IV2008#005.nb 2. à Black & Scholes ü S K t T t=t - t s : t : : : : : : ü CHtL = SNHd H+L L - K e -HT-tL NHd H-L L d H L ª H lnhs ê KL + H ÅÅÅÅ 2 s2 L HT - tl LëI s è!!!!!!!!! T - t M à (cashflow) cash flow cf. à cash flow
IV2008#005.nb 2 00 00 C = 00 n 6% 0 00 00 µ H + 0.06L = 06 2 00 µ H + 0.06L 2 = 2.6 00 µ H + 0.06L = 9.02 4 00 µ H + 0.06L 4 = 26.248 5 00 µ H + 0.06L 5 =.82 ª ª 0 00 µ H + 0.06L 0 = 79.085 C n FV C = 00 = 0.06 FV n = C H + L n futue value (pesent value) (discount ate) n C = 00 6% 0 00 00 êh + 0.06L = 94.96 2 00 êh + 0.06L 2 = 88.9996 00 êh + 0.06L = 8.969 4 00 êh + 0.06L 4 = 79.2094 5 00 êh + 0.06L 5 = 74.7258 ª ª 0 00 êh + 0.06L 0 = 55.895 n C PV
IV2008#005.nb PV = C êh + L n à 6 60 6% 0 00 00 0.5 00 µ H + 0.06 ê 2L = 0 00 µ H + 0.06L = 06 00 µ H + 0.06 ê 2L 2 = 06.09.5 00 µ H + 0.06 ê 2L = 09.27 2 00 µ H + 0.06L 2 = 2.6 00 µ H + 0.06 ê 2L 4 = 2.55 ª ª ª 5 00 µ H + 0.06L 5 =.82 00 µ H + 0.06 ê 2L 0 = 4.92 ª ª ª 0 00 µ H + 0.06L 0 = 79.085 00 µ H + 0.06 ê 2L 20 = 80.6 n 00 µ H + 0.06L n 00 µ H + 0.06 ê 2L 2 n n 00 µ H + 0.06 ê 4L 4 n n 00 µ H + 0.06 ê 2L 2 n n 00 µ H + 0.06 ê 65L 65 n ï k FV n = H + ÅÅÅÅ k Lkn k ö ê k m ö H + ÅÅÅÅÅ m Lm öe = 2.7828 FV n = H + ÅÅÅÅ k Lkn = I + ÅÅÅÅÅÅÅ kê Mkn = H + ÅÅÅÅÅ m Lmn = IH + ÅÅÅÅÅ m Lm M n ö e n m = k ê FV n = e n = exphnl 6% 0 00 00 00 00 00 06 06.09 06.6 06.68 00 µ e 0.06µ = 06.84 2 2.6 2.55 2.649 2.76 00 µ e 0.06µ2 = 2.75 ª ª ª ª ª ª 5.82 4.92 4.686 4.885 00 µ e 0.06µ5 = 4.986
IV2008#005.nb 4 ª ª ª ª ª ª 0 79.085 80.6 8.402 8.94 00 µ e 0.06µ0 = 82.22 f HxL = e x f HxL = exphxl 2 exponential.5 0.5 2 4 6 8 0 2 0 C t FV = Ce t C = 2 futue value.5 0.5 2 4 6 8 0 tem HyL t C 0 PV = Ce - t C =
IV2008#005.nb 5 pesent value 0.8 0.6 0.4 0.2 2 4 6 8 0 tem HyL (discount facto) t C 0 PV = Ce - t C = pesent value 0.8 0.6 0.4 0.2 2 4 6 8 0 tem HyL DF n n DF n = H + L -n DF n = e - n = exph - nl à 5.00%
IV2008#005.nb 6 F a = H + L T = H + 0.05L =.05 F c = exphtl = exph0.05 µ L =.0527 F a = H + L T = H + 0.05L H + 0.05 µ ÅÅÅÅÅ 2 L =.06 F c = exphtl = exph0.05 µ H + 0.25LL =.06449 TBTBLIBOR LIBOR LIBOR.2% LIBOR C C H + 0.02 µ ÅÅÅÅÅ 2 L = Ce µ ÅÅÅÅÅÅ 2 C H + 0.02 µ ÅÅÅÅÅ 2 L = C exph µ ÅÅÅÅÅ 2 L H + 0.02 µ ÅÅÅÅÅ 2 L = exph µ ÅÅÅÅÅ 2 L lnh + 0.02 µ ÅÅÅÅÅ 2 L = H µ ÅÅÅÅÅ 2 L ln e lnh + 0.02 µ ÅÅÅÅÅ 2 L = µ ÅÅÅÅÅ 2 = ÅÅÅÅÅ 2 µ lnh + 0.02 µ ÅÅÅÅÅ L = 4lnH + 0.02 µ ÅÅÅÅÅ 2 2 L PV n n C FV n C n FV n = C H + L n = C H + L n PV n = C êh + L n = C H + L -n FV n = Ce n = C exph + nl PV n = C ê e n = Ce - n = C exph - nl à
