Mathematics of Finance - Essay Example

Geometric Brownian Motion In Share Price Analysis Geometric Brownian Motion is considered to be one of the most useful and easy way of modelling the expected future price of stocks and indices. It is also known as a lognormal process of analysis. Over the span of time, it has become one of the most widely accepted models for forecasting. Also know as Wiener process, it was first applied in physics to describe the motion of a particle that is subject to a large number of small molecular shocks and was called Brownian motion (Hull, 2000). The mathematical description of the process was later developed by Wiener. Gradually, this concept was introduced in financial domain. The model assumes the fact that the stock prices follow a random movement as it happens with molecules. These movements take into account the existing information and also the past experience while determining the stock price in future. Hence, the present value of the share or index plays the most important role in predicting the future price. The Black Sholes Model for pricing options also takes into account the geometric Brownian motion of share prices. This model provides a quite simple analytical evaluation for asymmetric risk. The prime assumption of this model is similar to that of ‘efficient market hypothesis’, which says that Simple models of asset prices assume that (i) The past history is fully reflected in the present price (ii) The present price is affected only by new information about the asset This means that knowing the past history of an asset price movement gives no additional information about its future value that is already contained in its current value. Choice for Study The S&P 500 Index has been chosen for analysis with a perspective that it was one of the most volatile indexes during the sub prime crisis. This also has all the major banks listed which impacted the financial market in a big way. The aim of choosing this index for analysis would prove if the trends in the initial month of 2008 could predict the further downslide that happened in the mid of 2008 and beginning of 2009. Also this analysis aims to check if the slight upside in the later part of 2009 could be predicted through the model. Citigroup has been selected as a choice of study of the share price. Citibank was one of the most badly hit banks in the crisis. As already mentioned above, the aim of this study was to critically evaluate if Geometric Brownian Motion Model, could predict the crashing stock price for Citigroup. Historical Volatility and GBM Model for S&P 500 For S&P 500, the total number of days in the calendar year was taken as 365. The data was collected from 1st of January 2008 to 31st December 2009. The historical volatility was computed with an interval of 50 days. The average volatility for the complete range of data was 38.45. The collected prices , is said to follow a GBM with drift parameter and volatility parameter , if for all non-negative values of t and , the random variable is independent of all prices up to time t, and if in addition, the relative change in the asset price or return, , is a normal random variable with mean and variance . Provided is not too large so that the change in the price could be assumed small, may be approximated by , and we say that the ratio of future price to its present value follows a lognormal probability