MatricesOperaciones con Matrices en ExcelCaptulo 21 de Simon Benninga (1998) Financial Modeling, MIT PressSuma de Matrices130.101.1330230260A =6-9B =8-33.4C =14-42.4511-150-10117122.331.29.3313.2Transpuesta de una Matriz-13-8-3-13-8-30D =-810-1DT =-810-112-3-111-3-11130123Multiplicacin de MatricesA =2-6B =69-12C =426-48-93-524-69-75120Inversa de una Matriz0.100.020.040.0510.8801-0.8882-0.8100-0.7569A =0.020.200.040.01A-1 =-0.88825.1745-0.44410.06180.040.040.400.10-0.8100-0.44412.7200-0.37840.050.010.100.60-0.75690.0618-0.37841.7918Comprobacin de la Inversa de una Matriz1000AA-1 =-01-00= I-0010-0001Solucin de Sistemas de Ecuaciones Lineales Simultneas3 x1 + 4 x2 + 66 x3 = 641.3- 33 x2 + x3 = -215.342x1 + 3 x2 + 2 x3 = 227.6Ax=b3466x1641.3A =0-331x =x2b =-215.34232x3227.6x=A-1b-0.00080.00210.0239A-1 =0.0005-0.0303-0.00000.01520.0017-0.0011x14.5x =x2=6.8x39.1

PortafolioIntroduccin a los Modelos de PortafolioCaptulo 5 de Simon Benninga (1998) Financial Modeling, MIT PressDos ActivosPrecioRendimiento discreto (%)Rendimiento contnuo (%)MesAccin AAccin BAccin AAccin BAccin AAccin B025.0045.00124.1244.85-0.0352-0.0033-0.0358-0.0033223.3746.88-0.03110.0453-0.03160.0443324.7545.250.0591-0.03480.0574-0.0354426.6250.870.07560.12420.07280.1171526.5058.50-0.00450.1500-0.00450.1398628.0057.250.0566-0.02140.0551-0.0216728.8862.750.03140.09610.03090.0917829.7565.500.03010.04380.02970.0429931.3874.380.05480.13560.05330.12711036.2578.500.15520.05540.14430.05391137.1378.000.0243-0.00640.0240-0.00641236.8878.12-0.00670.0015-0.00680.0015EstadsticosPromedio29.1360.450.03410.04880.03240.0460Varianza22.25167.230.00250.00380.00230.0034Desv. Estndar4.7212.930.05020.06170.04780.0584Covarianza0.000360.00034Correlacin0.117190.12032Media y Varianza del PortafolioRendimiento contnuo (%)MesAccin AAccin BE[rt]1-0.0358-0.0033-0.01962-0.03160.04430.006330.0574-0.03540.011040.07280.11710.09505-0.00450.13980.067660.0551-0.02160.016770.03090.09170.061380.02970.04290.036390.05330.12710.0902100.14430.05390.0991110.0240-0.00640.008812-0.00680.0015-0.0026E[rA] =0.03240.0460= E[rB]E[rp] =0.0392VAR[rA] =0.00230.0034= VAR[rB]VAR[rp] =0.00159SD[rA] =0.04780.0584= SD[rB]SD[rp] =0.0399COV[rA rB] =0.0003Frontera EficienteProporcinVAR[rp]SD[rp]E[rp]0.000.003410.05840.04600.050.003120.05580.04530.100.002850.05340.04460.150.002600.05100.04390.200.002380.04880.04330.250.002190.04680.04260.300.002020.04490.04190.350.001870.04330.04120.400.001760.04190.04050.450.001660.04080.03990.500.001590.03990.03920.550.001550.03940.03850.600.001530.03910.03780.650.001540.03920.03710.700.001570.03960.03650.750.001630.04030.03580.800.001710.04130.03510.850.001820.04260.03440.900.001950.04410.03380.950.002110.04590.03311.000.002290.04780.0324Vector de Rendimiento Esperado y Matriz de Varianza-CovarianzaE(r) =E[rA]=0.0324E[rB]0.0460VAR-COV(r) =VAR[rA]COV[rA,rB]=0.002290.00034COV[rB,rA]VAR[rB]0.000340.00341CORR(r) =CORR[rA,rA]CORR[rA,rB]=1.000000.12032CORR[rB,rA]CORR[rB,rB]0.120321.00000Cuatro ActivosDados0.060.1-0.10.20.2E(r) =0.08VAR-COV(r) =-0.10.30.10.020.100.20.10.40.060.150.20.020.060.5Portafolio 1Portafolio 20.20.20.30.10.40.10.10.6E(r)VAR (r)COV (r)CORR (r)Portafolio 10.20.30.40.10.09100.15800.12480.622238Portafolio 20.20.10.10.60.12000.2546ProporcinVAR[rp]SD[rp]E[rp]0.00.25460.50460.12000.10.23030.47990.11710.20.20920.45740.11420.30.19140.43750.11130.40.17680.42050.10840.50.16560.40690.10550.60.15750.39690.10260.70.15280.39080.09970.80.15120.38890.09680.90.15300.39110.09391.00.15800.39750.09101.10.16630.40780.08811.20.17780.42170.0852