IV2008#005.nb 7 PV n FV n n C C n ê k PV n = C êh + L n = C H + L -n PV n = H + ÅÅÅÅ kn L- PV k n = Ce - n = C exph - nl FV n = C H + L n = C H + L n FV n = H + ÅÅÅÅ k Lkn FV n = Ce + n = C exph + nl PV n = C êh + L n = C H + L -n FV n = C H + L n = C H + L n ê k PV n = H + ÅÅÅÅ kn L- k FV n = H + ÅÅÅÅ k Lkn PV n = Ce - n = C exph - nl FV n = Ce + n = C exph + nl à ü y = f HxL x = f - HyL f ghyl = f - HyL f - ë f = f ë f - = % 2 2 FHL = H + L 2 2 F % 2 HFL = è!!!! F - 2 Ç 6% 0 µ H + 0.06L = 06 2 µ H + 0.06L 2 =.26 % 2 F = f HL = H + L 2 % t f H, tl = H + L t
IV2008#005.nb 8 y = f HxL x = ghyl % 2 F = f HL = H + L 2 è!!!! F = H + L = è!!!! F - f - HFL = è!!!! F - è!!!!!!!!! FHL =-H + L F > 0 > 0 F = f HL = H + L 2 = è!!!! F - F = f I è!!!! F - M = I + I è!!!! F - MM 2 = I è!!!! F M 2 = F f ë f - HFL = F f - HFL = è!!!! F - F F = H + L 2 f - HFL = è!!!! F - = "############## H + L 2 - = H + L - = f - ë f HL = 2 Ç 6% 0 exph0.06 µ L =.0684 2 exph0.06 µ 2L =.275 % 2 F = f HL = exph µ 2L % t f H, tl = exph µ tl
IV2008#005.nb 9 y = f HxL x = ghyl % 2 F = f HL = exph µ 2L lnhfl = lnhexph µ 2LL = µ 2 = ÅÅÅÅ 2 lnhfl f - HFL = ÅÅÅÅ 2 lnhfl à f HxL = e x e = lim nø H + ÅÅÅÅ n Ln = 2.7828 e 0 = e x e y = e x+y He x L y = e xy e ÅÅÅÅÅ x = e x-y e y à ü y = e x y = log e x e (base) y = log e x = ln x y = e x ln e = ln e x = x ü ln = 0 ln Hx yl = ln x + ln y ln I ÅÅÅÅ x M = ln x - ln y y ln x y = y ln x ln è!!! y x = lnx êy = ÅÅÅÅÅÅÅÅ ln x y
IV2008#005.nb 0 f@x_d := Log@xD; Plot@f@xD, 8x, 0.0, 5<D - 2 4 5-2 - -4 f@x_d := Exp@xD; Plot@f@xD, 8x, Log@0.0D, Log@5D<D 5 4 2-4 - -2 - P t = log t P e ÅÅÅÅÅÅÅÅ P t- = ln ÅÅÅÅÅÅÅÅ t P t- cf. t = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ P t-p t- P t-
IV2008#005.nb lnh + xl º x» x» ` P ln ÅÅÅÅÅÅÅÅ t P t- = lni + ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ P t-p t- P t- M º ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ P t-p t- P t- à EXCEL ü. 0 2. 0 PV n = C êh + L n = C H + L -n PV n = Ce - n = C exph - nl FV n = C H + L n = C H + L n FV n = Ce + n = C exph + nl 2.00% ~ 0.00% % 0 20 ü EXCEL TIPS EXCEL =EXP()