distribution with mean and variance . The volatility graph for S&P is given in the figure below. A GBM model analysis was done, where the drift factor was considered to be about 10 percent. The distribution of the price is found to be almost normal with a trend of being positively skewed. Actual Price versus GBM Model Price For S&P 500 An analysis was done to check the validity of GBM model versus the actual price movement over the span of entire period. The figure below brings more clarity to the analysis. The above figure shows that there are deviations in both the prices. However, the trend is almost similar, especially during the downturns. A closer look at the prices during the crisis shows that the model at times can exaggerate the prices however is conservative during the sliding prices. At some instances, the downslide was predicted much before. Historical Volatility and GBM Model for Citigroup Inc A similar analysis was done for Citigroup shares, which is a component in the S&P 500 Index. The days in the calendar year was taken as 250 trading days for the ease of calculating the option prices. The data was collected from 1st of January 2008 to 29th march 2010. The historical volatility was computed with an interval of 50 days. The average volatility for the complete range of data was 41.55% The findings were almost similar as that of the Index. The GBM prices are depicted in the figure below. The histogram shows that the prices are also normally distributed. Black Scholes Model for Option Pricing- Citigroup Inc A call was selected which is trading at $ 5 strike price. A similar put was taken for analysis with the same strike price. Both the call and the put are expiring on 16th April 2010. The call price was computed in excel with the following formula in mind. , (2) where, , , and denote the standard normal distribution function: and . In the above, is price of the asset today; T is time to expiration (years); X is the strike price; r is the risk-free interest rate (assumed constant); is the volatility parameter (also constant). The put price was computed with call put parity. The figures found after analysis are as follows. Black-Scholes Option Pricing Model Inputs: Stock Price (S) $4.18 Strike Price (X) $5.00 Volatility () 50.20% Risk-free Rate 5.00% Time to expiration (T) 0.047222222 yrs Dividend Yield 0.00% # of Options (000) 1 Tax Rate 30.00% Output: D1 -1.56585 D2 -1.67494 N(D1) 0.05869 N(D2) 0.04697 Call Price $0.01102 Put Price $0.81923 Value of Call Options (000) $0.01 After-tax Option Value (000) $0.01 Actual Call Price $0.03 Actual Put Price $0.83 Though the call price was trading above the model price, the put price is hovering almost at the expected level. Conclusion Though GBM is considered to be one of the most practical way of predicting prices for the distant time horizon, it may often fail in predicting the exact price. This can be due to the fact that sometimes information takes time to be readily available to all investors. For example, during the sub prime crisis, there were people in the industry who were aware of the upcoming havoc. However, this information was not available to many people, especially the small investors. Hence, the gradual exit of few people from the market might not have brought in a general selling trend for a long time. Hence, it can be concluded that GBM model can be used as a supporting analysis