x52x22x6dx3x5ax5b x1ax5fx8x62 x1x6ex2x1ax18x3f Black–Scholesx48x49x5a x68x6fx70x2bx74x63x71x58x59(?;F) x72x4x62x1fx79?xax2bx74x12x7bx22x16x4cx62 F xax60?x23x22x45x16x61x11x55x22 x77x5ax67x33x1ex7cx5ax6ax6bx62x52x6ax13x1x4fx35x5dx40 x2ax20x59x21x6bx22x6dx6bx21x6bx62 x1exc -x77xb (filtration): (Ft)t>0, x1fx79x53x74Ft x43xa?x22x45x16x61x11x55x22 -x77x62x5cx6ax65x66x74x75x10x2b 0 < t1 < t2 =) F0 ‰F1 ‰F2 ‰F: 231 x1fxax68x6fx73x1fx2bx4ex18x3fx23x61x46x78x22x60x1cx2cx71x33x20x61x5ax55x22 -x77xbx22x4f x4x5ax6cxax68x6fx39x76x1ax3ex35 x1cx2cx71x33x20 x16x17x1ax3ex35xfx5a x13x58x73x63x71x58x59 (?;F)x23x22x3cx52x6ax7dx23x71x78P, x16x7ex62x17x13x1fx2bx4ex76x6ex18x3fx22x16x17x62 x3e x3fx22x31x3cx6cx7dx23x2cx71x26x14x53x74Ftx62x4ax74x24x72x43x2bx40x27x28x23x35x25x4x2ax22 x73x43x1ax5ax14xax1dx5fx1bx1fx4fx35x22F0 xex6dx2cxax2bx40x10x3x78x3cx22x2cx71(F0 x18 x57x75 ? x53;), x22x57x75x6cx7dx23x2cx71x53x7dx23x52x2bx22x2cx71x5ax6dx2ex67x68x6fx4ax1f x22x18x3fx2bx17x62x68x6fx4ax6ex39x24x22xax13x14x1ax7bx3cx2bx40xdx5ex5fx7bx52x22x51x3dx7d x23x71x78x62x70x72x73x68x6fx22x29x2ax23x62x2bx5dx22xax5dx68x7bx62 x6cx7dx23x2cx71x3x5dxa x2bx74x7bx3cx22x3fx61xdFt x22x45x16x5a xdx6ax63x71x58x59(?;F)x62xex63x3cx52x40x41x5dx5d X (x79Xx52x3cx52x73?x68x22 x63x71x6cx6dx62 xfx26x14x12x14x27x6dc, f! 2 ?jX(!) < ag ‰ F, x2bx72xe fX < cg xax2bx74x2cx71)x53x40x41x2ex5a(Xt)t>0 (x79x26x14x53x74t, Xt xax40x41x5dx5d)x5ax2bx74 x40x41x2ex5a(Xt) x26x14x3bx3cx22! 2 ?x1axe, t 7! Xt(!)xatx22x6cx6dx62 xfx49x52 x7dx2ex5ax22x74x15x5ax5axfx40x1x6dx2bx3cxax18x19x22x5ax6cxax52x6ax5bx25x6dx3x68x22x2 x28x68x22x7fx36x62x68x6fx4ax6ex75x31x3cx61x18x3fx22x40x41x2ex5ax22x5ax5bx61xdx22x5axfx43 xax18x19x22x5ax1fx26x14x1dx1ex7ex7fx22x70x1ex18x3fx6ex48x66x30x5a x67x1fx69x2bx17x62x68x6fx62x17x63x4ax3cx52x58x12x2ex5ax5ax40x41x2ex5a(Xt)t>0 x49x52 x4cx4dx70x4bxax50x26x14x12x14 t, Xt xaFt-x63x71x22x5ax58x12x2ex5ax43xax52x26 -x77xb x72x1fx22x5ax6cxax26x2bx74x3bx3cx22x40x41x2ex5ax1axex62x60xfx6ex63x2ax55x2bx74 -x77xbx62 x79x7d (Xs;s 6 t) = Ft (xfx5fx60x26x14Xs (s 6 t) x1axex63x76x4x2ax22x2cx71x16 x4cx61x2ax55x22 -x77)x62 x16x7e (Xt) x26x14x1fx74 -x77xbx2bx3cxax58x12x2ex5ax5ax6dx2ex60 x71x47x2ax22Ft x6dx2bx3cx57x75x61xdx22x6cx7dx23x2cx71x5ax52x71x62x22x54x38x61xdx22x6cx7d x23x2cx71x6ex67x78x53x74Ft x23x58x5ax73x33x1ex48x49x23x62 -x77xbx27x28x68xexax1dx1ex22 x3dx52x2ex5ax61x5ax55x22x5ax6cx16x20x6cx7dx23x2cx71x67x6dx63x76x2cx71xax2bx7dx2cx62x70x72 x6dx1ex50x68x1cx69x54x5a x13x58x2cx2dx40x41x2ex5ax22x5axfxax6dx18x19x22x62 x16x7ex68x6fx6ex63x3cx52x63x4fx2e x5ax4ax4bx4ax7cx2bx40x2ex5ax5ax6cxax26x14x5axfxax18x19x22x40x41x2ex5ax1axex62 x1cx1fx2bx74 x20x62x20x38x6ax22xex11xexcxdx7bx52x5ax44x71x62x73x68x6fx4ax33x18x3fx22x48x49x23x62 x70x7 x6ex79xax58x12x2ex5ax5a x1bx1x62x73x30x3dx51x22x48x49x23x62 x70x7x79x39xax63x4fx2ex5ax5a x73x1fx2bx4ex22x33x1ex48x49x23x62x1dx1ex3dx52x22x5dx5cx73x53x2bx20x62x43xdx3bx3bx63 x76x5a x1bx20x62x5dx5cx66x30x37x62x1dx1ex3dx52x73x3bx3cx20x62x42x52x40x41x5dx5dx7cx5ex14x3b x36x51x7ex5ax72x1fx3bx43x1ax22x18x19x5cx62xex33x1dx1ex3dx52x22x5dx5cx54x5dx13x61x13Brown x5x3ax1ax62x4dx5ax7dx23x58x59(?;F;P)x68x22x40x41x2ex5a(Bt)t>0 x49x52Brown x53 x69xax50x2b 1. (x27x28x4e)x2b (Bt)x22x5ax5bx61xdx22x5axfxax18x19x22x61 2. (x54x6ex43x31x32x4e)x2b x13x58s < t, x16x7e Bt ?Bs x26 Fs = (Bu; u 6 s)x45 x31x61 232 3. (x54x6ex43x1cx55x4e)x2b x13x58s < t, x16x7e Bt ?Bs x67Bt?s ?B0 xdx62x17x22 x7dx23x51x7ex5a x63x4ax1dx68x62 x13x58(Bt)t>0 xaBrownx5x3ax62 x16x7e Bt?B0 xaxax17x52?t, x32x76 x52 2tx22x3bx36x40x41x5dx5dx62x4ax23?x53 x52x3fx6dx5ax72x26x140 6 t1 < ￠￠￠ < tn, (Bt1;:::;Btn)xax3bx36x40x41x5ex5dx5ax1fx79xcxdx5bx72x22x1dx68x22x27x68x62x3bx13x68x6f x73x68x2bx4ex23x61x50x72x22x62x76x13x40x41x48x2cx47x0x22x39x7cx5ex14x3bx36x40x41x5dx5dx2b x17x5a B0 = 0 (x73x7dx23 1 x22x7bx52x33 (a.s.)), ? = 0, = 1 x22 Brown x5x3ax49 x52x8x10Brownx5x3ax5ax4ax6ex68x6fx2bx9x75x31x3cx61x18x3fx22Brownx5x3axax8x10 Brownx5x3ax62x62x20x62 xfx22x67x2fx74 -x77 (xfx2x24xexax60x7dBrownx5x3ax61x2a x55x22) xcx3ax73x2bx50x5ax1fx20x62x61x13 Ft-Brown x5x3axexax2bx74x58x12x2ex5ax62x5c x6ax26x14s < t, Bt ?Bs x26 Fs x45x31x5a x26x14x30 -x77xb(Ft)t > 0x22x7dx23x58x59(?;F;P)x68x22x40x41x2ex5ax62x68x6f x62x17x63x4ax3cx52x70x71x6dx3x41x6bx62x4ax4bx3cx52x6ex5ax63x6dx58x12x2ex5a(Mt)t>0 x49x52 x6d(x1fx79x1cx63x6dx20x7bx29x2ax26x14x12x14 t, E[jMtj] < +1),xax50x26x14x12x14 s 6 t, E[MtjFs] = Ms. x6ax6bx69x6ex63x3cx52x68x6ex53x33x6ex5a x26x14x18x19x6e (Mt) x1axex62 x68x6fx4ax6ex1x3fx5dx1dx13x33x0x2bx5dx22Doob x71x17x2ex2b E ? sup 06t6T jMtj2 ? 6 4E[jMTj2]: x4ax1dx68x70x7x5a x13x8x10Ft-Brownx5x3ax63x4ax2ax55x31x5ax3bx6ex61 x79x68x6fxdx2b x58x5c10.1 x8(Bt)t>0 x20Ft-x28x29Borwnx75x2cx27x37x5d 1. (Bt)x20Ft-x75x2d 2. (B2t ?t)x20Ft-x75x2d 3. exp( Bt ?( 2=2)t)x20Ft-x75x27 x10x11 1. x44x52x1b s < tx20x62 Bt ?Bs x26 Fs x45x31x62 x78 E[Bt ?BsjFs] = E[Bt ?Bs] = 0. x44x71x62 (Bt)xaFt-x6e. 2. x26x14s < t,x60 E￡B2t ?B2sjFs? = E￡(Bt ?Bs)2 +2Bs(Bt ?Bs)jFs￠ = E￡(Bt ?Bs)2jFs?