Math 489/889 Stochastic Processes and Advanced Mathematical Finance Practice Problems Homework 10 Steve Dunbar Monday, December 7, 2009
<Text-field style="Heading 1" layout="Heading 1">Initialization of Necessary Packages</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRKGZpbmFuY2VGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiO0YnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Ny5JLWFtb3J0aXphdGlvbkc2IkkoYW5udWl0eUdGJEktYmxhY2tzY2hvbGVzR0YkSSpjYXNoZmxvd3NHRiRJLmVmZmVjdGl2ZXJhdGVHRiRJLGZ1dHVyZXZhbHVlR0YkSS9ncm93aW5nYW5udWl0eUdGJEkyZ3Jvd2luZ3BlcnBldHVpdHlHRiRJLGxldmVsY291cG9uR0YkSStwZXJwZXR1aXR5R0YkSS1wcmVzZW50dmFsdWVHRiRJMHlpZWxkdG9tYXR1cml0eUdGJA== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYjLUYsNiVRK1N0YXRpc3RpY3NGJ0YvRjIvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVA== 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 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder SSNfUkc2JSUqcHJvdGVjdGVkR0kvUmFuZG9tVmFyaWFibGVHNiJJOV9Qcm9iYWJpbGl0eURpc3RyaWJ1dGlvbkdGJg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Problem 1</Text-field> Find the mode (the value of the independent variable with the highest probability) of the lognormal probability density function. (Use parameters \316\274 and \317\203.) 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I define the lognormal in terms of the parameters \316\274 and \317\203^2, but Maple uses the parameters \316\274 and \317\203 to define the same distribution. So be careful to get the distribution you really want! 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<Text-field style="Heading 1" layout="Heading 1">Problem 3</Text-field> What is the price of a European call option on a non-dividend-paying stock when the stock price is \$52, the strike price is \$50, the risk-free interest rate is 12% per annum (compounded continuously), the volatility is 30% per annum, and the time to maturity is 3 months? 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<Text-field style="Heading 1" layout="Heading 1">Problem 4</Text-field> What is the price of a European call option on a non-dividend paying stock when the stock price is \$30, the exercise price is \$29, the risk-free interest rate is 5%, the volatility is 25% per annum, and the time to maturity is 4 months? 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# input placeholder JCIqKHA5REQhIik=
<Text-field style="Heading 1" layout="Heading 1">Problem 5</Text-field> <Text-field style="Heading 1" layout="Heading 1"></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic= QyQ+SSNWMUc2IiomSSJBR0YlIiIiSSJTR0YlRihGKA== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiZJIkFHNiIiIiJJIlNHRiRGJQ== PkkxYmxhY2tzY2hvbGVzVGVzdEc2IiwqLUklZGlmZkclKnByb3RlY3RlZEc2JEkjVjFHRiRJInRHRiQiIiIqKEkmc2lnbWFHRiQiIiNJIlNHRiRGLy1GJzYkLUYnNiRGKkYwRjBGLCNGLEYvKihJInJHRiRGLEYwRixGM0YsRiwqJkY3RixGKkYsISIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IiIh QyQ+SSNWMkc2IiomSSJBR0YlIiIiLUkkZXhwR0YlNiMqJkkickdGJUYoSSJ0R0YlRihGKEYo LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder KiZJIkFHNiIiIiItSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IyomSSJyR0YkRiVJInRHRiRGJUYl PkkxYmxhY2tzY2hvbGVzVGVzdEc2IiwqLUklZGlmZkclKnByb3RlY3RlZEc2JEkjVjJHRiRJInRHRiQiIiIqKEkmc2lnbWFHRiQiIiNJIlNHRiRGLy1GJzYkLUYnNiRGKkYwRjBGLCNGLEYvKihJInJHRiRGLEYwRixGM0YsRiwqJkY3RixGKkYsISIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder IiIh JSFH
JSFH
<Text-field style="Heading 1" layout="Heading 1">Problem 6</Text-field> Black Scholes theory does not apply. The Black-Scholes derivation explicitly assumes that the changes in price are continuous. Here the price will immediately change discontinuously by the amount \$10.00.
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<Text-field style="Heading 1" layout="Heading 1">Problem 7</Text-field> QyQ+SS5wdXRDYWxsUGFyaXR5RzYiLywmLUkiQ0dGJTYkSSJTR0YlSSJ0R0YlIiIiLUYrRiohIiIsJkYrRi0qJkkiS0dGJUYtLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJC1JInJHRiU2IywmSSJUR0YlRi1GLEYvRi9GLUYvRi0= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmLUkiQ0c2IjYkSSJTR0YmSSJ0R0YmIiIiLUYoRichIiIsJkYoRioqJkkiS0dGJkYqLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiMsJC1JInJHRiY2IywmSSJUR0YmRipGKUYsRixGKkYs QyQ+SS5kZWx0YVJlbGF0aW9uRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkucHV0Q2FsbFBhcml0eUdGJUkiU0dGJSIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1JIkNHNiI2JEkiU0dGKkkidEdGKkYsIiIiLUYlNiQtRixGK0YsISIiRi4= QyQ+SS5nYW1tYVJlbGF0aW9uRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JUkucHV0Q2FsbFBhcml0eUdGJUkiU0dGJUYrIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkiQ0c2IjYkSSJTR0YsSSJ0R0YsRi5GLiIiIi1GJTYkLUYlNiQtRi5GLUYuRi4hIiIiIiE= JSFH
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