along with other fundamental analysis. Annexure Historical Volatility for S&P 500 Index Date Mid Price Px Change Volatility 1/2/2008 1447.16 1/3/2008 1447.16 0.00% 1/4/2008 1411.63 -2.49% 1/7/2008 1416.18 0.32% 1/8/2008 1390.19 -1.85% 1/9/2008 1409.13 1.35% 1/10/2008 1420.33 0.79% 1/11/2008 1401.02 -1.37% 1/14/2008 1416.25 1.08% 1/15/2008 1380.95 -2.52% 1/16/2008 1373.2 -0.56% 1/17/2008 1333.25 -2.95% 1/18/2008 1325.19 -0.61% 1/22/2008 1310.5 -1.11% 1/23/2008 1338.6 2.12% 1/24/2008 1352.07 1.00% 1/25/2008 1330.61 -1.60% 1/28/2008 1353.96 1.74% 1/29/2008 1362.3 0.61% 1/30/2008 1355.81 -0.48% 1/31/2008 1378.55 1.66% 2/1/2008 1395.42 1.22% 2/4/2008 1380.82 -1.05% 2/5/2008 1336.64 -3.25% 2/6/2008 1326.45 -0.77% 2/7/2008 1336.91 0.79% 2/8/2008 1331.29 -0.42% 2/11/2008 1339.13 0.59% 2/12/2008 1348.86 0.72% 2/13/2008 1367.21 1.35% 2/14/2008 1348.86 -1.35% 2/15/2008 1349.99 0.08% 2/19/2008 1348.78 -0.09% 2/20/2008 1360.03 0.83% 2/21/2008 1342.53 -1.30% 2/22/2008 1353.11 0.78% 2/25/2008 1371.8 1.37% 2/26/2008 1381.29 0.69% 2/27/2008 1380.02 -0.09% 2/28/2008 1367.68 -0.90% 2/29/2008 1330.63 -2.75% 3/3/2008 1331.34 0.05% 3/4/2008 1326.75 -0.35% 3/5/2008 1333.7 0.52% 3/6/2008 1304.34 -2.23% 3/7/2008 1293.37 -0.84% 3/10/2008 1273.37 -1.56% 3/11/2008 1320.65 3.65% 3/12/2008 1308.77 -0.90% 3/13/2008 1315.48 0.51% 3/14/2008 1288.14 -2.10% 27.84% 3/17/2008 1276.6 -0.90% 27.89% 3/18/2008 1330.74 4.15% 29.64% 3/19/2008 1298.42 -2.46% 30.28% 3/20/2008 1329.51 2.37% 30.68% 3/24/2008 1349.88 1.52% 30.74% 3/25/2008 1352.99 0.23% 30.66% 3/26/2008 1341.13 -0.88% 30.53% 3/27/2008 1325.76 -1.15% 30.49% 3/28/2008 1315.22 -0.80% 29.83% 3/31/2008 1322.7 0.57% 29.86% 4/1/2008 1370.18 3.53% 30.34% 4/2/2008 1367.53 -0.19% 30.29% 4/3/2008 1369.31 0.13% 30.11% 4/4/2008 1370.4 0.08% 29.59% 4/7/2008 1372.54 0.16% 29.47% 4/8/2008 1365.54 -0.51% 29.17% 4/9/2008 1354.49 -0.81% 28.88% 4/10/2008 1360.55 0.45% 28.86% 4/11/2008 1332.83 -2.06% 29.37% 4/14/2008 1328.32 -0.34% 29.00% 4/15/2008 1334.43 0.46% 28.82% 4/16/2008 1364.71 2.24% 29.37% 4/17/2008 1365.56 0.06% 27.99% 4/18/2008 1390.33 1.80% 28.29% 4/21/2008 1388.17 -0.16% 28.24% 4/22/2008 1375.94 -0.88% 28.32% 4/23/2008 1379.93 0.29% 28.30% 4/24/2008 1388.82 0.64% 28.28% 4/25/2008 1397.84 0.65% 28.11% 4/28/2008 1396.37 -0.11% 27.85% 4/29/2008 1390.94 -0.39% 27.87% 4/30/2008 1385.59 -0.39% 27.90% 5/1/2008 1409.34 1.70% 28.17% 5/2/2008 1413.9 0.32% 27.93% 5/5/2008 1407.49 -0.45% 27.90% 5/6/2008 1418.26 0.76% 27.74% 5/7/2008 1392.57 -1.83% 28.15% 5/8/2008 1397.68 0.37% 28.16% 5/9/2008 1388.28 -0.67% 28.12% 5/12/2008 1403.58 1.10% 27.19% 5/13/2008 1403.04 -0.04% 27.19% 5/14/2008 1408.66 0.40% 27.18% 5/15/2008 1423.57 1.05% 27.27% 5/16/2008 1425.35 0.12% 26.49% 5/19/2008 1426.63 0.09% 26.34% 5/20/2008 1413.4 -0.93% 26.08% 5/21/2008 1390.71 -1.62% 24.76% 5/22/2008 1394.35 0.26% 24.61% 5/23/2008 1375.93 -1.33% 24.89% 5/27/2008 1385.35 0.68% 24.19% 5/28/2008 1390.84 0.40% 24.03% 5/29/2008 1398.26 0.53% 21.41% 5/30/2008 1400.38 0.15% 20.21% 6/2/2008 1385.67 -1.06% 19.52% 6/3/2008 1377.65 -0.58% 19.19% 6/4/2008 1377.2 -0.03% 19.19% 6/5/2008 1404.05 1.93% 19.68% 6/6/2008 1360.68 -3.14% 21.28% 6/9/2008 1361.76 0.08% 21.16% 6/10/2008 1358.44 -0.24% 21.13% 6/11/2008 1335.49 -1.70% 19.37% 6/12/2008 1339.87 0.33% 19.40% 6/13/2008 1360.03 1.49% 19.83% 6/16/2008 1360.14 0.01% 19.83% 6/17/2008 1350.93 -0.68% 19.90% 6/18/2008 1337.81 -0.98% 20.03% 6/19/2008 1342.83 