+2BsE[Bt ?BsjFs]; 233 x4ax4bx60(Bt)xax6ex62 E[Bt ?BsjFs] = 0,x63x64 E￡B2t ?B2sjFs? = E￡(Bt ?Bs)2jFs?: x2x60x21x5dx22x74xax10x53x8x10x10x62x64x78 E￡(Bt ?Bs)2jFs? = E￡B2t?s? = t?s: x44x71x62 E[B2t ?tjFs] = B2s ?s. 3. x38x36x7ax7bx78x62 x26x14x8x10x3bx36x51x7ex40x41x5dx5d g,x68x6fxd E￡e?g? = 1p2… Z +1 ?1 e?xe?x22 dx = e?22 : x62x20x62 x13x58s < t, x16x7e E ? e Bt ? 2t 2 flfl flFs ? = e Bs? 2t2 E￡e (Bt?Bs)jFs?; x5cx6ax60x14Bt ?Bs x26 Fs x45x31x62x68x6fxd E￡e (Bt?Bs)jFs? = E￡e (Bt?Bs)? = E￡e Bt?s? = E h e gpt?s i = e 2(t?s) 2 : 2 x73x33x1ex7cx5ax33x62x68x6fx6bx52x1dx1ex4bx4cx22x73x40x0x46x70x7`x33x22x42x43x3d x17x52 ?Vn(`) = V0 + nX j=1 `j??Sj: x38xfx18x19x5cx6ex62x1fx2bx5ex53x5bx79x5dx52x6dx51x5b Rt0 `sd?Ss. x1fxex4x4bx40x41x2ex5a x22x6dx51x5ax16x4axax1fx79x22x42x43x3dx52x79xax60Brownx5x3ax2ax55x22x6ex62x72x75x6fx7a xex6bx52Brownx5x3ax22x5ax5bx61xdx22x5axfx43xax37x37x6dx63xax22x62x4ax24x1fx74x6d x51x22x3cx52x35x36x1bx1cx48x49x5ax701920x63x1dx76x7cx6dx3x66Wiener (x60x14Wiener x22x45x22x62 Brownx5x3ax6ex49x52Wienerx70x4b)x22x2ex42x50x62x2bxdx781940x63x1d x2cx28x6dx3x66It?o (x71x72x48)x46x6x38x40x41x6dx51x22x75x52x47x48x5ax33x69x68x6fx1x1cx1f x32x69x22x2bx40x3cx5dx6bx41x5ax4ax1dx68x43x70x7x5a x32 (Bt)t>0 xax30 -x77xbx22x7dx23x58x59 (?;F;(Ft)t>0;P) x68x22 Ft-x8x10 Brown x5x3ax5ax68x6fx79x26x2bx6ax58x12x2ex5a (Ht) x3cx52x33x0x5ax5bx22x40x41x6dx51 234 Rt 0 HsdBs. x13x62x3cx52x3ex3fx22Lebesguex6dx51x16x17x62x68x6fx63x36x26 x1cx46x76x58x12x2ex5ax20 (Ht)x1ax3cx52x40x1x22x6dx51x5a (Ht)06t6T x49x52x77x17x18x4ex70x4bx62xax50 Ht = `i; t 2 (ti?1;ti]; 0 = t0 < t1 < ￠￠￠ < tp = T; x4ax23`ixaFti?1-x63x71x22xdx15x40x41x5dx5dx5ax1fx79x60x14x1fx17x22x70x51x2ex5ax6dx2xa x18x19x22x62x58x12x67x63x4fxexdx61x5cx20x5a x26x14x1fx17x22x70x51x2ex5ax62 x1b t 2 (tk;tk+1] x20x63x4ax3cx52 Z t 0 HsdBs = kX i=1 `k(Bti ?Bti?1)+`k+1(Bt ?Btk): x58x5c10.2 x1dx16(Ht)06t6T xfx61x6fx1ax13x33x37x5d 1. ?Rt 0 HsdBs · 06t6T x20Ft-x38x39x75x2d 2. E " Z t 0 HsdBs ?2# = E ?Z t 0 H2sds ? ; 3. E " sup t6T flfl flfl Z t 0 HsdBs flfl flfl 2# 6 4E ?Z T 0 H2sds ? : x7d H = ‰ (Ht)06t6T x52x26 (Ft)t>0 x58x12x22x2ex5ax62 E ?Z T 0 H2sds ? < +1 : x16x7ex26x14x1fx2bx6ax58x12x2ex5ax62x63x4ax3ex2ex33x0x57x54x1ax3cx52x40x41x6dx51x2b x58x5c10.3 x8(Bt)xfFt-x28x29Brownx75x2cx27x37x5dx63x2dx62x29x3Hx2 x36x33x2d[0;T]x25x3Ft-x38x39x75x3x5x38J,x7bx7c 1. x1dx16(Ht)t6T x20x61x6fx1ax13x33x37x5dx51x60x31x32t 2 [0;T], J(H)t = Z t 0 HsdBs a:s:; 2. x51x60x31x32t 6 T, E[J(H)2t] = E hRt 0 H 2 sds i : 235 x28xdx62x29xdxfx20x2dx64x1c1 (a.s.) x3x5cx33x9x69x36x3x27 x60x71x62x68x6fxex26Hx23x22x58x12x2ex5ax3cx52x6ax40x41x6dx51Rt0 HsdBs = J(H)t. x26x14x1fx17x22x40x41x6dx51x62x33x0Doobx6dx51x5bx39x1x55x31x2b E " sup 06t6T flfl flfl Z t 0 HsdBs flfl flfl 2# 6 4E ?Z T 0 H2sds ? : x26x14x66x2bx6ax30x4x22x58x12x2ex5ax2b ?H = ‰ (Ht)06t6T x52x26 (Ft)t>0 x58x12x22x2ex5ax62 Z T 0 H2sds < +1 a:s: : x6ex76x6ax6bx69x3cx52x40x41x6dx51x62x6cx4ax4fx12x22(Rt0 HsdBs)06t6T x6dx2x2bx3cxax6ex62 x72x18xax18x19x40x41x2ex5ax5ax6dx2exfxdx33x0x7bx52x33x22x18x19x10x2b x13x58fHng‰ ?H x65x66RT0 (Hns )2dsx3cx7dx23x7cx14x6cx62 x16x7e sup06t6T jRt0 Hns dBsjx6ex3cx7dx23x7cx14 x6cx5a xdx6ax1fx40x10x50x4ax6ex62x68x6fxex63x26x33x0x5ax5bx22x40x41xax51x32x5ax67x72x68 x7bx22x6dx3x3cx52x2b dXt = b(t;Xt)dt+ (t;Xt)dBt; X0 = Z: x13x58 = 0,x1fxexax8x3ex22xax51x32x5ax62x73x5cx3x68xfx62x4dx6ax53x2bx20x62x22x5 x3ax38x78(x65x59x53x32x5e)x22x5dx5cx5axdx6ax6ex2bx39x6ex62x1fx2bx5x3ax38x78x66x78x6ax40 x41x45x4bx5axdx23x68x6bx5bx35x31x3ex35x62x6cx73x6dx3x68x4ax1fx35x36xex64x35x7bxex5ax3c x52x6ax40x41x6dx51x4ax6ex62x1fx17x22x40x41xax51x32x5ax63x4ax3ex2ex61x13 It?o x2ex5ax1ax65 x1cx5a x32(?;F;(Ft)t>0;P)x52x2bx30 -x77xbx22x7dx23x58x59x5a (Bt)t>0x52Ft-Brown x5x3ax5a x16x7ex33x0x5ax5bx22x27x17x40x41x2ex5a(Xt)06t6T x49x52It?ox70x4bx2b 8t 2 [0;T]; Xt = X0 + Z t 0 Ksds+ Z t 0 HsdBs a:s:; x4ax23X0 x52F0-x63x71x61 (Kt)x53(Ht)x52x26 Ft x22x58x12x2ex5ax62 x6ax65x66 Z T 0 jKsjds < +1 a:s:; Z T 0 H2sds < +1 a:s:: x63x4ax1dx68x62x33x0x57x54x55x31x2b x58x5c10.4 x8(Mt)06t6T x20x38x39x75x33x3a Mt = Z t 0 Ksds; Z T 0 jKsjds < +1 a:s:; 236 x37x5d 8t 6 T; Mt = 0 a:s:: x3bxax60x14xdx1fx2bx57x54x62 It?o x2ex5ax73x5ax5bx37x37x22x7bx52x33xdx6cx2bx5fx60x61 x79x62 x13x58 Xt = X0 + Z t 0 Ksds+ Z t 0 HsdBs = X00 + Z t 0 K0sds+ Z t 0 H0sdBs; x16x7ex60x14X0, X00 x53x3bx74x40x41x6dx51x39x43xax18x19x6ex62x63x36x9x72x5ax5bx37x37xd Ks = K0s, x2x9x72 X0 = X00 a:s:. x76x6ex62 x2x64x78x5ax5bx37x37xdHs = H0s. It?ox2ex5ax33x1fx69x46x78x22x40x41xax51x32x5axdx6ax7bxex22x75x52x5a It?ox1dx68x62 x13 x58 Z T 0 jb(s;Xs)jds < +1 a:s:; Z T 0 j (s;Xs)j2ds < +1 a:s:; x 7! b(t;x)x53x 7! (t;x)xaLipschitzx6cx6dx62 bx53 xdx15x62x4ax4bE[Z2] < +1, x16x7ex73x5ax5bx37x37x22x7bx52x33x62x7dx40x41xax51x32x5axdx6cx2bx35 (Xt)06t6T, x6a E￡sup06s6T jXsj2? < +1: x40x41xax51x32x5ax22x79x7ax22x1ex50x6ax40x41x51x3dx23x6fx52x2bx5dx22It?ox15x5bx5ax1f