Portafolio

Sigma del PortafolioRendimiento Esperado del PortafolioFrontera Eficiente0.000.050.100.150.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.00

VCV-SIM

Sigma del PortafolioRendimiento Esperado del PortafolioFrontera Eficiente0.00.10.20.30.40.50.60.70.80.91.01.11.20.650.700.750.800.850.900.951.00

Log-NormalMatriz de Varianza-CovarianzaCaptulo 6 de Simon Benninga (1998) Financial Modeling, MIT PressRendimiento Medio anual para seis accionesAMRBSGEHRMOUKSP5001974-0.3505-0.1154-0.4246-0.2107-0.07580.2331-0.264719750.70830.24720.37190.22270.02130.35690.372019760.73290.36650.25500.58150.12760.07810.23841977-0.2034-0.4271-0.0490-0.09380.0712-0.2721-0.071819780.1663-0.0452-0.05730.27510.1372-0.13460.06561979-0.26590.01580.08980.07930.02150.22540.184419800.01240.47510.3350-0.18940.20020.36570.32421981-0.0264-0.2042-0.0275-0.74270.09130.0479-0.049119821.0642-0.14930.6968-0.26150.22430.04560.214119830.19420.36800.31101.86820.20660.26400.2251Promedio0.20320.05310.15010.15290.10250.12100.1238Varianza0.20600.07900.08670.44350.00830.03920.0359Desv. Estndar0.45380.28110.29450.66600.09120.19800.1896Matriz de Exceso de RendimientoAMRBSGEHRMOUK1974-0.5537-0.1686-0.5747-0.3635-0.17840.112119750.50510.19400.22180.0698-0.08120.235919760.52970.31340.10490.42860.0250-0.04291977-0.4066-0.4802-0.1991-0.2466-0.0313-0.3931A =1978-0.0369-0.0984-0.20740.12220.0347-0.25551979-0.4691-0.0374-0.0603-0.0736-0.08100.10441980-0.19080.42200.1849-0.34230.09770.24471981-0.2296-0.2574-0.1777-0.8956-0.0112-0.073119820.8610-0.20240.5467-0.41440.1217-0.07541983-0.00900.31490.16091.71540.10410.1430Transpuesta de la Matriz de Exceso de Rendimiento1974197519761977197819791980198119821983AMR-0.55370.50510.5297-0.4066-0.0369-0.4691-0.1908-0.22960.8610-0.0090BS-0.16860.19400.3134-0.4802-0.0984-0.03740.4220-0.2574-0.20240.3149AT =GE-0.57470.22180.1049-0.1991-0.2074-0.06030.1849-0.17770.54670.1609HR-0.36350.06980.4286-0.24660.1222-0.0736-0.3423-0.8956-0.41441.7154MO-0.1784-0.08120.0250-0.03130.0347-0.08100.0977-0.01120.12170.1041UK0.11210.2359-0.0429-0.3931-0.25550.10440.2447-0.0731-0.07540.1430Matriz de Varianza-Covarianza de RendimientoAMRBSGEHRMOUKAMR0.20600.03750.10770.04930.02080.0059BS0.03750.07900.03550.10280.00890.0406ATA =GE0.10770.03550.08670.04430.01940.0148HR0.04930.10280.04430.44350.01930.0274MO0.02080.00890.01940.01930.0083-0.0015UK0.00590.04060.01480.0274-0.00150.0392Betas del Modelo de ndice Simple (Single-Index Model)Vector Beta1.48201.08401.31071.29910.26220.4939Matriz de Varianza-Covarianza para el Rendimiento de cada activo y el Rendimiento del Portafolio de SP500 (Modelo SIM)BetaBetaBetaBetaBetaBetaAMR1.48201.48201.48201.48201.48201.4820BS1.08401.08401.08401.08401.08401.0840GE1.31071.31071.31071.31071.31071.3107HR1.29911.29911.29911.29911.29911.2991MO0.26220.26220.26220.26220.26220.2622UK0.49390.49390.49390.49390.49390.4939AMRBSGEHRMOUKAMR0.07900.05780.06980.06920.01400.0263BS0.05780.04220.05110.05060.01020.0192GE0.06980.05110.06180.06120.01240.0233HR0.06920.05060.06120.06070.01220.0231MO0.01400.01020.01240.01220.00250.0047UK0.02630.01920.02330.02310.00470.0088Diferencia entre la Matriz de Varianza-Covarianza para el Rendimiento por activo y el Rendimiento del Portafolio de SP500 (Modelo SIM)y la Matriz de Varianza-Covarianza calculada a partir del RendimientoAMRBSGEHRMOUKAMR0.1270-0.02020.0379-0.02000.0069-0.0205BS-0.02020.0368-0.01560.0522-0.00130.0213GE0.0379-0.01560.0250-0.01690.0071-0.0084HR-0.02000.0522-0.01690.38290.00700.0043MO0.0069-0.00130.00710.00700.0058-0.0061UK-0.02050.0213-0.00840.0043-0.00610.0304