0.37% 19.94% 6/20/2008 1317.93 -1.87% 20.52% 6/23/2008 1318 0.01% 19.77% 6/24/2008 1314.29 -0.28% 19.76% 6/25/2008 1321.97 0.58% 19.79% 6/26/2008 1283.15 -2.98% 20.36% 6/27/2008 1278.38 -0.37% 20.37% 6/30/2008 1280 0.13% 19.68% 7/1/2008 1284.91 0.38% 19.73% 7/2/2008 1261.52 -1.84% 20.16% 7/3/2008 1262.9 0.11% 20.13% 7/7/2008 1252.31 -0.84% 20.08% 7/8/2008 1273.7 1.69% 20.61% 7/9/2008 1244.69 -2.30% 21.38% 7/10/2008 1253.39 0.70% 21.52% 7/11/2008 1239.49 -1.12% 21.66% 7/14/2008 1228.3 -0.91% 21.07% 7/15/2008 1214.91 -1.10% 21.12% 7/16/2008 1245.36 2.48% 22.41% 7/17/2008 1260.32 1.19% 22.58% 7/18/2008 1260.68 0.03% 22.16% 7/21/2008 1260 -0.05% 22.11% 7/22/2008 1277 1.34% 22.46% 7/23/2008 1282.19 0.41% 22.25% 7/24/2008 1252.54 -2.34% 22.99% 7/25/2008 1257.76 0.42% 23.00% 7/28/2008 1234.37 -1.88% 23.15% 7/29/2008 1263.2 2.31% 24.16% 7/30/2008 1284.26 1.65% 24.69% 7/31/2008 1267.38 -1.32% 24.79% 8/1/2008 1260.31 -0.56% 24.51% 8/4/2008 1249.01 -0.90% 24.55% 8/5/2008 1284.88 2.83% 25.70% 8/6/2008 1289.19 0.33% 25.63% 8/7/2008 1266.07 -1.81% 25.98% 8/8/2008 1296.32 2.36% 26.81% 8/11/2008 1305.32 0.69% 26.90% 8/12/2008 1289.59 -1.21% 26.94% 8/13/2008 1285.83 -0.29% 26.92% 8/14/2008 1292.93 0.55% 26.98% 8/15/2008 1298.2 0.41% 26.42% 8/18/2008 1278.6 -1.52% 25.40% 8/19/2008 1266.69 -0.94% 25.49% 8/20/2008 1274.54 0.62% 25.57% 8/21/2008 1277.72 0.25% 25.22% 8/22/2008 1292.2 1.13% 25.41% 8/25/2008 1266.84 -1.98% 25.55% 8/26/2008 1271.51 0.37% 25.58% 8/27/2008 1281.66 0.80% 25.66% 8/28/2008 1300.68 1.47% 25.89% 8/29/2008 1282.83 -1.38% 26.11% 9/2/2008 1277.58 -0.41% 25.66% 9/3/2008 1274.98 -0.20% 25.66% 9/4/2008 1236.83 -3.04% 26.89% 9/5/2008 1242.31 0.44% 26.86% 9/8/2008 1267.79 2.03% 26.30% 9/9/2008 1224.51 -3.47% 27.89% 9/10/2008 1232.04 0.61% 27.95% 9/11/2008 1249.05 1.37% 28.20% 9/12/2008 1251.7 0.21% 27.77% 9/15/2008 1192.7 -4.83% 30.66% 9/16/2008 1213.6 1.74% 31.00% 9/17/2008 1156.39 -4.83% 33.18% 9/18/2008 1206.51 4.24% 34.75% 9/19/2008 1255.08 3.95% 36.35% 9/22/2008 1207.09 -3.90% 37.74% 9/23/2008 1188.22 -1.58% 37.90% 9/24/2008 1185.87 -0.20% 37.79% 9/25/2008 1209.18 1.95% 37.56% 9/26/2008 1213.27 0.34% 37.41% 9/29/2008 1106.42 -9.22% 44.83% 9/30/2008 1166.36 5.28% 47.26% 10/1/2008 1161.06 -0.46% 47.09% 10/2/2008 1114.28 -4.11% 48.23% 10/3/2008 1099.23 -1.36% 47.99% 10/6/2008 1056.89 -3.93% 48.96% 10/7/2008 996.23 -5.91% 51.06% 10/8/2008 984.94 -1.14% 50.53% 10/9/2008 909.92 -7.92% 54.00% 10/10/2008 899.22 -1.18% 53.99% 10/13/2008 1003.35 10.96% 62.49% 10/14/2008 998.01 -0.53% 62.48% 10/15/2008 907.84 -9.47% 66.39% 10/16/2008 946.43 4.16% 67.62% 10/17/2008 940.55 -0.62% 67.54% 10/20/2008 985.4 4.66% 68.57% 10/21/2008 955.05 -3.13% 68.83% 10/22/2008 896.78 -6.30% 70.50% 10/23/2008 908.11 1.26% 70.70% 10/24/2008 876.77 -3.51% 71.02% 10/27/2008 848.92 -3.23% 71.24% 10/28/2008 940.51 10.25% 77.26% 10/29/2008 930.09 -1.11% 77.26% 10/30/2008 954.09 2.55% 77.67% 10/31/2008 968.75 1.52% 77.85% 11/3/2008 966.3 -0.25% 77.71% 11/4/2008 1005.75 4.00% 78.59% 11/5/2008 952.77 -5.41% 79.68% 11/6/2008 904.88 -5.16% 80.53% 11/7/2008 930.99 2.84% 80.89% 11/10/2008 919.21 -1.27% 80.88% 11/11/2008 898.95 -2.23% 80.99% 11/12/2008 852.3 -5.33% 81.93% 11/13/2008 911.29 6.69% 84.15% 11/14/2008 873.29 -4.26% 84.66% 11/17/2008 850.75 -2.61% 84.48% 11/18/2008 859.12 0.98% 84.28% 11/19/2008 806.58 -6.31% 85.54% 11/20/2008 752.44 -6.95% 86.88% 11/21/2008 800.03 6.13% 88.95% 11/24/2008 