x74x15x5bx21x22x68x6fx62 x26x14It?ox2ex5ax1axex62x8x3ex22x3dx4cx6cx6dx5ex9x15x5bx6dx2x55 x31x5ax1x2bx74x76x1x74x22x5cx45x1ax21x62x32(Bt)xax8x10Brownx5x3ax5a x13x58x26xf x12x13x8x3ex22x3dx4cx6cx6dx5ex9x15x5bx2b f(t)2 = f(0)2 +2Rt0 f(s)df(s), x16x7ex12x7d xdB2t = 2Rt0 BsdBs. x6cx1fxax6dx63x76x22x62x44x52x3ax37xax6ex62x4axax17x52x6cx61 x72 x48x37x22xax17xat. x3bx7bx22x5ex9x15x5bx60x33x0It?ox2dx2ex67x72x2b x3x410.1 x8(Xt)06t6T x20It?ox1ax13x18 Xt = X0 + Z t 0 Ksds+ Z t 0 HsdBs; f(￠)x20x3bx78x38x39x4ax2cxax14x33x37x5d f(Xt) = f(X0)+ Z t 0 f0(Xs)dXs + 12 Z t 0 f00(Xs)dhX;Xis; x24x1a hX;Xit = Z t 0 H2sds; Z t 0 f0(Xs)dXs = Z t 0 f0(Xs)Ksds+ Z t 0 f0(Xs)HsdBs: 237 x1dx16t 7! f(t;x)x29x78x38x39x4ax2cx33 x 7! f(t;x)x3bx78x38x39x4ax2cx33x37x5dx68x29 x50x6bx3a f(t;Xt) = f(0;X0)+ Z t 0 f0s(s;Xs)ds+ Z t 0 f0x(s;Xs)dXs+12 Z t 0 f00xx(s;Xs)dhX;Xis: x68x6fx6dx73x1fx79x67x72x1fx2bx15x5bx22x1dx68x62x6cxax70x5ax5bx68x1ax21x62x68x6fx73 x68x2bx4ex48x1x50x72x13x14x1ax3ex35x1fx2bx15x5bx5a x4ax68x22x18x3fx43xax2bx57x22x5a x26x14x5ex5dx40x41x2ex5ax62x26x45x4fx4x68x69x22x61 xdx18x3fx5a 10.3 x43x73x68x6fx1ax18x3fx1x37x22Black–Scholesx48x49x5ax73x1fx74x48x49x23x62 x20x59x5dx5dxax18x19x22x62 x6ax18xdx3bx3bx1dx1ex5ax3cx2bx3bx1dx1exax2fx72x73x1dx1ex62x4f x1bx14x8x7x5dx5ex5ax4ax49x4ax14x20x59tx22x3dx52 S0t x65x66x33x0(x3f) xax51x32x5ax2b dS0t = rS0t dt; x4ax23 r xax1cx67x20x20x8x7x1dx23x62x31x32xfxax3bx3fx6dx5ax6dx12x5b S00 = 1, x16x7e S0t = ert. x66x2bx3bx1dx1exaxdx72x73x1dx1ex5a xfx22x3dx52x65x66x33x0x40x41xax51x32 x5ax2b dSt = St(?dt+ dBt); x4ax23?x53 xax3fx6dx62 (Bt)xax8x10Brownx5x3ax5ax68x6fx79x38x69x54x30x0x73xd x30x20x59x5cx59 [0;T] x68x1ax79x7ax62x4ax23 T xa (x65x5b) x41x57x78x41x22x20x62x5ax3bx13 x68x69x22x18x3fx23x61x1cx62x1fx2bx32x5ax22x35x63x5fx60x52 St = S0 exp ?t? 2 2 t+ Bt ? ; x4ax23S0x52x7dx1dx1ex22x70x6ax3dx52x5ax1fx17x62 x26x14x3bx3cx22t, Stxax26x6dx3bx36x51x7e x40x41x5dx5dx62x79x62 logSt xax3bx36x51x7ex40x41x5dx5dx5a x30x7bxex69xex62 (logSt) xa Brownx5x3a (x6dx2bx3cxax8x10x22)x5a x26x1fx17x22x7ex7fx48x49x62x68x6fx62x17x63x4ax3cx52x51x52` = (`t)06t6T = ((H0t ;Ht)); xfxax2bx74x56x57x22 (x26 Brown x5x3a (Bt) x2ax55x22 -x77xbFt) x22x58x12x2ex5ax62 x4ax23H0t x53Ht x51x20x5fx60x73x20x62 tx22x26x3bx3bx1dx1ex22x61x45x5dx5a xfx61x5ax55x22 x1dx1ex4bx4cx73x20x62 tx22x3dx17x52 Vt(`) = H0t S0t +HtSt: x70x7`x52x55x41x1ax22x70x71x43x73x5dx52x2b dVt(`) = H0t dS0t +HtdSt: 238 x6cxax52x6ax33x1fx74x51x5bxdx7bx52x62x4ax24(Vt(`))x55x52It?ox2ex5ax62 x54x31x32 Z T 0 jH0tjdt < +1 a:s:; Z T 0 H2t dt < +1 a:s:: x1fx17x62x6dx51 Z T 0 H0t dS0t = Z T 0 H0t rertdt x53x40x41x6dx51 Z T 0 HtdSt = Z T 0 (HtSt?)dt+ Z T 0 HtStdBt x43xdx3cx52x5ax44x71x62 x70x7` = ((H0t ;Ht))x52x40x0x46x22x7bxex3cx52x12x7dxax33x0 x3bx74x70x71x2b 1. RT0 jH0tjdt+RT0 H2t dt < +1 a:s:, 2. x26x14x12x14 t 2 [0;T],xd H0t S0t +HtSt = H00S00 +H0S0 + Z t 0 H0udS0u + Z t 0 HudSu a:s:: x67x4ax1fx2bx17x62x68x6fx4a ?St = e?rtSt x5fx60x72x73x1dx1ex22x42x43x3dx52x5a x16x7e x68x6fx62x17xdx33x0x57x54x2b x58x5c10.5 x8` = ((H0t ;Ht))06t6T xfx3bx34x63x4ex1ax13x33x3ax62x69RT0 jH0tjdt+R T 0 H 2 t dt < +1 a:s:. Vt(`) = H 0 t S 0 t + HtSt, ?Vt(`) = e ?rtVt(`). x37x5d`xf x46x60x25x59x5ax6ex3ax35x6e 8t 2 [0;T]; ?Vt(`) = V0(`)+ Z t 0 Hud?Su a:s:: x68x6fx48x1x6bx52x62 (?St) = (e?rtSt)x2bx9x6dx2bx3cxax6ex5a x13x58x79x7ax33x1ex48 x49x23x22x8x6ax62x68x6fx22x76x8x79xax5ex72x2fx3bx51x3dx7dx23x71x78x62 x33x64(?St)xax6ex5a x14xax60x71xex63x26x41x57x4ax24x2bx9x22x13x2ax1dx1ex1ax3cx3dx5a x26x14x2bx9x22x63x71x58 x59x62 x3bx3bx7dx23x71x78x51x3dxax50xfx6fx50x4fx26x26x32x2dx26x18x19x62 x2bx72x63x4axex62 x26 x1fx3bx3bx7dx23x71x78x1axex62 xfx6fx22x6cx7dx23x2cx71xax2bx17x22x5ax72x5dx38 (?St)x1fx17x22 x5ax5bx22x58x12x2ex5ax5dx48x55x2bx74Brownx5x3ax62 x54x5dx12x13x33x0x2bx5dx22Girsanov x3cx3ex2b 239 x3x410.2 x8( t)06t6T xfx62x69RT0 2sds < 1 a:s:x3x63x4ex1ax13x33x3ax1d x9x36x33x3(Lt)06t6T x20x75x18 Lt = exp ? Z t 0 sdBs ? 12 Z t 0 2sds ? : x37x5dx6dx51x60LT x3x64x1cx7bx78x30x36x33x3x64x1cx77x78P(L) x42x6cx33x36x33xfWt = Bt +Rt0 sdsx3x1ax13(Wt)06t6T x20x28x29Brownx75x2cx27 x60x14 d?St = ?re?rtStdt+e?rtdSt = ?St ((??r)dt+ dBt); x68x6fx63x73x1fx74x3cx3ex23x5b t = (??r)= , x16x7ex4fx26x14x1fx74 x22(Lt)x14x1 xax6ex5ax70x72x62 x7d Wt = Bt +(??r)t= , x16x7exfxex26x2fx74x51x3dx7dx23x71x78P? x52x8x10Brownx5x3ax62x5cx6a d?St = ?St dWt; ?St = ?S0 exp( Wt ? 2t=2); x79(?St) x26 P? x6exax6ex5a xdx6ax1fx74x51x3dx7dx23x71x78x4ax6ex62x68x6fxex63x26 (x65x5b)x41x57(x4dx3cx57x22) x3cx3dx69x54x67x72x7dx12x5ax1fx79x41x57x3cx52x52x2x44x22FT-x63x71x22x40x41x5dx5dx62x5cx6a x22x5dx5ex26x68x69x3cx52x22x7dx23P?x74x32x63x6dx5ax19x44x73x14x73x1fx17x22x70x71x33x62x63 x4ax1dx68x62 x53x74x1fx17x22x41x57h x43x63x4ax13x2fx74x40x0x46x70x7`x1ax8x78x5ax14xa hx73x20x62 tx22x3cx3dxex63x3ex2e Vt = E?￡e?r(T?t)hjFt? x1ax67x72x5ax16x4axax62 x13x58h = f(ST), x16x7e Vt = E?￡e?r(T?t)f(ST)jFt? = E? h e?r(T?t)f ? Ster(T?t)e (WT?Wt)?