Explorations of the log-normal distributionFeb 5th, 2011When a stock follows a Brownian Motion of the form 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, then at the time LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiVEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= the distribution of stock prices is given by the log-normal distribution. This pdf 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(as you open this work sheet, please remember to execute the command above, in order to enter the formula into Maple's memory.)Mean and Standard DeviationThis distribution allows us to calculate certain quantities. For example, if the parameters areQyo+SSNTMEc2IiIkKyIhIiI+SSJUR0YlIiIiRic+SSJyR0YlJCIiKSEiI0YnPkkmc2lnbWFHRiUkIiNJRi9GJw==then the mean is:PkklbWVhbkc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUkkaW50RzYkRidJKF9zeXNsaWJHRiQ2JComSSJTR0YkIiIiLUkkcGRmR0YkNiNGL0YwL0YvOyIiIUkpaW5maW5pdHlHRic=which indeed is the same as PkklbWVhbkc2IiomSSNTMEdGJCIiIi1JJGV4cEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjKiZJIlRHRiRGJ0kickdGJEYnRic=the variance and standard deviation areQyU+SSl2YXJpYW5jZUc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUkkaW50RzYkRihJKF9zeXNsaWJHRiU2JComLCZJIlNHRiUiIiJJJW1lYW5HRiUhIiIiIiMtSSRwZGZHRiU2I0YxRjIvRjE7IiIhSSlpbmZpbml0eUdGKEYyPkkmc3RkZXZHRiUtSSVzcXJ0R0YsNiNGJA==which by theory isPkkmc3RkZXZHNiIqKEkjUzBHRiQiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSJUR0YkRidJInJHRiRGJ0YnLUklc3FydEdGKjYjLCYtRik2IyomSSZzaWdtYUdGJCIiI0YvRidGJyEiIkYnRic=which, incidentally, can be approximated byKihJI1MwRzYiIiIiLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YkNiMqJkkiVEdGJEYlSSJyR0YkRiVGJSwmKiZJJnNpZ21hR0YkRiUtSSVzcXJ0R0YoNiNGLUYlRiUtSSJeR0YpNiRGMCIiJCNGJSIiJUYlJSFHProbabilitiesThis distribution allows us to calculate certain probabilities. For example, the chance that the stock price ends up higher than $100 is justLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRpbnRHNiRGJEkoX3N5c2xpYkc2IjYkLUkkcGRmR0YqNiNJIlNHRiovRi87IiQrIkkpaW5maW5pdHlHRiQ=higher than $150LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRpbnRHNiRGJEkoX3N5c2xpYkc2IjYkLUkkcGRmR0YqNiNJIlNHRiovRi87IiRdIkkpaW5maW5pdHlHRiQ=higher than $200LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRpbnRHNiRGJEkoX3N5c2xpYkc2IjYkLUkkcGRmR0YqNiNJIlNHRiovRi87IiQrI0kpaW5maW5pdHlHRiQ=below $50LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRpbnRHNiRGJEkoX3N5c2xpYkc2IjYkLUkkcGRmR0YqNiNJIlNHRiovRi87IiIhIiNdNormal versus Log-Normal
The fact that S is log-normally distributed means that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjbG5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYkLUYjNiQtRiw2JVEiU0YnL0YwUSV0cnVlRicvRjNRJ2l0YWxpY0YnRjJGMkYy is normally distributed. Its mean is 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, its variance is 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hence its pdf 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with our parameters we getQyo+SSNTMEc2IiIkKyIhIiI+SSJUR0YlIiIiRic+SSJyR0YlJCIiKSEiI0YnPkkmc2lnbWFHRiUkIiNJRi9GJw==PkklbWVhbkc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUkkaW50RzYkRidJKF9zeXNsaWJHRiQ2JComSSJYR0YkIiIiLUklcGRmMkdGJDYjRi9GMC9GLzsiIiFJKWluZmluaXR5R0YnQyU+SSl2YXJpYW5jZUc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUkkaW50RzYkRihJKF9zeXNsaWJHRiU2JComLCZJIlhHRiUiIiJJJW1lYW5HRiUhIiIiIiMtSSVwZGYyR0YlNiNGMUYyL0YxOyIiIUkpaW5maW5pdHlHRihGMj5JJnN0ZGV2R0YlLUklc3FydEdGLDYjRiQ=JSFH