851.81 6.27% 90.34% 11/25/2008 857.39 0.65% 90.17% 11/26/2008 887.68 3.47% 90.12% 11/28/2008 896.24 0.96% 89.26% 12/1/2008 816.21 -9.35% 91.43% 12/2/2008 848.81 3.92% 91.93% 12/3/2008 870.74 2.55% 92.31% 12/4/2008 845.22 -2.97% 92.52% 12/5/2008 876.07 3.58% 92.97% 12/8/2008 909.7 3.77% 93.70% 12/9/2008 888.67 -2.34% 90.77% 12/10/2008 899.24 1.18% 89.52% 12/11/2008 873.59 -2.89% 89.75% 12/12/2008 879.73 0.70% 89.27% 12/15/2008 868.57 -1.28% 89.27% 12/16/2008 913.18 5.01% 89.95% 12/17/2008 904.42 -0.96% 88.63% 12/18/2008 885.28 -2.14% 88.75% 12/19/2008 887.88 0.29% 86.18% 12/22/2008 871.63 -1.85% 86.26% 12/23/2008 863.16 -0.98% 80.75% 12/24/2008 868.15 0.58% 80.79% 12/26/2008 872.8 0.53% 76.73% 12/29/2008 869.42 -0.39% 75.83% 12/30/2008 890.64 2.41% 76.14% 12/31/2008 903.25 1.41% 75.12% 1/2/2009 931.8 3.11% 75.19% 1/5/2009 927.45 -0.47% 73.21% 1/6/2009 934.7 0.78% 73.16% 1/7/2009 906.65 -3.05% 73.00% 1/8/2009 909.73 0.34% 72.44% 1/9/2009 890.35 -2.15% 67.10% 1/12/2009 870.26 -2.28% 67.30% 1/13/2009 871.79 0.18% 66.90% 1/14/2009 842.62 -3.40% 67.29% 1/15/2009 843.74 0.13% 67.30% 1/16/2009 850.12 0.75% 66.33% 1/20/2009 805.22 -5.43% 66.34% 1/21/2009 840.24 4.26% 66.12% 1/22/2009 827.5 -1.53% 65.70% 1/23/2009 831.95 0.54% 65.67% 1/26/2009 836.57 0.55% 65.46% 1/27/2009 845.71 1.09% 63.95% 1/28/2009 874.09 3.30% 61.92% 1/29/2009 845.14 -3.37% 61.52% 1/30/2009 825.88 -2.31% 61.43% 2/2/2009 825.44 -0.05% 61.36% 2/3/2009 838.51 1.57% 59.05% 2/4/2009 832.23 -0.75% 55.85% 2/5/2009 845.85 1.62% 53.56% 2/6/2009 868.6 2.65% 51.31% 2/9/2009 869.89 0.15% 51.28% 2/10/2009 827.16 -5.04% 52.17% 2/11/2009 833.74 0.79% 52.15% 2/12/2009 835.19 0.17% 45.55% 2/13/2009 826.84 -1.00% 44.36% 2/17/2009 789.17 -4.66% 45.47% 2/18/2009 788.42 -0.10% 44.83% 2/19/2009 778.94 -1.21% 43.72% 2/20/2009 770.05 -1.15% 42.36% 2/23/2009 743.33 -3.53% 42.90% 2/24/2009 773.14 3.93% 44.26% 2/25/2009 764.9 -1.07% 43.73% 2/26/2009 752.83 -1.59% 43.79% 2/27/2009 735.09 -2.38% 44.08% 3/2/2009 700.82 -4.77% 43.16% 3/3/2009 696.33 -0.64% 43.15% 3/4/2009 712.87 2.35% 43.59% 3/5/2009 682.55 -4.35% 44.80% 3/6/2009 683.38 0.12% 44.69% 3/9/2009 676.53 -1.01% 44.69% 3/10/2009 719.6 6.17% 48.11% 3/11/2009 721.36 0.24% 48.07% 3/12/2009 750.74 3.99% 49.50% 3/13/2009 756.55 0.77% 49.03% 3/16/2009 753.89 -0.35% 48.80% 3/17/2009 778.12 3.16% 48.83% 3/18/2009 794.35 2.06% 49.26% 3/19/2009 784.04 -1.31% 49.24% 3/20/2009 768.54 -2.00% 48.89% 3/23/2009 822.92 6.84% 52.57% 3/24/2009 806.12 -2.06% 52.55% 3/25/2009 813.88 0.96% 52.32% 3/26/2009 832.86 2.31% 52.73% 3/27/2009 815.94 -2.05% 52.22% 3/30/2009 787.53 -3.54% 53.05% 3/31/2009 797.87 1.30% 53.14% 4/1/2009 811.08 1.64% 51.29% 4/2/2009 834.38 2.83% 50.55% 4/3/2009 842.5 0.97% 50.45% 4/6/2009 835.48 -0.84% 50.48% 4/7/2009 815.55 -2.41% 50.88% 4/8/2009 825.16 1.17% 50.89% 4/9/2009 856.56 3.73% 51.12% 4/13/2009 858.73 0.25% 50.29% 4/14/2009 841.5 -2.03% 50.20% 4/15/2009 852.06 1.25% 50.31% 4/16/2009 865.3 1.54% 50.30% 4/17/2009 869.6 0.50% 50.26% 4/20/2009 832.39 -4.37% 51.49% 4/21/2009 850.08 2.10% 51.30% 4/22/2009 843.55 -0.77% 51.34% 4/23/2009 851.92 0.99% 49.54% 4/24/2009 866.23 1.67% 49.69% 4/27/2009 857.51 -1.01% 49.77% 4/28/2009 855.16 -0.27% 49.70% 4/29/2009 873.64 2.14% 48.25% 4/30/2009 872.81 -0.10% 48.25% 5/1/2009 877.52 0.54% 48.10% 5/4/2009 907.24 3.33% 48.66% 5/5/2009 903.8 -0.38% 47.53% 5/6/2009 919.53 1.73% 46.67% 5/7/2009 907.39 -1.33% 46.73% 5/8/2009 