( 2=2)(T?t) iflfl flFt · : x44x52St xaF-x63x71x22x62 x6ax73x7dx23x71x78P? x33x62 WT ?Wt x26 Ft x45x31x62x3a Vt = F(t;St),x68x6fx64x78 F(t;x) = E? h e?r(T?t)f ? xer(T?t)e (WT?Wt)?( 2=2)(T?t) ·i = e ?r(T?t) p2… Z +1 ?1 f ? xe(r?( 2=2))(T?t)+ y pT?t· e?y 2 2 dy: 240 x26x14x65x5bx59xcx41x57x62 f(x) = (x?K)+,x60x71x5bxex76x44x6x64x78x4ax3dx52 c(t;x) x52x2b c(t;x) = xN(d1)?Ke?r(T?t)N(d2); x4ax23 d1 = log( x K)+(r + 2 2 )(T ?t) pT ?t ; d2 = d1 ? pT ?t: N(d) = 1p2… Z d ?1 e?y2=2dy: x72x26x14x65x5bx2ex72x41x57x62 f(x) = (K ? x)+, x60x71x63x64x78xfx22x3dx52 p(t;x) x52x2b p(t;x) = Ke?r(T?t)N(?d2)?xN(?d1): xfx6fxexax1x37x22Black–Scholesx15x5bx5a x4ax68x68x6fx3ex2ex67x33x1ex48x49x22x6ax6bx9x11x62x9x64x6ax1x37x22Black–Scholes x15x5bx5a x13x58x54x72x26x40x41x51x3dx1ax1bx7cx7dx62x1fx79x22x6dx3x4fx9x6bx5bxcxdx5ax1ex5a x6cxax26x1ax1fx2bx4ex23x22x18x3fx62x68x6fx63x4ax4x43x62 x13x58x5dx52x31x3ax12x22x18x19x48 x49x22x3cx3dx3ex3fx62x68x6fx22x54x5dx52x31x2fx1cx1dx3fx63x2ax26x7ex7fx3f x10x3x7ex7fx51x2b x3ax0x7dx3fx62x4ax4bxdx39x22x2bx3ax0x3cx3ex62x46x76x33x1fx79x22x1x2x8x9x64x78x1ax1b x10x11x22x1bx1cx5a x71x10x62 Black–Scholesx15x5bx22x63x12x13x14x2dx37x4ax7cx6dx62x14x65x5bx47x48x41 x57x22x7cx5ax5ax4ax23xdx2bx40x7cx5ax43x6dx54x5dx79x3cx15x5bx5ax5cx13x62 x26x14x6bx41x6cx35 x46x1ex22x75x41x41x57x62x63x4ax19x77x6dx3ax69x12x7ex19x1ax22x15x5bx62 x18xax38x4ax23x22x47 x48x3dx52x3cx52x46x1ex3dx52x5ax6cxaxdx40x7cx5ax54x5dx26x19x15x5bx75x42x79x3cx5ax5cx13x62 x13 x58x47x48xaxdx51x23x22x62x5cx6ax31x3cx4ax51x23x4ax18x19x51x23x23 d x1ax66x7x62 x16x7ex4f x12x22Black–Scholesx15x5bx12x7dx38x51x23x22x3dx17x53x6dx73x25x5ax1fx20x15x5bx5dx52x2b c(t;x) = xe?d(T?t)N(d1)?Ke?r(T?t)N(d2); x4ax23 d1 = log( xe?d(T?t) K )+(r + 2 2 )(T ?t) pT ?t ; d2 = d1 ? pT ?t: p(t;x) = Ke?r(T?t)N(?d2)?xe?d(T?t)N(?d1): x1fx2bx15x5bx6ex1x3fx13x14x50x6dx41x57x7cx5ax5ax1fx20x62x63x38x50x6dx21x42x2bx74xdx18x19 x51x23x22x47x48x5ax22x63x4ax13x14x10x57x41x57x22x7cx5ax5ax1fx20x62 dxax10x57x1dx23x62 xxa x57x23x5a x13x71x51x51x5a 241 10.4 x1x72x62x73x1x37x22Black–Scholesx22x19x6ax3fx7fx23x62x7cx6fx2cx22x2xa x66x2bx70x6ax5ax4ax23x30x37x22xax1dx13x6ax15xax51x32x5ax42x52x2ex4cx5a x7cx6fx22x7ax11xax36x52 (x65x5bx59xc) x41x57x3dx52x9x72x2bx74x40x41xax51x32x5ax5a x2cx27x68x62x60It?ox15x5bx62x31x79x63x64 dc(t;St) = ct(t;St)+cx(t;St)?St + 12cxx(t;St) 2St ? dt +cx(t;St) StdB(t); c(T;ST) = (ST ?K)+: x7ax7bx78dSt x22x3cx52x62 xfx6ex63x5bx55 dc(t;St) = cx(t;St)dSt +ct(t;St)dt+ 12cxx(t;St) 2S2t dt; c(T;ST) = (ST ?K)+: x1fxax74x70x8x37x20x59T x72x4x1ax5ex72 T x1fx69x22x20x62 tx22x61x13x1cx2fx5ex40x41xa x51x32x5ax20x5ax6dx2ex1fx17x22x2fx5ex32x5ax2fx11x5ex35x62x44x52x68x6fx22xcxdx38xfx67x2f x72x73x1dx1ex4fxcx3ax62x6dx76x44x6xfx22x42x43x3dx17x5ax4ax25x62 Blackx53Scholesx70 x53x6dx41x57xdx5bx25x72x73x22x42x13x72x4x72x39x52x62x73x20x62 tx4ax3dx52x52St x2ex72x2b x56x47x48x62 x3cx53x62x20x59x6e 1=cx(t;St) x56x41x57x1ax4cx17x5ax1fxax44x52x1bx47x3dx5dx5c dSt x20x62x41x57x3dx52xex36x6bx69x5dx5c cx(t;St)dSt. x3bx39x5dx5cx3bx31x4fx7fx5ax2ex72 cx(t;x(t)) x56x47x48x62 x59x6ex2bx56x41x57x22x7cx4bx6ex2bx17x5ax14xax20x3dx52 fl(t) = cx(t;St)St ?c(t;St) x4ax3dx52x5dx5cx52 dflt = cx(t;St)dSt ?dc(t;St) x22x1dx1exax2bx39x2fx72x73x1dx1ex62x70x72xfx12x65x66x3fxax51x32x5a dflt = fltrdt: x60x71x2x38 St x1dx58x52x,xex63x9x64x62 ct(t;x)+ 12cxx(t;x) 2x2 = ?rcx(t;x)x+rc(t;x): x1fxax2bx74x4atx53xx52x5dx5dx22x15xax51x32x5ax62x5cx6ax63x4ax1x2ex5dx48x6ex5cx52x7c x6bx22x53x1ex32x5ax5a x2x67x68x8x37x70x71c(T;x) = (x?K)+, xfxex63x4ax5ex35x62x5c x6ax62x17x64x78x68x69x22Black–Scholesx15x5bx5a x13x58x68x6fx13u = u(t;x)x1ax1dx58c(t;x)x42x52x2bx9x22x65x5bx47x48x13x2ax1dx1e x22x3cx3dx6cx6dx62x5cx6ax38x15x9x6dx3ax3bx5bx64x30x68x7bx62 x16x7ex68x6fx63x4ax4x43x62x33 242 x0x61x13Black–Scholes xdx4bx26x14x12x14x65x5bx47x48x13x2ax1dx1ex22x3cx3dx43xax55 x31x22x2b @u(t;x) @t + 2x2 2 @2u(t;x) @x2 +rx @u(t;x) @x ?ru(t;x) = 0: x61x6dx62x22x2cx2cxax8x37x70x71x5ax60x71x62x68x6fxex63x4ax13x15xax51x32x5ax32x11x5ex64 x65x3bx65x5bx47x48x41x57x22x3cx3d(x4dx21x68x69)x5a x7ex6ax65x5bx41x57x4ax10x62x22xdx2dx37x4ax7cx5ax5bx22x41x57x5ax4ax23x76x3fx6ax22xa x76x5bx41x57x5ax61x13x76x5bx41x57xax78x41x2cx4ax1fx43x63x4ax66x7x22x41x57x5ax76x5bx41x57 x22x3cx3dx1ax65x5bx41x57x22x3cx3dx5dx3dx51x64x37x5ax6cxax19x3exax2bx17x22x5ax70x15xax51 x32x5ax22x77x78x1axex62 xfx61x12x7dx65x66x22x32x5ax39x1xa Black–Scholes x32x5ax62x6c xax8x37x70x71x6dx2x2bx17x5ax60x14xfx5cx6dx2bx3cx78x41x66x7x62x70x72x4ax66x7x20x59x6e xax54x5dx7bx3cx22x5ax1fx3bx6ax49x22x15xax51x32x5ax69x54x49x52x40x60x59x15x69x54x62x4ax5e x35x32x11x1ax1bx3dx51x5a x1x72x62x1fx2bx4ex23x46x72x22x56x9x33x32x11x39x1x73x1fx3bx7cx4b x33x63x64x19x12x13x5a x76x1fx62x56x9x33x32x11x67x15xax51x32x5ax22x44x6x32x11x38x59x22x39 x3ax62x6exax67x71xdx39x22x2bx74x66x78x2bx2ax22x79x7ax49x54x5a Black–Scholes x41x57x3cx3dx48x49x18xax18x19x20x59x7fx0x3x23x22x2bx74x37x49x3f x54x5a x2cx27x68x62x68x6fx1fx69x18x3fx2ex22x61xdx3fx54(x46x47x3cx3dx27x28x3cx3ex3f