Histograma Log-NormalDistribucin LognormalCaptulo 12 de Simon Benninga (1998) Financial Modeling, MIT PressSimulacin de un Proceso Log-Normal para el precio de una accin0.150.300.00435tZPrecio035.0011.7136.1820.7436.7130.7337.2540.5637.6750.5638.106-1.7936.8470.2837.068-0.1536.9891.3137.93101.3638.9411-1.4037.94120.9838.6813-2.3637.0114-0.0636.99150.9337.6716-2.7135.80170.2335.98180.1836.13190.6236.5820-0.3036.40212.0937.8922-0.6737.4323-1.1936.6224-1.0435.9225-0.5435.58260.0835.65271.0936.42280.6836.9129-1.3835.9830-0.6635.5631-2.6633.83320.1233.9333-0.3133.75341.2034.5535-0.8933.9936-0.7733.5137-0.4733.2338-1.0732.59390.1732.7140-0.4732.4541-0.0532.4342-1.3631.6343-0.4831.36440.4231.6345-0.1631.55460.8932.11470.7132.5748-1.2731.8149-0.7231.4050-0.9030.8951-0.2930.73520.5531.0853-1.1430.4354-0.5430.1355-0.5529.84560.6730.2457-1.3529.4958-0.6429.16591.2629.88601.4330.72610.9831.3162-0.3131.1563-0.4630.9064-1.6729.95651.0130.5566-1.5329.69670.8930.22681.5231.1269-1.9530.00700.0230.03711.1030.68722.6032.25731.5333.22740.8833.8075-0.0933.7776-1.0333.13770.8833.7178-0.6733.30790.6433.7380-0.2033.6281-0.4333.36821.3334.2383-0.2134.1284-0.2433.98851.7035.12862.3336.7387-0.6236.33883.0438.5189-0.1538.42900.4538.77911.4839.90920.4940.2993-0.2940.1094-1.4539.04951.4540.15960.1840.31970.4540.6898-0.9140.01990.4140.351000.9841.131010.1941.30102-0.5440.901031.1641.84104-1.0441.04105-1.1440.191061.0341.011070.3641.311080.1341.44109-0.9640.711100.5741.181110.5841.661120.5942.151130.0242.20114-0.7941.59115-0.5341.201161.2042.17117-0.8341.541180.8042.20119-0.2742.01120-0.6441.52121-1.3540.50122-0.2840.311230.5240.731241.3441.81125-0.7141.271260.0741.35127-1.2640.391280.4140.73129-0.7240.20130-0.3439.97131-0.1239.901320.9140.62133-0.0140.64134-1.2839.691351.5640.90136-0.3440.671370.6541.191380.0541.26139-0.8340.641400.0740.72141-0.0840.681421.0441.51143-1.7040.21144-1.2539.29145-0.7538.76146-0.1838.65147-0.8738.041480.8638.69149-0.0438.681500.8839.36151-0.3639.121520.2539.33153-0.1039.28154-1.5538.161552.1539.771560.0839.861570.1039.971581.7241.311591.4242.47160-0.3742.19161-0.8641.53162-1.5540.351631.1041.23164-0.2041.09165-0.1141.03166-0.1940.91167-1.9139.47168-0.1039.43169-1.0838.651700.8239.28171-0.2239.14172-1.0038.43173-0.6038.021740.3338.281750.8138.901761.5840.11177-0.1740.01178-1.8238.68179-1.1137.89180-0.0537.88181-1.3836.93182-1.4235.97183-0.4935.661840.2435.85185-0.7535.361860.5935.781871.7737.03188-0.3536.811891.3837.80190-1.1936.981910.1137.08192-0.0537.061930.9437.75194-0.5337.40195-0.6036.991960.2837.211970.0137.241980.7437.791990.6338.26200-0.0538.24201-1.4137.262022.4639.072030.9639.812040.1539.952050.9840.732060.6541.252071.3942.38208-1.1541.492090.2941.752100.2141.94211-1.6340.69212-0.7140.172131.4541.31214-0.8040.72215-0.4040.442161.6141.722171.2042.712180.1742.87219-0.8242.24220-1.2741.26221-1.2440.322221.1541.23223-0.8140.63224-1.0339.87225-0.8739.24226-1.1638.41227-0.5738.02228-0.5737.63229-3.2635.40230-1.1934.632312.4336.282320.2936.502330.0236.542342.0237.992350.5638.412362.3340.182370.6140.672380.3040.92239-0.1540.83240-0.7840.252410.8840.962420.1241.08243-0.2840.88244-0.0840.852450.9441.61246-1.0440.822470.4841.222480.5441.67249-0.7441.11250-0.3340.88