929.23 2.38% 46.74% 5/11/2009 909.24 -2.17% 46.65% 5/12/2009 908.35 -0.10% 44.43% 5/13/2009 883.92 -2.73% 45.18% 5/14/2009 893.07 1.03% 44.91% 5/15/2009 882.88 -1.15% 43.17% 5/18/2009 909.71 2.99% 43.67% 5/19/2009 908.13 -0.17% 43.50% 5/20/2009 903.47 -0.51% 40.77% 5/21/2009 888.33 -1.69% 41.18% 5/22/2009 887 -0.15% 40.01% 5/26/2009 910.33 2.60% 40.46% 5/27/2009 893.06 -1.92% 40.88% 5/28/2009 906.83 1.53% 40.28% 5/29/2009 919.14 1.35% 40.09% 6/1/2009 942.87 2.55% 40.30% 6/2/2009 944.74 0.20% 39.77% 6/3/2009 931.76 -1.38% 35.89% 6/4/2009 942.46 1.14% 35.39% 6/5/2009 940.09 -0.25% 35.38% 6/8/2009 939.14 -0.10% 34.95% 6/9/2009 942.43 0.35% 34.38% 6/10/2009 939.15 -0.35% 32.77% 6/11/2009 944.89 0.61% 32.68% 6/12/2009 946.21 0.14% 32.48% 6/15/2009 923.72 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0.07% 25.64% 8/3/2009 1002.63 1.52% 25.36% 8/4/2009 1005.65 0.30% 25.34% 8/5/2009 1002.72 -0.29% 24.54% 8/6/2009 997.08 -0.56% 23.94% 8/7/2009 1010.48 1.33% 23.86% 8/10/2009 1007.1 -0.34% 23.70% 8/11/2009 994.35 -1.27% 23.10% 8/12/2009 1005.81 1.15% 23.27% 8/13/2009 1012.73 0.69% 22.94% 8/14/2009 1004.09 -0.86% 22.94% 8/17/2009 979.73 -2.46% 23.97% 8/18/2009 989.67 1.01% 24.09% 8/19/2009 996.46 0.68% 24.13% 8/20/2009 1007.37 1.09% 24.24% 8/21/2009 1026.13 1.85% 24.65% 8/24/2009 1025.57 -0.05% 24.65% 8/25/2009 1028 0.24% 23.61% 8/26/2009 1028.12 0.01% 23.26% 8/27/2009 1030.98 0.28% 23.24% 8/28/2009 1028.93 -0.20% 23.21% 8/31/2009 1020.62 -0.81% 23.38% 9/1/2009 998.04 -2.24% 22.56% 9/2/2009 994.75 -0.33% 22.61% 9/3/2009 1003.24 0.85% 22.65% 9/4/2009 1016.4 1.30% 22.24% 9/8/2009 1025.39 0.88% 22.29% 9/9/2009 1033.37 0.78% 22.27% 9/10/2009 1044.14 1.04% 22.17% 9/11/2009 1042.73 -0.14% 22.19% 9/14/2009 1049.34 0.63% 20.38% 9/15/2009 1052.63 0.31% 20.38% 9/16/2009 1068.76 1.52% 19.62% 9/17/2009 1065.49 -0.31% 19.65% 9/18/2009 1068.3 0.26% 19.65% 9/21/2009 1064.66 -0.34% 19.63% 9/22/2009 1071.66 0.66% 18.80% 9/23/2009 1060.87 -1.01% 19.14% 9/24/2009 1050.78 -0.96% 18.05% 9/25/2009 1044.38 -0.61% 18.11% 9/28/2009 1062.98 1.77% 18.58% 9/29/2009 1060.61 -0.22% 18.45% 9/30/2009 1057.08 -0.33% 18.51% 10/1/2009 1029.85 -2.61% 20.00% 10/2/2009 1025.21 -0.45% 19.16% 10/5/2009 1040.46 1.48% 19.52% 10/6/2009 1054.72 1.36% 19.80% 10/7/2009 1057.58 0.27% 19.77% 10/8/2009 1065.48 0.74% 19.76% 10/9/2009 1071.49 0.56% 19.59% 10/12/2009 1076.19 0.44% 19.61% 10/13/2009 1073.19 -0.28% 19.28% 10/14/2009 1092.02 1.74% 19.76% 10/15/2009 1096.56 0.41% 19.73% 10/16/2009 1087.68 -0.81% 19.81% 10/19/2009 1097.91 0.94% 19.67% 10/20/2009 1091.06 -0.63% 19.74% 10/21/2009 1081.4 -0.89% 19.56% 10/22/2009 1092.91 1.06% 19.52% 10/23/2009 1079.6 -1.23% 19.83% 10/26/2009 1066.95 -1.18% 19.96% 10/27/2009 1063.41 -0.33% 18.71% 10/28/2009 1042.63 -1.97% 19.42% 10/29/2009 1066.11 2.23% 20.20% 10/30/2009 1036.19 -2.85% 21.57% 11/2/2009 1042.88 0.64% 21.06% 11/3/2009 1045.41 0.24% 21.07% 11/4/2009 1046.5 0.10% 21.06% 11/5/2009 1066.63 1.91% 21.66% 11/6/2009 1069.3 0.25% 21.66% 11/9/2009 1093.08 2.20% 22.39% 11/10/2009 1093.01 -0.01% 22.25% 11/11/2009 1098.51 0.50% 21.28% 11/12/2009 1087.24 -1.03% 21.49% 11/13/2009 1093.48 0.57% 21.44% 11/16/2009 1109.3 1.44% 21.49% 11/17/2009 1110.32 0.09% 21.41% 11/18/2009 1109.8 -0.05% 21.35% 11/19/2009 1094.9 -1.35% 21.58% 11/20/2009 1091.38 -0.32% 21.60% 11/23/2009 1106.24 1.35% 21.82% 11/24/2009 1105.65 -0.05% 21.81% 11/25/2009 1110.63 0.45% 21.48% 11/27/2009 1091.49 -1.74% 22.01% 11/30/2009 1095.63 0.38% 22.03% 12/1/2009 1108.86 1.20% 22.21% 12/2/2009 1109.24 0.03% 22.16% 12/3/2009 1099.92 -0.84% 22.10% 12/4/2009 1105.98 0.55% 21.95% 12/7/2009 1103.25 -0.25% 21.89% 12/8/2009 1091.94 -1.03% 21.61% 12/9/2009 1095.95 0.37% 21.62% 12/10/2009 1102.35 0.58% 21.63% 12/11/2009 1106.41 0.37% 20.33% 12/14/2009 1114.11 0.69% 20.31% 12/15/2009 1107.93 -0.56% 20.08% 12/16/2009 1109.18 0.11% 19.79% 12/17/2009 1096.08 -1.19% 20.08% 12/18/2009 1102.47 0.58% 20.05% 12/21/2009 1114.05 1.04% 20.18% 12/22/2009 1118.02 0.36% 20.17% 12/23/2009 1120.59 0.23% 20.15% 12/24/2009 1126.48 0.52% 19.67% 12/28/2009 1127.78 0.12% 19.64% 12/29/2009 1126.2 -0.14% 19.51% 12/30/2009 1126.42 0.02% 19.36% 12/31/2009 1115.1 -1.01% 19.49% Geometric Brownian Simulations of S&P 500 Time Normal (0,1) X(t) logX(t) - logX(t-.01) .00 1447.16 .01 -.540510 1413.076003 -.023834 .02 -.070239 1408.737610 -.003075 .03 .424530 1435.423310 .018766 .04 1.285320 1519.260292 .056764 .05 1.197243 1601.754087 .052876 .06 .979565 1672.577923 .043267 .07 1.776623 1809.078380 .078451 .08 -1.046180 1727.476282 -.046156 .09 -.921680 1658.645581 -.040660 .10 -.166938 1646.509933 -.007343 .11 -.907399 1581.902261 -.040030 .12 -.243383 1565.037932 -.010718 .13 .891680 1627.910556 .039387 .14 1.273027 1722.055218 .056221 .15 1.109263 1808.523110 .048992 .16 -.224319 1790.749257 -.009876 .17 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4178030000 1094.9 11/20/2009 1094.66 1091.38 3751230000 1091.38 11/23/2009 1094.86 1106.24 3827920000 1106.24 11/24/2009 1105.83 1105.65 3700820000 1105.65 11/25/2009 1106.49 1110.63 3036350000 1110.63 11/27/2009 1105.47 1091.49 2362910000 1091.49 11/30/2009 1091.07 1095.63 3895520000 1095.63 12/1/2009 1098.89 1108.86 4249310000 1108.86 12/2/2009 1109.03 1109.24 3941340000 1109.24 12/3/2009 1110.59 1099.92 4810030000 1099.92 12/4/2009 1100.43 1105.98 5781140000 1105.98 12/7/2009 1105.52 1103.25 4103360000 1103.25 12/8/2009 1103.04 1091.94 4748030000 1091.94 12/9/2009 1091.07 1095.95 4115410000 1095.95 12/10/2009 1098.69 1102.35 3996490000 1102.35 12/11/2009 1103.96 1106.41 3791090000 1106.41 12/14/2009 1107.84 1114.11 4548490000 1114.11 12/15/2009 1114.11 1107.93 5045100000 1107.93 12/16/2009 1108.61 1109.18 4829820000 1109.18 12/17/2009 1106.36 1096.08 7615070400 1096.08 12/18/2009 1097.86 1102.47 6325890000 1102.47 12/21/2009 1105.31 1114.05 3977340000 1114.05 12/22/2009 1114.51 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Date Open Close Adj Close 1/2/2008 29.73 27.34 27.34 1/3/2008 29.22 27.35 27.35 1/4/2008 28.61 26.69 26.69 1/7/2008 28.34 26.71 26.71 1/8/2008 28.46 25.65 25.65 1/9/2008 27.15 25.99 25.99 1/10/2008 26.98 26.57 26.57 1/11/2008 27.89 27 27 1/14/2008 28.95 27.47 27.47 1/15/2008 28.26 25.47 25.47 1/16/2008 27 24.8 24.8 1/17/2008 26.52 23.59 23.59 1/18/2008 24.53 23.11 23.11 1/22/2008 22.36 23.06 23.06 1/23/2008 23.98 24.92 24.92 1/24/2008 26.8 25.83 25.83 1/25/2008 28 25.18 25.18 1/28/2008 26.43 26.14 26.14 1/29/2008 27.9 26.38 26.38 1/30/2008 27.69 26.35 26.35 1/31/2008 27 26.94 26.94 2/1/2008 28.28 28.39 28.39 2/4/2008 29.34 27.94 27.94 2/5/2008 28.65 25.87 25.87 2/6/2008 27.31 25.74 25.74 2/7/2008 26.75 25.53 25.53 2/8/2008 26.61 24.89 24.89 2/11/2008 26.23 24.68 24.68 2/12/2008 26.15 25.06 25.06 2/13/2008 26.58 25.19 25.19 2/14/2008 26.19 24.61 24.61 2/15/2008 25.49 24.36 24.36 2/19/2008 25.83 24.21 24.21 2/20/2008 