Modigliani– Miller x3cx3ex3fx1dx1ex4bx4cx4dx4ex3f CAPMx3fx2bx9x1x2xaxbx51x51) x43x63x4axdx18 x19x20x59x22x29x2ax5axdx50x51x22x54x72x63x4dx0 Merton x22x19x5 [15]x5ax33x69x62x68x6f x79x4ax62x17x22x8x6ax1ax13x18x19x20x59x48x49x1ax18x3fx1dx23x22x41x30x59x11x5a 10.5 xdx78x43x73x52x10x62x68x6fx18x3fx23x61x1x4x78x22x1dx23x42x52x2fx72x73(x1b) x21x22x23x62 xfx73x48x49x23xax74x6dx5dx22x3fx6dx5ax1dx23xax42x52x2fx3bx3ax3bx44x5dx27x10 x1ax1excx22x5ax60x14x41x57x3cx3dx51x18x3fx61x4x4bx22x20x59x24x4fx26x1bx75 (x2bx9x24x37 x5ax74x5f)x62 x78x38x1dx23x42x52x3fx6dx37x3ex69x54x6dx65x5ax6cxax43x27x34x15x23x22x8x7x1d x23x3fx46x1ex1dx23xax40x20x59x5dx5cx72x5dx5cx22x5ax44x71x62x73x53x6dx20x59x1bx6bx22x7fx0x47 x46x3cx3dx69x54xex6dx76x2x4ax2bx39x6dx5dx22x2fx72x73x21x22x23x1ax42x27x10x5ax61x13x27x6a x43x1ax2x34x35 (term structure of interest rates)xexax50x1dx23x67x41x30x22x39x3ax5a xex13x68x6fx3ex3fx61x3ex35x22x16x17x62x2bx9x6bx41x1dx23x5dx1ax75x41x1dx23x5dx4c (x73x2f x40x13x3fx7cx4bx33x6ex5ex4fx13)x5ax1fx3bx1dx23x67x41x30x22x39x3ax3ex2ex8x9x14x15 (yield curve) y = y(s)x1ax65x1cx62x5cx13x62x73x76x7cx62 x53x34x43x5dx15x7ex7cx60x1ex22x21x22xa xfx62x79x62 3x74x5fx3f 6x74x5fx3f 1x63x3f 2x63xdx2430x63x22x7cx60x1ex1dx23x17x5a xfxe x1x3fx42x52x65x3bx7fx0x47x46x3cx3dx22x27x10x5ax1fx69x18x3fx22x41x57x3cx3dx69x54x73x19x5f x68x43x63x4fx4x78x1dx23x6exdx40x41x5dx5cx22x7cx5ax5ax6cxax1fx4ex23x62x68x6fx5dx18x3fx22 xax2b x1bx1dx23xdx40x41x5dx5cx20x62x6cx17x1ax52x46x1ex3cx3dx5a x52x6ax3ex35x6cx17x13x21x22xaxfx1ax52x46x1ex3cx3dx62x68x6fx36x1ax18x3fx2bx74x1x74 x22x46x1ex3cx3dx69x54x5ax34x68x46x1ex3bx6ax20x79x3x38x5ax6cxax27x28x68x63x51x52x78x41x3d x28x22x35x53x51x41x3dx28x22x35x3bx3bx5ax1fx17x62x3ex3fx68x62x4x52x22x4x70x5x31 (coupon 243 bond) x63x4ax21x42x76x27x28x22x46x1ex62 xfx22x5ex23xax51x41x3dx28x22x35x2bx72x51x41x3d x35x5ax78x41x3dx28x22x35x22x46x1ex63x4ax21x42xfx22x5ex5cx2b x6x70x5x31 (zero-coupon bond)x5ax27x10x21x22xaxf y = y(s) x23x22x40x5dx5d s x63x21x42x6cx35x46x1ex22x41x30x62 yxax4fx12x22x1dx23x62x4ax23x1dx23x22x74x4fx43x3x42x6x52x13x12x2bx22x74x4fx41x30x21x22x1a x44x6x5ax14xax62 x13x58s = 0;1;:::;T, x16x7e s = 0x20x22x2bx77x57x62x78s = 1x20x62 xfx79x5dx52(1+y(1))x77x62x78s = 2x20x62 xfx79x5dx52(1+y(2))2x77x62 ￠￠￠￠￠￠x62x2b x9x7cx4bx33x62x78sx20x62 xfx79x5dx52(1+y(s))s x77x5ax62x17x62 s = 1x20x22x2bx77x57x62 x79x51x14s = 0x20x221=(1+y(1))x77x62 s = 2x20x22x2bx77x57x62x79x51x14s = 0x20x22 1=(1 + y(2))2 x77x62 ￠￠￠￠￠￠x62x2bx9x7cx4bx33x62 sx20x22x2bx77x57x62x79x51x14s = 0x20 x221=(1+y(s))s x77x5ax1fx4fx1bx14x53x54x5x31 (discount bond,x78x41x3dx48x69x3dx17 x22x46x1e)x22x4x7x3dx44x6x5ax31x3cy(s) x43x6dxax40x41x5dx5dx62 x16x7ex68x6fx6ex63x38xf x21x42x1cx4dx1ax35x36x20x52x20x62 s = 1;2;:::;T x22x40x41x5dx5dx5ax1fx20x62x62x17x73x2fx1c x1dx31x32x33x62 1=(1+y(1)), 1=(1+y(2))2, ::: 1=(1+y(s))s, :::, 1=(1+y(T))T xex43x63x4ax21x42x65x74x1cx35x36x3dx52x20x5ax60x71xex63x26x12x14x46x1ex3cx3dx5ax5cx13x62x2f x3bx3dx35x46x1e ˉx x73 s = T x20x78x41x62 x16x20x46x1ex64xdx72x79x64x78 K x77x62x6cx73 s = 1;2:::;T ?1x77x62x46x1ex64xdx72x22x76x53x25x64x78(x1dx35) Rx77x5a x16x7e xx22 x1bx1fx3dx52x52 p(ˉx) = T?1X s=1 R (1+y(s))s + K (1+y(T))T : x30x2bx9x22x73 T x78x41x22x3dx35x46x1ex63x13 T x57x5ex5d x = (x1;x2;:::;xT) x1ax5f x60x5a x16x7ex73x27x10x21x22xaxfy = y(s)x33x62x4ax1bx1fx3dx52x52 p(x) = TX s=1 xs (1+y(s))s: x68x6fx63x4ax4x43x3fx6ax22x3fx5x23x22x61xdxdx39x46x1ex3fx47x48x3ex2ex1dx23x1ax1x74x3c x3dx22x15x5bx43xax1fx74x15x5bx22x5ex5cx5ax1fx79x68x6fx2x24x22x63x53x6dx7x21x51x44x34x5a x6dx36x21x72x62x1fx2bx15x5bx22x63x4fx4x78y(s)x52xdx30x35x36x22x40x41x5dx5dx22x7cx5ax5a x1dx23x41x30x59x11x22x3ex3fx79x7ax76x1fx2bx9x13x18x19x20x59x48x49x1ax18x3fx5ax31x3c (x18x19x3dx4cx79x20) x1dx23 r(t) x40x20x59 t x5dx5cx5a x16x7ex46x1ex22x3dx17x40x20x59x22x5d x5c V = V(t)x12x7dx65x66x33x0xax51x32x5ax2b dV(t) dt = r(t)V(t); V(0) = V0: x60x71x63x64 V(t) = V0e Rt 0 r(s)ds: x13x58x53x6dx22xax42x43x46x1ex62x48x6bx22xax78x41x17V(T) = VT, x16x7ex79xd V(t) = VTe? RT t r(s)ds: 244 x68x1cx21x22xaxfx3ex3fxax26x74xax1dx23x72x1fx22x5ax3ax60t(< T)x78T x22x74xax1dx23 x52R(t;T),x79 VT = V(t)e(T?t)R(t;T): x16x7ex1ax1bx68x1cx3bx5bx63x64 R(t;T) = 1T ?t Z T t r(s)ds: x7d s = T ?t. x16x7e y = y(s) = R(t;T)xexax1fx69x61xex22x21x22xaxfx5axdx20x68 x6fxex5bVT = 1. x1fx20x3a V(t) = P(t;T) = e? RT t r(s)ds xex63x21x42x42x43x46x1ex73 x20x62 tx22x3dx52x5ax1fx20x79xd y(T ?t) = R(t;T) = ?logP(t;T)T ?t : x1x72x62x1dx23r(t)x2bx9x40x20x59x40x41x5dx5cx5a x3ex3fx31x3cxfxax2bx74Ft-x58x12 x2ex5ax62x5cx65x66x33x0x5ax5bx22x40x41xax51x32x5ax2b dr(t) = u(t;r(t))dt+w(t;r(t))dBt; (7) x4ax23Bt x52x8x10Brownx5x3ax62 ux53w x52(t;r)x22x7bx3cx6cx6dx5ax73x1fx3bx7cx4b x33x62x46x1ex22x3dx17x6dx6cx49x4ax14x20x59x62x22x49x4ax14x1dx23x22x40x41x5dx5cx62x79 V = V(t;r(t)). x52x67x72 V x22x5fx8x5bx62x2bx3bx7ax11xax1dx13x2fx1cx1dx31x32x67x72x2bx74x26 x14x46x1ex7ex7fx22x51x3dx7dx23x6ex71x78E?,x5cx5b R(t;T) = E? ? 