Histograma Log-Normal

PrecioPeriodoPrecio de la accinSimulacin de trayectoria del precio de una accin

Parmetros Log-NormalHistograma de la Distribucin LognormalCaptulo 12 de Simon Benninga (1998) Financial Modeling, MIT PressElaboracin de un Histograma de Distribucin Log-Normal para el Rendimiento de una accin0.150.30step10.151tZPrecio00.381.30Datos Agrupados10.091.19Intervalo SFrecuencia21.071.60003-1.310.780.15041.521.830.305-0.471.010.4536-0.231.080.6117-0.331.050.755580.131.210.91309-1.410.761.0516610-0.630.961.216111-0.121.121.35151120.271.261.512413-0.331.051.6579140.621.401.85415-1.440.751.953416-0.241.082.11317-0.990.862.25918-0.121.122.44190.061.182.555200.611.392.7221-0.550.982.850221.031.583023-0.141.11241.041.5925-2.000.6426-0.461.01270.541.36280.861.50290.811.4830-1.010.86311.311.72320.771.4633-0.630.96341.962.09351.271.7036-0.081.14371.301.72382.502.46390.541.3740-0.860.90411.781.98421.051.59432.452.4344-2.270.5945-0.980.8646-0.291.0747-0.071.1448-1.420.76490.161.22501.121.6251-0.710.94520.131.2153-1.140.83541.271.7055-0.980.87560.081.1957-1.330.7858-0.031.1559-1.170.8260-0.960.87610.151.2262-0.011.16630.641.4164-0.141.12650.481.34661.571.8667-1.330.78680.561.3769-0.081.1370-0.640.9671-0.540.99720.181.23732.712.62742.052.1575-0.521.00760.861.50770.101.20780.891.5279-0.930.88801.241.69810.051.1882-0.940.8883-1.420.76840.381.3085-1.130.83860.411.3187-0.451.02880.131.21890.851.5090-0.361.0491-0.501.00920.141.21930.961.5594-1.020.8595-0.970.87960.991.5697-0.151.11980.351.2999-1.350.77100-0.670.95101-0.910.881020.031.17103-2.090.621040.401.311051.491.821060.121.211071.071.601081.481.811090.981.56110-1.360.771110.411.311120.061.18113-1.250.801140.251.25115-0.231.08116-0.411.03117-0.770.921181.161.64119-0.331.051200.641.411211.932.071220.231.25123-0.580.98124-1.720.691250.661.421260.141.211271.491.82128-1.310.79129-1.730.691301.852.031310.111.201320.051.18133-1.510.74134-1.570.73135-0.970.871360.201.24137-0.061.141380.321.281392.132.20140-2.170.61141-2.110.62142-1.540.731431.131.631440.511.35145-0.680.951461.291.711470.511.35148-0.351.05149-0.451.02150-0.271.071510.541.371520.371.301530.271.26154-0.271.071551.661.91156-0.441.02157-0.590.97158-0.031.151592.522.471600.781.471610.371.301620.351.291631.351.74164-0.540.99165-0.311.061660.161.221670.111.201681.741.96169-0.451.02170-0.281.07171-0.391.031721.141.64173-0.151.11174-0.690.941751.321.731761.061.60177-0.201.091780.711.44179-3.170.45180-1.460.751810.361.29182-0.900.89183-1.050.851840.201.231850.171.221860.481.34187-0.031.151880.621.401890.271.261901.791.99191-0.600.971920.311.28193-1.890.66194-1.830.671952.262.29196-0.720.941971.411.77198-1.080.841991.461.80200-1.160.82201-0.890.89202-0.411.03203-0.600.97204-1.800.68205-1.740.692060.571.38207-0.610.97208-1.450.75209-0.401.03210-0.011.162110.611.40212-0.511.002130.781.47214-0.650.95215-1.000.862160.121.212171.471.802180.161.222190.871.51220-0.750.932211.201.672220.551.37223-0.301.062240.611.402250.771.46226