25.11 24.37 24.37 2/21/2008 25.63 23.95 23.95 2/22/2008 25.09 24.02 24.02 2/25/2008 24.65 23.66 23.66 2/26/2008 24.46 23.86 23.86 2/27/2008 24.7 24.59 24.59 2/28/2008 25.46 23.92 23.92 2/29/2008 24.66 22.67 22.67 3/3/2008 23.72 22.08 22.08 3/4/2008 22.23 21.13 21.13 3/5/2008 22.32 21.18 21.18 3/6/2008 21.95 20.24 20.24 3/7/2008 20.75 19.99 19.99 3/10/2008 20.97 18.83 18.83 3/11/2008 20.9 20.55 20.55 3/12/2008 21.63 20.28 20.28 3/13/2008 20.69 20.15 20.15 3/14/2008 21.47 18.91 18.91 3/17/2008 18.25 17.81 17.81 3/18/2008 19.84 19.8 19.8 3/19/2008 21 19.52 19.52 3/20/2008 20.72 21.52 21.52 3/24/2008 22.96 22.25 22.25 3/25/2008 23.09 22.4 22.4 3/26/2008 22.64 21.08 21.08 3/27/2008 22.13 20.84 20.84 3/28/2008 21.86 19.92 19.92 3/31/2008 20.8 20.48 20.48 4/1/2008 22.61 22.8 22.8 4/2/2008 24.05 22.97 22.97 4/3/2008 23.7 23.29 23.29 4/4/2008 24.44 23.03 23.03 4/7/2008 24.85 23.52 23.52 4/8/2008 24.31 22.72 22.72 4/9/2008 24.11 22.55 22.55 4/10/2008 23.51 22.67 22.67 4/11/2008 23.29 22.34 22.34 4/14/2008 23.13 21.52 21.52 4/15/2008 22.78 21.8 21.8 4/16/2008 23.4 22.41 22.41 4/17/2008 23.05 22.98 22.98 4/18/2008 25.73 24.01 24.01 4/21/2008 24.82 23.93 23.93 4/22/2008 25.04 24.02 24.02 4/23/2008 25.07 23.55 23.55 4/24/2008 24.7 24.63 24.63 4/25/2008 26.2 25.44 25.44 4/28/2008 26.7 25.64 25.64 4/29/2008 26.8 25.17 25.17 4/30/2008 25.51 24.16 24.16 5/1/2008 25.23 25.17 25.17 5/2/2008 26.82 25.56 25.56 5/5/2008 26.4 24.94 24.94 5/6/2008 25.5 25.06 25.06 5/7/2008 26.03 23.71 23.71 5/8/2008 24.77 23.53 23.53 5/9/2008 24.41 22.89 22.89 5/12/2008 23.72 22.9 22.9 5/13/2008 23.7 22.3 22.3 5/14/2008 23.21 22.52 22.52 5/15/2008 23.29 22.98 22.98 5/16/2008 23.81 22.39 22.39 5/19/2008 23.22 22.27 22.27 5/20/2008 22.8 21.41 21.41 5/21/2008 22.13 20.4 20.4 5/22/2008 21.25 21.04 21.04 5/23/2008 21.75 20.45 20.45 5/27/2008 21.24 20.98 20.98 5/28/2008 22.04 20.92 20.92 5/29/2008 21.53 21.35 21.35 5/30/2008 22.27 21.2 21.2 6/2/2008 21.7 20.78 20.78 6/3/2008 21.64 20.75 20.75 6/4/2008 21.35 20.42 20.42 6/5/2008 21.29 20.55 20.55 6/6/2008 21.04 19.43 19.43 6/9/2008 20.15 18.98 18.98 6/10/2008 19.5 19.62 19.62 6/11/2008 20.39 18.6 18.6 6/12/2008 19.48 19.26 19.26 6/13/2008 20.2 19.83 19.83 6/16/2008 20.41 20.17 20.17 6/17/2008 21.2 19.82 19.82 6/18/2008 20.13 19.76 19.76 6/19/2008 20.5 19.53 19.53 6/20/2008 19.74 18.69 18.69 6/23/2008 19.4 17.97 17.97 6/24/2008 18.55 18.26 18.26 6/25/2008 19.13 18.26 18.26 6/26/2008 18.06 17.11 17.11 6/27/2008 17.81 16.71 16.71 6/30/2008 17.19 16.23 16.23 7/1/2008 16.57 16.59 16.59 7/2/2008 17.27 16.31 16.31 7/3/2008 16.96 16.29 16.29 7/7/2008 17.05 15.88 15.88 7/8/2008 16.52 16.84 16.84 7/9/2008 17.46 15.92 15.92 7/10/2008 16.1 15.77 15.77 7/11/2008 15.86 15.68 15.68 7/14/2008 16.76 14.74 14.74 7/15/2008 14.99 14.1 14.1 7/16/2008 15.13 15.95 15.95 7/17/2008 17.15 17.4 17.4 7/18/2008 19.8 18.74 18.74 7/21/2008 19.94 19.07 19.07 7/22/2008 19.04 20.23 20.23 7/23/2008 20.63 20.45 20.45 7/24/2008 21.12 18.46 18.46 7/25/2008 19.21 18.26 18.26 7/28/2008 18.82 16.88 16.88 7/29/2008 16.95 17.87 17.87 7/30/2008 18.9 18.22 18.22 7/31/2008 18.4 18.41 18.41 8/1/2008 18.95 18.59 18.59 8/4/2008 18.53 18.55 18.55 8/5/2008 19.37 19.63 19.63 8/6/2008 19.64 19.41 19.41 8/7/2008 19.29 18.2 18.2 8/8/2008 18.41 19.1 19.1 8/11/2008 19.24 19.53 19.53 8/12/2008 19.69 18.27 18.27 8/13/2008 18.42 17.55 17.55 8/14/2008 17.73 17.81 17.81 8/15/2008 18.36 18.28 18.28 8/18/ Read More