1 T ?t Z T t r(s)ds flfl flFt ? : x60x71xex63x9x64V(t;r(t)) = VTe(T?t)R(t;T): x66x2bx3bx7ax11xax1dx13x72x73x26x3x5ax6cxax67x41x57x3cx3dx69x54x44x28x6dx62x38x37 x73x14rx5cx2xaV x22x8x22x46x47x62 V x6ex5cx2x13x1ax26x3rx22x72x73x5ax52x26x3x72 x73x62x68x6fx12x7dx13x3bx3bx6dx62x41x30x22x46x1ex22x58x1bx4bx4cx1ax26x3x72x73x5ax31x32xd x3bx3bx78x41x2cx51x20xaT1 x53T2 x22x46x1ex5ax1fx72x22x3dx52 V1,x6ex72x22x3dx52x52V2. x68x6fx13x13x33x22x4bx4cx1ax1cx41x4cx17x2b Π = V1 ??V2; x4ax23?xax2bx74x58x1bx22x6dx5ax1fx2bx4bx4cx73x20x59dT x23x22x5dx5cx52 dΠ = @V1@t dt+ @V1@r dr + 12w2@ 2V1 @r2 dt?? @V 2 @t dt+ @V2 @r dr + 1 2w 2@ 2V2 @r2 dt ? ; 245 x4ax23x26 rx53tx22x6cx6dx12x13x6aIt?ox15x5bx5ax60x71x63x4ax21x78x5b ? = @V1=@r@V 2=@r ; xex76x5x7edΠx23x22x40x41x44x34x62 x33x64 dΠ = @V 1 @t + 1 2w 2@ 2V1 @r2 ? @V1=@r @V2=@r @V 2 @t + 1 2w 2@ 2V2 @r2 ?? = rΠdt = r V1 ? @V1=@r@V 2=@r V2 ? dt; x4ax23x68x6fx1dx13x6ax2fx72x73x1dx23x52rx22x2cx27x5ax1fx17x62x68x6fx64x78x15xax51x32x5a @V 1 @t + 1 2w 2@ 2V1 @r2 ? @V1=@r @V2=@r @V 2 @t + 1 2w 2@ 2V2 @r2 ?? = r V1 ? @V1=@r@V 2=@r V2 ? : x38x30V1 x22x39x48x73x48x37x62 x38x30V2 x22x39x48x73x3ax37x62x60x71x63x64 @V 1 @t + 1 2w 2@ 2V1 @r2 ?rV1 ? =@V1@r = @V 2 @t + 1 2w 2@ 2V2 @r2 ?rV2 ? =@V2@r : x1fxax2bx74x32x5ax62 x3bx74x4dx6bx6cx6dV1 x53V2. x6cxaxfx6fx51x20x73x3bx37x62x44x71x3b x37x43x12x51x14x67V x2fx39x22x6cx6da(t;r). x52x27x6ex12x13x22x32x31x62x6dx12x3a a(t;r) = w(t;r)?(t;r)?u(t;r); x60x71x79x64x6x70x5x31x3x1dxdx4bx2b @V @t + 1 2w 2@ 2V @r2 +(u??w) @V @r ?rV = 0: (8) x4ax8x37x70x71x52V(r;T) = VT,x6cx59x15x70x71x5dx5bx6fx14ux53wx22x5ax5bx5a x1fx79x22?xdx4ax5ex5fx7bx52x5a x2cx27x68x62x60It?ox15x5bx62 dV = w@V@r dBt + @V @t + 1 2w 2@ 2V @r2 +u @V @r ? dt: x70x72x60(8)x63x64 dV = w@V@r dBt + w?@V@r +rV ? dt; x2b dV ?rVdt = w@V@r (dBt +?dt): 246 x1fx2bx51x5bxex68x62 x6cx35x46x1ex5cx2xax2fx72x73x1dx1ex62x44x52x2fx72x73x1dx1ex22x3ax37 x12x7dxax6cx5ax3ax37x63x4ax35xfx52x47x8x72x73x22x68x4ax21x22x62x72x6cx6d? x3ex3fx49x52x46 x47x43x42x43x1dx4bx62 xfxax6ax6bx14Sharpex1ax16x17x22x5dx5ax1fx63x4ax3ex2ex38x32x5a(8) x67Black–Scholes x32x5ax4fx1ax1bx1ax21x72x5ax73Black–Scholes x32x5ax23x67u??w x4fx1bx22xax2fx72x73x1dx23rx67x47x3dxx22x1dx6dx62 ux4fx1bx14?x, wx4fx1bx14 x. x14 xa?xex4fx1bx14(??r)= . x1fx67x31xexax3cx20x4ex23x46x78x22Sharpex1ax5a x4ax68xax26x2bx9x22x1dx23x40x41xax51x32x5a (7) x1ax18x3fx22x5ax67x41x57x3cx3dx3e x3fx6dx62x22xax62x1dx23x40x41xax51x32x5ax2bx9x43x6dxax1cx3fx21x22x3f x3fx74x3ax23x20x16x17 x22x5ax5bx5ax7fx22x23x79x7ax1bx34xcx22x2bx40x1dx23x40x41xax51x32x5ax63x73 (7) x22x3ax3b x33x62x12x2bx3ax55x33x0x5ax5bx2b w(t;r) = p fi(t)r?fl(t) (9) u(t;r) = ? ? (t)r +–(t)+?(t;r) p fi(t)r?fl(t) · : (10) x1b fi = 0 x6ax61xdx4dx6dx43xax3fx6dx20x62x63x64 dr(t) = a(b?r(t))dt+ dBt; x4ax23a;b; x43xax3fx6dx5ax1fxaVasicek [15]x141977x63x46x72x22x48x49x5ax1fx2bx32 x5ax73xcxdBrwonx5x3ax45x4bx20x62x4ax35x52r(t) = r(0)e?at + b(1?e?at),x79x40 x2ax20x59x7cx5ex14x2fx30x65x62x1dx23x7cx5ex14x3fx6d b. x72 a x63x3ex35x52x1dx23x22x75x4e x38x78x5ax73xdBrownx5x3ax45x4bx20x62x65x66x1fx17x22x40x41xax51x32x5ax22x40x41x2ex5a xax61x13Ornstein–Uhlenbackx2ex5ax22x2bx74x5dx5ax5a xfx63x4ax5bx55x40x41x6dx51x22x5a x5bx2b r(t) = r(0)e?at +b(1?e?at)+ e?at Z t 0 easdBt: x60x71x63x64x32x76Var[r(t)] = 2(1?e?2at)=(2a). x1b fl = 0 x6ax61xdx4dx6dx43xax3fx6dx20x62x63x64 dr(t) = a(b?r(t))dt+ p r(t)dBt: xfx67 Vasicek x48x49x22x5cx20x2cx73x14x37x6a pr(t) x39x5ax1fxa Cox, Ingersoll x53 Ross [3] x14 1985 x63x46x72x22x5ax1fx74x48x49x5ax5bxax76x1fx76xdx6x22x1dx23x41x30x59 x11x22x48x49x5a x3fx49 CIRx48x49x5a x67xc pr(t)x22x76x22x73x14x5x7eVasicekx48x49x23 r(t)xdx63x76x5bx44x17x22x6dx4cx3ex7cx5ax5a x26x14x1fx3bx3bx48x49x62xdx63x76x38 R(t;T)x22x5fx8x5bx5bx72x1ax5a x2cx27x68x62x32 R(t;T) = 1T ?t[B(t;T)r(t)?A(t;T)]: 247 x16x7ex63x4ax5ex64x62 x26x14Vasicekx48x49x62 B(t;T) = 1?e ?a(T?t) a ; A(t;T) = (B(t;T)?tT)(a 2b? 2=2) a2 ? 2B(t;T)2 4a : x72x26x14Cox–Ingersoll–Rossx48x49x62 B(t;T) = 2(e (T?t) ?1) ( +a)(e (T?t) ?1)+2 ; A(t;T) = 2ab 2 log ? 2 e(a+ )(T?t)=2 ( +a)(e (T?t) ?1)+2 ? ; x4ax23 = pa2 +2 2. x68x1cx3bx3bx48x49x22x3ax6dx43x49x4ax14tx22x7cx5axaHullx67White [9]x731990 x63x52x26x22x5ax1fx20x79x64 dr(t) = a(t)(b(t)?r(t))dt+ (t)dBt; dr(t) = a(t)(b(t)?r(t))dt+ (t) p r(t)dBt: x26x14xfx6fx6ex76x3ex2ex6ax6bx22x32x11x64x78x4fx12x22x5fx8x5bx5ax24x14x2bx9x22x18x3fx63 x4dx21Duffie [5]x5a x66x2bx6ax1dx23x41x30x59x11x48x49xax70x9x1ax27x6ax72x4x22x5ax73x33x1ex7cx5ax33x62 x20x62 tx22x0x41x1dx23f(t)x3cx52x52 1+f(t) = (1+y(t+1)) t+1 (1+y(t))t : x1fx2bx3cx52x22x36x6bx18x19x5fx8x5bx52 e?Tf(t;T) = e RT+?T t r(s)ds eRTt r(s)ds ; x70x72x73x18x19x7cx5ax33x62x20x62 tx22x0x41x1dx23f(t;T)x3cx52x52 f(t;T) = lim ?T!0 Z T+?T t r(s)ds? Z T t r(s)ds ?T = ? @logP(t;T) @T : xfx67x79x20x1dx23r(t)x22x39x3ax52r(t) = f(t;t). x6cxax1fx43xax73x2x40x41x22x7cx5a x33x5ax73x40x41x7cx5ax33x62x68x69x22x2bx40x48x49x43xax26 rx46x72x40x41x48x49x5a 1987x63x62 248 Heath, Jarrow