-0.191.10227-0.371.04228-0.481.00229-0.061.142300.751.45231-0.461.012320.021.17233-0.471.012340.641.412351.641.902360.651.412370.781.472380.181.23239-3.450.41240-1.660.71241-1.140.832421.221.68243-0.411.032440.031.17245-0.341.05246-1.760.682470.321.28248-0.990.862491.161.65250-0.061.142510.511.352520.141.212530.901.52254-0.720.942551.271.70256-0.201.092570.461.332580.421.32259-1.370.77260-1.300.792610.851.502621.491.82263-1.060.84264-0.570.98265-0.021.16266-0.211.09267-0.660.95268-0.980.87269-0.121.122700.111.202711.371.75272-1.220.812731.071.60274-0.081.14275-0.720.942760.121.21277-1.570.732780.381.30279-0.021.15280-1.490.742810.211.24282-0.101.13283-1.170.82284-0.311.062851.251.692860.351.292870.361.302880.301.272891.041.592901.641.902910.791.472921.031.582930.981.562940.891.52295-0.331.05296-0.391.03297-0.131.12298-0.501.002990.351.29300-0.580.983010.601.39302-0.181.10303-0.820.913040.191.233050.731.45306-1.440.76307-0.061.14308-0.261.08309-1.900.66310-1.710.703110.421.323120.611.403130.061.183140.931.533150.411.313161.301.72317-0.511.003181.992.113190.781.473200.251.253210.691.433220.341.293230.021.17324-0.091.133251.141.633260.951.55327-0.431.02328-0.211.093290.461.33330-1.010.86331-0.750.933320.671.42333-1.160.823342.032.143350.051.183360.551.37337-0.111.123380.861.50339-1.540.73340-2.390.573412.122.19342-1.280.793431.091.613440.191.233450.331.283460.591.39347-0.311.063480.821.493490.361.29350-0.191.103510.611.393520.751.46353-0.890.893541.541.843550.471.34356-0.001.163570.771.473581.701.933590.041.17360-0.620.96361-2.100.62362-0.520.993631.791.993641.621.893651.451.80366-1.330.783671.601.883680.741.453690.211.243700.611.403710.281.26372-2.110.62373-0.960.873741.251.693750.821.48376-0.620.963770.571.38378-1.560.73379-0.700.94380-0.331.05381-1.410.76382-1.330.783831.471.81384-1.890.663850.681.43386-0.231.08387-1.290.79388-0.291.07389-2.960.483900.631.403910.891.523922.722.633930.161.22394-0.690.953951.221.68396-1.220.80397-0.021.153980.521.36399-1.060.85400-0.141.114011.481.81402-0.411.03403-0.401.034040.331.284051.701.944061.882.04407-1.100.84408-0.351.054091.601.88410-1.350.77411-0.041.15412-0.201.09413-0.670.954140.451.334150.661.42416-0.760.924171.341.744182.362.36419-1.460.754200.601.394210.271.26422-0.091.134230.151.224240.241.25425-0.461.01426-0.421.03427-1.630.71428-0.560.984290.551.37430-0.221.094311.471.804320.171.224330.551.374340.191.23435-0.890.894360.721.444370.111.204380.191.23439-0.800.914400.571.38441-0.520.99442-1.010.86443-0.840.904440.551.374451.491.82446-0.910.88447-0.051.14448-0.241.08449-1.230.80450-1.270.79451-0.181.10452-0.840.904530.731.454540.651.41455-0.491.004560.661.424570.191.234580.221.244590.341.29460-0.021.15461-0.331.054620.131.21463-1.150.82464-0.301.064650.001.164660.511.35467-1.160.824681.231.684690.511.364700.801.48471-1.290.794721.161.654730.281.26474-0.441.024751.091.614760.831.494771.451.80478-1.070.84479-0.740.934800.721.44481-0.361.044821.431.784830.831.49484-1.130.83