x53Morton [6] x52x26x26x0x41x1dx23f(t;T)x46x72x40x41x48x49x62x79 x31x3cx0x41x1dx23x65x66x33x0x5ax5bx22x40x41xax51x32x5ax2b df(t;T) = fi(t;T)dt+ (t;T)dBt; x4ax23Brownx5x3ax39dBt x22x63x31x3cx52x37x57x22x5ax1fx2bx48x49x22x31x37x73x14x62x40 x2aT x22x6dx62x63x64x78x6dx62x22x59x58x5ax60x71x64x72x22x21x22xaxfx1ax74x4fx70x62x2bx74 x79x20x1dx23 r x22x40x41x48x49x64x72x22x21x22xaxfx5dx4cx3ex64x37x5a x1bx1x62x4ax1dx3dxa x6dx3x68x6ex5dx3dx51x64x37x5a x71x10x62x4ax1dx23x42x52x8x22x46x47x22x13x2ax1dx1ex3bx6ax6ex35x37x5ax5cx13x62x46x1e x41x57x62 x55x41(swap, x3bx3bx6dx62x41x30x22x46x1ex22x50x48)x51x51x5a x26xfx6fx22x3cx3dx39 x1x63x4ax13x67x4ax1fx2bx17x22x32x11x5ax6ax6bx22x18x3fx22x1x3fx26x57x23x6ex7x5a x2ax2b x48x40x1cBlack–Scholesxdxex1ex6cx54x6x30x1x3ex3fx361900x56xcBachelierxc x48x1dxex3ax54x7x2ex4cx8x9x22x23x8x9x48x1cx55x27x6cx17x18x6ax54xex7ex2cx37x75x35x6cx3x17x18x55x27x22x23 x48x6xcx73x7dx54x13x32x33x34Black–Scholesxcx8x39xex3a [2]x31Mertonxcx8x39xex3a [15]xf 1973x56x53x5dx39x68x16x1ax55x54 Blackx21 Scholesx2fx40x69x38x2excxex3ax54xex41x41x2ax1fx53x5d(x32 x31x54x7dx4axcx53x5dx62x4ax2Millerx6ex31x76x42x43x3e)x16x1cx0x49x64x38x2exfx21 Mertonxcx6x1fx60 xex54x65Mertonx1fx76x0x21x22x51x6x7xcxfx10x16x71x7dx54x10x1bxex3ax4ex4fx66x55x53x5dx16 x3x17x18x55x27x22x23x48x6x59x74x1ex44x45x50x20x3cx76x40x48x7ax13x16x7fx70x17x7x2ex48x44x7fx6x48x79 x45x18xcx46x40x47x16 Huang–Litzenberger [7] x31 Jarrow [11] x5x0x35x7ex6cx48x5ax19x49x70x59x16 Ingersoll [10] x31 Hull [8] x7cx18x6x56x3excx4cx37x4axexfx59x5axcx40x48x7ax13x16x40x48x6cx4bx12x2b x9xcx55x56x7eDuffie [4], [5]x54xex5x40x48x35x36x2ax78xcx41x1ex4ax31x54x7fx10x7ex6ex3bxax2ax4dx69x41x16 x4bx12x69x41xcx14x54x2cx79x2dx2ex1ex6cx4cx8xexfx2xcx54x7e Baxter–Rennie [1]x16x2dx2excx55x56x7c x3fx4bx12x12x13xf Lamberton–Lapeyre [12]x16x2dx2exbx4ax2dx2excx55x56x4ax4bx12x25x26x12x4bx54xf x2ax4bx40x48x6cxcx16x4cx16 x44x15x17x18x55x27x22x23x48x6ex46x5axcx44x45x50x20x54x71x64x5axcx13x7ex6x42x35x7ex2fx30x4 Brown x3ax47x4bx42x4bx44x45x3bx50x4bx44x45x4fx50x4cx58x4b It?ox36x37x42x16x5x7fx42x2fx30xcx6x7xdx15x2bx2ax26x13 x1x4dx16x7x11x5x7fx42x2fx30x11x6x1dxcxdx15x54x69x1fx2dx4ex17x18x55x27x22x23x48x31x4fx1cx31x42x16x13x16x3d xfx73x10x50x51xdx11x54x5x6x7xcx41x1ex4ax31x2bx2ax59x5ax16 x34x8x9x3fx20x4ax31x54x7fx63x71x29x7excx13x7ex3ax3bx1dx5fx35x7ex1dxdx16x79x52x60xex55x7bx5ax7ex10 xax4cx3ex4x52x52x42x5fx2fx43x42x7x59x31x52x52(x17x18xc)x37x1x2x2x4x28x1bx16x6cx1exdxex6cx48x1fx6 x55x6ex31Girsanovx1dxdx54x15x16x32x18x55x4cx48x1fx73x10x2fx43x3ax39x54x1x48x5aMonte Carlox16x52 x38x39x4cx3ex16x7dx1ex8x0x9x66x67x1ax24x76x15x17x4fx50x4cx58x54x1x4ax2x17x4fx50x4cx58xcx40x58x4cx3ex4a x52x15x16xex7ex6cx6x28x64xcx7ax3cx3dx4cx3ex5xfx17x18x55x27x22x23x48x4ax31x54x37xex7exdxex6cx13x7ex15 x16x38x39x6cx54x5x23xax7ex11x11xaxcx16 x6x43x49x6ex24x21x66x67x1cx17x18x55x27x22x23x48x64x4ex4fx21x53x72x2x4x1dx5fx66x67x6x33x64x5ax16xe x1c Huang–Litzenberger [7] x31 Jarrow [11] x64x4ex4fx77x11x5ex16 Ingersoll [10] x64x23x28x35 x2ax10x16x3x7ex69x62x15x18xc Hull [8] x28x35x2ax22x16x6x43x49x6ex24x21x66x67x7bx5ax7ex5ax76x6x43xcx44 x45x46x6ax21x22x16x17x16x7fx5x36x4x1dx5fx28x50x64x5ax16 249 x5ax5bx5cx5dx5ex5c 1. x1fx7exax40x41x48x2cx5fx1fx7exaBrownx5x3ax5fx1fx7exax16x3ex3fx5f 2. x73Black–Scholesx48x49x23xax6cx17x26x47x48x3dx52x42x72x31x3cx22x5f xfx22x1x2 x7bx52xax1fx7ex5f 3. x1fx7exax40x41xax51x32x5ax5fx1fx7exa It?o x15x5bx5fx6cx17x3ex35x40x41xax51x32x5a x67It?ox15x5bx5f 4. Black–Scholesx48x49x22x72x4x23xax1fx7ex5fx6cx17x26x42x43x3dx52x5ex72x51x3dx7d x23x6ex71x78x5f xfxax1dx13x1fx7ex6dx3x3cx3ex1ax5ex64x22x5f 5. x6cx17x1dx13x47x48x67x41x57x22x2fx72x73x4bx4cx1ax9x72 Black–Scholes x32x5ax5fx6c x17x73x4ax23x1dx13It?ox15x5bx5f 6. x1fx7exax6cx35x46x1ex3cx3dx32x5ax5f xfx67Black–Scholes x32x5axdx1fx7ex4fx62x53 x6dx62x38x37x5f 7. x1fx7exax1dx23x40x41xax51x32x5ax48x49x5f x3fx13x22x48x49xdx3cx40x5f x61x5bx62x63 [1] Baxter, M., and Rennie, 1996, A., Financial Calculus, An Introduction to Derivative Pricing, Cambridge University Press, Cambridge. [2] Black, F., and M. Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy, 81: 637–654. [3] Cox, J. C., Ingersoll, J. E. Jr., and Ross, S. A., 1985, A theory of the term structure of interest rates, Econometrica, 53: 385–408. [4] Duffie, D., 1988, Security Markets: Stochastic Models, Academic Press, Boston. [5] Duffie, D., 1997, Dynamic Asset Pricing Theory, Second ed., Princeton University Press, Princeton, N. J.. [6] Heath, D., R. Jarrow and A. Morton, 1992, Bond pricing and the term structure of interest rates, Economerica, 60: 77–106. [7] Huang, Chi-fu & R. H. Litzenberger, 1988, Foundations for Financial Economics, Prentice–Hall, Englewood Cliffs, N. J.. 250 [8] Hull, J. C., 1999, Options, Futures, and Other Derivatives, 4th ed., Prentice–Hall, N.J.. (x3cx3ax12xdx23xbx28x2bx14x15x62 J., 2000x62x41x57x3fx41 x3ax53x13x2ax1dx1ex62x54x16x17xbx62x18x19x72x12x13). [9] Hull, J. and