485-0.860.904862.102.18487-0.341.054881.041.594890.301.274900.921.534910.271.26492-0.501.00493-2.250.59494-0.441.024950.951.55496-0.221.094970.531.36498-2.090.62499-1.440.765001.051.59501-1.170.825020.371.30503-1.160.82504-1.820.675050.011.175061.121.62507-0.051.14508-1.940.655091.241.69510-0.900.895110.921.535121.431.78513-0.251.08514-0.770.925151.661.915161.201.675171.211.67518-1.280.795190.981.56520-0.131.12521-1.600.72522-0.171.105231.201.665240.031.175250.031.175260.941.545271.141.645280.231.25529-0.720.945300.961.55531-1.060.855320.571.38533-1.020.865341.311.725350.451.335361.681.925371.431.78538-0.610.97539-0.181.105400.771.465410.961.555421.231.68543-1.100.84544-1.040.855450.201.235461.711.94547-0.491.005482.472.44549-0.570.985501.401.775510.271.265520.291.275530.821.495540.141.21555-0.870.905560.291.275570.171.22558-0.700.94559-0.101.135601.311.725610.471.34562-1.110.835630.231.255641.331.73565-0.451.01566-0.491.005670.221.245680.781.47569-3.170.455700.811.48571-1.170.82572-0.051.155730.951.555740.741.455752.262.295761.071.605771.812.00578-0.391.03579-0.770.92580-1.590.725810.641.41582-2.200.605830.721.445840.621.405850.751.45586-0.680.955870.471.34588-1.040.855890.311.28590-0.810.915911.251.69592-1.000.865932.022.13594-1.510.74595-0.780.92596-1.580.725971.111.62598-0.960.875990.541.376000.061.18601-0.191.10602-0.411.036030.121.206041.011.57605-0.770.92606-1.590.72607-0.511.006080.291.276090.261.266100.131.21611-0.401.036120.551.37613-1.880.666140.491.356150.551.376161.671.926170.461.336181.351.746190.271.266201.711.946210.401.316221.371.75623-0.520.996241.431.78625-0.061.14626-0.461.016270.161.226280.171.22629-0.680.956301.892.056311.241.696321.131.636330.591.39634-0.131.126351.051.596361.141.64637-1.200.816380.351.29639-0.331.05640-0.071.14641-0.231.086420.411.326430.371.306440.931.54645-0.011.16646-0.081.14647-1.630.71648-0.910.88649-0.221.09650-0.481.01651-0.520.996521.191.666530.301.276540.011.166550.151.21656-0.471.01657-2.040.636581.261.706590.331.28660-0.391.036610.631.406621.992.116630.791.47664-1.600.726650.771.46666-1.330.78667-0.011.16668-0.560.986691.121.63670-0.351.05671-0.511.00672-0.910.88673-1.500.746740.801.48675-0.061.146761.151.646770.341.29678-0.161.11679-0.021.16680-1.900.66681-0.720.94682-0.041.15683-0.441.026840.301.276850.121.21686-0.680.956870.321.286880.091.19689-0.241.086901.571.86691-0.800.92692-0.760.93693-0.351.05694-0.570.986951.661.91696-0.830.916970.181.23698-1.180.82699-0.670.95700-0.590.97701-0.151.117020.821.49703-1.230.807040.701.43705-0.890.897061.441.79707-0.560.98708-1.430.76709-0.890.897101.001.57711-0.151.117120.311.27713-0.041.15714-1.930.657150.221.24716-0.231.08717-1.300.79718-0.650.967190.611.397200.811.48721-0.171.107220.811.487231.461.807240.401.31725-1.840.677260.781.47727-2.650.537280.731.45729-0.401.03730-0.900.89731-0.221.09732-0.021.16733-0.431.02734-1.220.81735-1.240.80736-1.110.83737-0.431.027381.471.817391.351.74740-0.361.04741-0.860.90742-0.900.89743-