White, A., 1990, Pricing interest-rate-derivatives securities, Review of Financial Studies, 3: 573–592. [10] Ingersoll, J. E. Jr., 1987, Theory of Financial Decision Making, Rowman & Littlefield Publishers, Maryland. [11] Jarrow, R. A., 1988, Finance Theory, Prentice-Hall, Englewood Cliffs, N. J.. [12] Lamberton, D., et B. Lapeyre, 1991, Introduction au Calcul Stochas- tique Appliqu′e `a la Finance, Ellipses, Paris (x1axbx28x2b Lamberton, D., & B. Lapeyre, 1996, Introduction to Stochastic Applied to Finance, Chapman & Hall, London). [13] Lo, A. W. & A. C. MacKinlay, 1988, Stock market prices do not follow random walks: evidence from a simple specification test, Review of Financial Study, 1: 41–66. [14] Merton, R., 1973, The theory of rational option pricing, Bell Journal of Economics and Management Science, 4: 141–183. [15] Merton, R. C., 1992, Continuous-Time Finance, Rev. ed., Blackwell, Oxford. [16] Vasicek, O. A., 1977, An equilibrium characterization of the term struc- ture, J. Fin. Econ. 5: 177–188.. x0x1x2b Black–Scholesxdx4bx43x72xa Black–Scholes x32x5axax50x33x0x32x5ax2b @u(t;x) @t + 2x2 2 @2u(t;x) @x2 +rx @u(t;x) @x ?ru(t;x) = 0: (11) x65x5bx59xcx41x57x22x3cx3dx69x54xexax26x1fx74x32x5ax73x33x0x8x37x70x71x33x5ex35x2b u(T;x) = (x?K)+ = max[0;x?K]: (12) 251 (11) x1fx17x6ax49x22x32x5ax49x52x21x59x49x32x5ax5ax2bx9x7cx4bx33x62x1fx3bx32x5ax35 x36x5ex64x35x3dx5fx8x5bx5axbx5x22xax62 x32x5a(11)xdx23x5ex20x62 xfx26x8x37x70x71(12) x22x35x1ex31x63x4ax5ex72x35x3dx5fx8x5bx62 x33x64Black–Scholesx15x5bx64x4ax69x34x5a x76x1x74x22x21x59x49x32x5a(xfx49x52x1cx2dx11x9x32x5ax20x3f x1cx2dx32x5ax20x2b x1cx53x1e x32x5ax20)xdx1fx17x22x5ax5bx2b @y(s;v) @s ? @2y(s;v) @v2 = 0; y(0;v) = y0(v): (13) x1fx74x32x5ax22x35x63x4ax13x6dx51x5fx8x2b y(s;v) = 12p…v Z +1 ?1 y0(v ?w)exp ?w 2 4s ? dw = 12p…v Z +1 ?1 y0(w)exp ?(v ?w) 2 4s ? dw: (14) x26 (11), (12)x22x5ex35x5dx1dx13x1fx2bx35x22x5fx8x5bx5ax52x71x62x32 u(t;x) = f(t)y(s(t);v(t;x)); (15) x4ax23 f(t), s(t), v(t;x) x53 y(s;v) x43xax32x3cx22x4dx6bx6cx6dx62x5cx6ax31x3c y = y(s;v)x65x66x53x1ex32x5a(13)x5ax14xa @u @t = f 0(t)y(s;v)+f(t)@y @ss 0(t)+f(t)@y @v @v @t; (16) @u @x = f(t) @y @v @v @x; (17) @2u @x2 = f(t) " @2y @v2 @v @x ?2 + @y@v @ 2v @x2 # : (18) x38x1fx40x15x9x6dx1dxc Black–Scholes x32x5a(11)x62x68x6fx64x78 0 = f0(t)y(s;v)+f(t) ?@y @ss 0(t)+ @y @v @v @t ? + 2x2 2 f(t) " @2y @v2 @v @x ?2 + @y@v @ 2v @x2 # +rxf(t)@y@v @v@x ?rf(t)y(s;v): (19) x31x32f = f(t)x65x66f0(t) = rf(t); f(T) = 1,x79f(t) = e?r(T?t). x16x7ex68x1c x32x5ax23x22f x43x63x46x72x1ax62 x33x64(19)x63x1x5cx52 @y @v @v @t + @y @ss 0(t)+ 2x2 2 " @2y @v2 @v @x ?2 + @y@v @ 2v @x2 # +rx@y@v @v@x = 0: (20) 252 x2x60y = y(s;v)x65x66(13),x68x6fx2x63x31x32 s0(t) = ?a2 = ? 2x2 2 @v @x ?2 ; (21) x33x64x68x1cx32x5ax2x25x55x3bx39x5ax60x71x2x31x32s(T) = 0,x63x64 s(t) = a2(T ?t): (22) x1fx17x2bx1ax62 x32x5ax54x5cx52vx22x13x33x32x5ax2b 2x2 2 @2v @x2 +rx @v @x + @v @t = 0: (23) x1dx13a2 x22x5fx8x5b (21)x62x68x6fx63x5ex64 @v @x = p2? a x · ; (24) x2b v(t;x) = p2 ?a · log ? x K · +b(t): (25) x1dxc (23)x64x78 2x2 2 ? 1x2 ?p 2 ?a · +rxp2 ?a · 1 x ? +b0(t) = 0; (26) x2b b0(t) = 12a p2?rp2 ?a · : (27) x1fx17x62x31x32b(T) = 0,xex63x64 b(t) = p2 ?a ·? r? 2 2 ? (T ?t) (28) xfx33x64 v(t;x) = a p2 ? log ? x K · + r? 2 2 ? (T ?t) ? (29) x43x73x1dx13u = u(t;x)x22x5fx8x5bx62x63x64 u(x;T) = f(T)y(s(T);v(x;T)) = y ? 0; a p2 log ? x K ·! : (30) 253 xfx3cx53x65x66 u(x;T) = y(0;k) = 8 < : K ? exp k ap2 ? ?1 ? ; x1b k > 0; 0; x4ax7cx7cx5ax62 (31) x4ax23 k = a p2 log ? x K · ; x = K exp k ap2 ? : (32) x12x13x1fx74x70x71x62 x35x22x5fx8x5b (14)x5dx52 y(s;v) = K2p…s Z v ?1 ? exp (v ?w) ap2 ? ?1 ? exp ?w 2 4s ? dw: (33) x7d w = ?qp2sx62 xfx2x5dx52 y(s;v) = Kp2… Z 1 ?v=p2s " exp ?v +qp2s￠ ap2 ?1 # exp ?q 2 2 ? dq: (34) x1dx13x32x5a(22)x53(29),x68x6fx64x78 vp 2s = ap2 ? log ? x K · + r? 2 2 ? (T ?t) ?? 1 ap2pT ?t ? = log ? x K · + r? 2 2 ? (T ?t) pT ?t = d2: (35) x4ax4b ap2 ?? v +qp2s · = ap2 (? ap2 !? log ? x K · + r? 2 2 ? (T ?t) ? +qap2pT ?t ) = log ? x K · + r? 2 2 ? (T ?t)+q pt?T: (36) x38x1fx40x17x1dxc (34)x62x68x6fxd y(s(t);v(t;x)) = Kp2… Z +1 ?d2 exp ? log ? x K · + r? 2 2 ? (T ?t)+q pT ?t? q 2 2 ? dq ? Kp2… Z +1 ?d2 exp ?q 2 2 ? dq: (37) 254 xfx6ex63x4ax3ax52 u(t;x) = f(t)y(s(t);v(t;x)) = xexp h ? 2 2 (T ?t) i p2… Z +1 ?d2 exp(q pT ?t)exp ?q 2 2 ? dq ?Ke ?r(T?t) p2… Z +1 ?d2 exp ?q 2 2 ? dq: (38) x2x42x5dx5dx58x48 p = ?qx62 x16x7exd u(t;x) = xp2… Z d2 ?1 exp ? 2 2 (T ?t)?p pT ?t? p2 2 ? dp?Ke?r(T?t)N(d2): (39) x7d z = p+ pT ?tx62x5f xp 2… Z d2 ?1 exp ? 2 2 (T ?t)?p pT ?t? p2 2 ? dp = xp2… Z d2 ?1 exp ?12 ? p+ pT ?t ·2? dp = xp2… Z d2+ pT?t ?1 exp ?z 2 2 ? dz = xN(d1): (40) x44x71x62 u(t;x) = xN(d1)?Ke?r(T?t)N(d2): (41) x4ax23 d1 = log ? x K · + r + 2 2 ? (T ?t) pT ?t ; (42) d2 = log ? x K · + r? 12 2 ? (T ?t) pT ?t ; (43) N(d) = 1p2… Z d ?1 exp ?12z2 ? dz: (44) x1fx17x62x76x6exex35x64x65x5bx59xcx41x57x22Black–Scholesx15x5bx5a 255