0.600.97744-1.040.857450.501.357460.061.187470.291.277480.601.39749-0.540.997500.491.357510.381.307520.911.537531.121.637541.101.62755-0.530.99756-2.160.61757-1.090.847580.931.54759-1.660.717600.281.26761-1.960.657621.541.84763-2.180.60764-0.700.947651.291.717660.791.47767-0.520.997680.071.18769-0.511.00770-0.051.15771-0.770.92772-1.630.717730.511.357740.991.56775-0.161.117760.101.207771.511.83778-0.191.10779-0.011.16780-0.780.927810.101.20782-0.361.047830.371.30784-0.880.897850.201.23786-2.680.52787-0.740.937881.361.757891.351.74790-0.770.927911.031.58792-0.181.107930.501.35794-0.231.09795-0.750.93796-1.040.857970.531.367980.851.507990.181.228000.641.418010.151.218022.512.47803-0.730.938040.051.188050.801.488060.101.20807-1.250.80808-0.261.07809-0.620.978100.471.348110.331.288120.381.308130.281.268140.811.488150.781.47816-0.860.90817-0.111.128180.531.368191.391.768200.411.318210.121.20822-2.240.598230.911.538241.591.878251.211.67826-0.530.998270.221.248280.701.43829-0.481.018300.761.46831-0.800.92832-0.810.918330.111.208340.741.45835-1.280.798361.401.77837-0.371.048380.891.528391.021.588400.541.378410.661.42842-1.260.808430.251.25844-0.381.048450.441.338461.351.74847-0.321.068480.291.278490.491.358500.531.368510.611.408520.671.42853-0.311.06854-0.221.09855-1.310.78856-0.181.10857-0.041.15858-1.300.79859-0.321.068600.781.47861-0.341.05862-1.360.778630.341.298641.151.64865-0.980.87866-1.150.828670.681.42868-0.720.93869-0.101.13870-0.231.08871-0.530.99872-0.241.088730.711.448740.331.288751.481.81876-0.291.06877-0.960.87878-1.080.848791.011.578800.041.18881-0.570.98882-0.710.94883-0.770.92884-1.460.75885-1.110.838861.041.59887-1.160.82888-0.051.14889-1.470.758900.321.28891-0.091.13892-1.080.84893-0.620.96894-0.191.10895-2.670.52896-0.850.90897-0.261.08898-2.290.588992.282.31900-0.840.90901-0.321.069020.961.559030.511.36904-1.790.68905-0.061.14906-1.550.739070.161.229081.812.00909-0.381.04910-0.580.98911-0.630.96912-0.610.979130.991.569140.141.219151.171.659160.961.55917-1.130.839180.011.169190.611.399200.471.34921-2.140.619220.521.36923-0.630.96924-0.930.88925-0.820.919261.141.63927-0.730.939280.831.49929-0.211.099300.411.31931-0.950.879320.881.51933-1.100.849340.121.209350.541.379360.631.409370.581.389380.341.299391.211.67940-0.880.89941-0.530.999420.801.47943-0.101.13944-2.190.609451.151.64946-0.071.149471.161.65948-0.471.01949-0.780.92950-0.840.909510.791.479520.641.41953-0.810.91954-0.321.069550.711.449561.942.08957-1.160.829580.021.179590.251.25960-1.340.789610.211.24962-0.131.129630.931.54964-0.071.14965-0.590.97966-0.341.059670.151.22968-1.130.83969-1.010.86970-1.420.769711.451.809720.731.459730.721.44974-2.690.52975-0.710.94976-1.210.819771.842.02978-0.700.94979-0.630.96980-1.740.69981-0.491.009821.251.69983-0.620.96984-1.640.71985-1.220.81986-0.221.099872.162.229881.031.589890.311.279901.121.63991-0.491.00992-0.051.149931.001.579941.581.869951.251.69996-1.010.86997-0.710.94998-0.411.03999-2.220.6010000.011.16

Parmetros Log-Normal

PrecioPeriodoPrecio de la accinSimulacin de trayectoria del precio de una accin

Black-Scholes

FrecuenciaIntervalo de PrecioFrecuenciaHistograma de FrecuenciaDistribucin Log-Normal

Parmetros de la Distribucin LognormalMesPrecioRendimiento MensualRendimiento Mensual (%)012.00112.900.07237.23212.35-0.0436-4.36312.880.04204.20413.800.06906.90513.20-0.0445-4.45613.550.02622.62712.65-0.0687-6.87814.500.136513.65915.350.05705.701014.85-0.0331-3.311115.250.02662.661215.500.01631.63Promedio mensual =0.02132.13Desviacin estndar mensual =0.05745.74Promedio anual =0.255925.59Desviacin estndar anual =0.198819.88Otros mtodos para calcular los parmetros dela distribucin log-normal:Extrapolacin de los rendimientos futuros mediante Promedio Mvil sobre la historia del rendimiento en el pasadoDeterminacin de la volatilidad implcita mediante la frmula de Black-Scholes

Modelo de Black-ScholesCaptulo 13 de Simon Benninga (1998) Financial Modeling, MIT PressS25Precio de la accinX25Precio de ejercicior6%Tasa de inters libre de riesgoT0.5Periodo de maduracin de la opcin (en aos)Sigma30%Volatilidad de la accind10.2475d20.0354N(d1)0.5977N(d2)0.5141N(-d1)0.4023N(-d2)0.4859Precio call2.47Precio put1.73SXrTSigmad1d2N(d1)N(d2)N(-d1)N(-d2)CallIntrnseco10.0250.060.50.3-4.0719-4.28410.00000.00001.00001.00000.000010.012.5250.060.50.3-3.0200-3.23220.00130.00060.99870.99940.000890.015.0250.060.50.3-2.1606-2.37270.01540.00880.98460.99120.016260.017.5250.060.50.3-1.4339-1.64600.07580.04990.92420.95010.116400.020.0250.060.50.3-0.8044-1.01660.21060.15470.78940.84530.458750.022.5250.060.50.3-0.2492-0.46130.40160.32230.59840.67771.217180.025.0250.060.50.30.24750.03540.59770.51410.40230.48592.470670.027.5250.060.50.30.69680.48470.75700.68600.24300.31404.174282.530.0250.060.50.31.10700.89480.86580.81460.13420.18546.213175.032.5250.060.50.31.48431.27220.93110.89830.06890.10178.467077.535.0250.060.50.31.83361.62150.96660.94750.03340.052510.8440810.0C: Precio de la opcin de compra (call)S: Precio de la accin subyacenteX: Precio de ejercicio de la opcin de compra (call)T: Periodo de ejercicio de la opcin de compra (call)R: Tasa de intersSigma: Desviacin estndar del logaritmo del rendimiento de la accinSupuesto: La accin no paga dividendos antes del periodo de ejercicio T.

CallIntrnsecoPrecio B&S vs. Valor intrnseco

MBD0002CDCC.unknown

MBD001C04C8.unknown

MBD00213701.unknown

MBD002799F2.unknown

MBD001D7177.unknown

MBD0002CDCE.unknown

MBD001AB833.unknown

MBD001AD128.unknown

MBD00039C1D.unknown

MBD0002CDCD.unknown

MBD00010626.unknown

MBD0001FB44.unknown

MBD0002CDCB.unknown

MBD000124FD.unknown

MBD0001BBB0.unknown

MBD00008A49.unknown

MBD00009E89.unknown

MBD00008940.unknown