Math 489/889 Stochastic Processes and Advanced Mathematical Finance Final Exam Solutions Steve Dunbar Thursday, December 17, 2009 JSFH
<Text-field style="Heading 1" layout="Heading 1">Initializations</Text-field> QyQtSSV3aXRoRzYiNiNJKGZpbmFuY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiUiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Ny5JLWFtb3J0aXphdGlvbkc2IkkoYW5udWl0eUdGJEktYmxhY2tzY2hvbGVzR0YkSSpjYXNoZmxvd3NHRiRJLmVmZmVjdGl2ZXJhdGVHRiRJLGZ1dHVyZXZhbHVlR0YkSS9ncm93aW5nYW5udWl0eUdGJEkyZ3Jvd2luZ3BlcnBldHVpdHlHRiRJLGxldmVsY291cG9uR0YkSStwZXJwZXR1aXR5R0YkSS1wcmVzZW50dmFsdWVHRiRJMHlpZWxkdG9tYXR1cml0eUdGJA== LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSStTdGF0aXN0aWNzR0Yn LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder 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 QyQ+SSJaRzYiLUkvUmFuZG9tVmFyaWFibGVHRiU2Iy1JJ05vcm1hbEdGJTYkIiIhIiIiRi0= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder SSNfUkc2JSUqcHJvdGVjdGVkR0kvUmFuZG9tVmFyaWFibGVHNiJJOV9Qcm9iYWJpbGl0eURpc3RyaWJ1dGlvbkdGJg==
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<Text-field style="Heading 1" layout="Heading 1">Problem 1</Text-field> A stock has a constant volatility of 18% and the risk-free interest rate (compounded continuously) is 6%. What is the value of an option to buy the stock for \$25 in two years time, given the current stock price is \$20? 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<Text-field style="Heading 1" layout="Heading 1">Problem 3</Text-field> PkkiQkc2IiwmLUkvUmFuZG9tVmFyaWFibGVHRiQ2Iy1JKkJlcm5vdWxsaUdGJDYjJCIiJiEiIiIiI0YuIiIi LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LCZJJF9SMEc2JSUqcHJvdGVjdGVkR0kvUmFuZG9tVmFyaWFibGVHNiJJOl9Qcm9iYWJpbGl0eURpc3RyaWJ1dGlvbjBHRiciIiMhIiIiIiI= QyQ+SSZmbGlwc0c2Ii1JJ1NhbXBsZUdGJTYkSSJCR0YlIiQrIiIiIg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder LSZJJ1ZlY3Rvckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSRyb3dHRig2Iy9JJCVpZEdGKCIqM0ZbXiI= QyY+JkkiU0c2IjYjIiIhRighIiI/KEkiaUdGJiIiIkYsIiQrIkkldHJ1ZUclKnByb3RlY3RlZEc+JkYlNiNGKywmJkYlNiMsJkYrRixGKUYsRiwmSSZmbGlwc0dGJkYyRixGKQ== NyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQmSSJTRzYiNiMsJEkiaUc2IiIjNS9JImlHNiI7IiIhIiM1 LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder Ny0iIiEkISIlRiMkISInRiMkISM1RiMkISIpRiNGKkYmRipGJkYoRio= QyQ+SSNkdEc2IiQiIiMhIiIiIiI= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIiIyEiIg== NiNJImpHNiI= NiMmSSJYRzYiNiNJImpHNiI= NiMqJCkmSSJYRzYiNiNJImpHNiIjIiIiIiIjIiIi NiNJI2RXRzYi NiNRQGRYfj0oMS1YW2pdKSpkdH4rfnNxcnQoWFtqXSkqZFc2Ig== NiMvJkkiWEc2IjYjLCZJImpHNiIiIiIiIiIiIiIsJiZJIlhHNiI2I0kiakc2IiIiIkkjZFhHNiIiIiI= NiMiIiE= NiMiIiI= NiMiIiM= NiMiIiQ= NiMiIiU= NiMiIiY= NiMkIiIhIiIh NiMkIiIhIiIh NiMkISsrKysrZyEjNQ== NiMkIiIjISIi NiMkIiIjISIi NiMkIiIjISIi NiMkIitiZjhzVyEjNQ== NiMkISsrKysrPyEjNQ== NiMkIishNEdkMCghIzY= NiMkIis0R2QwRiEjNQ== NiMkIis0R2QwRiEjNQ== NiMkIisnNDc6PyYhIzU= NiMkIisrKysrPyEjNQ== NiMkIitkeT0qXCMhIzU= NiMkIittMXcvXyEjNQ== NiMkIitkeT0qXCMhIzU= NiMkIisieig9KipcISM1 NiMkIiIhIiIh NiMkIitIQzsrOiEjNQ== NiMkIisnR10kKipSISM1 NiMkIisnR10kKipSISM1 NiMkIitiOy9DaiEjNQ== NiMkISsrKysrPyEjNQ== NiMkISkpUXlZJyEjNQ== NiMkIispKj1uTVIhIzU= NiMkIispKj1uTVIhIzU=
JSFH
<Text-field style="Heading 1" layout="Heading 1">Problem 5</Text-field> Qyc+SSJyRzYiJCEiIkYnIiIiPkkmc2lnbWFHRiUiIiNGKD4mSSJ4R0YlNiMiIiEkIiQjRyEiIw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCEiIkYj IiIj JCIkI0chIiM= LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiM3JSooLCYmSSJ4RzYiNiMiIiEiIiJJInJHRisiIiVGLkkmc2lnbWFHRishIiItSSVzcXJ0RzYkRiRJKF9zeXNsaWJHRis2I0YwRjIqKCwmRilGLkYvIiIpRi5GMUYyLUY0NiNGOkYyKigsJkYpRi5GLyIjN0YuRjFGMi1GNDYjRj9GMg== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder NyUkIisrKytdZyEjNSQiK1cjKikzZCRGJSQiKyJmbyNRQkYl QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJENERkc2IjYkSSJaR0YpLCQqKCwmJkkieEdGKTYjIiIhIiIiSSJyR0YpIiIlRjNJJnNpZ21hR0YpISIiLUklc3FydEc2JEYlSShfc3lzbGliR0YpNiNGNUY3RjdGMw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIrJClcKmVzIyEjNQ== QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJENERkc2IjYkSSJaR0YpLCQqKCwmJkkieEdGKTYjIiIhIiIiSSJyR0YpIiIpRjNJJnNpZ21hR0YpISIiLUklc3FydEc2JEYlSShfc3lzbGliR0YpNiNGNUY3RjdGMw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIreWg3ME8hIzU= QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJENERkc2IjYkSSJaR0YpLCQqKCwmJkkieEdGKTYjIiIhIiIiSSJyR0YpIiM3RjNJJnNpZ21hR0YpISIiLUklc3FydEc2JEYlSShfc3lzbGliR0YpNiNGNUY3RjdGMw== LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JVE1b3V0cHV0fnJlZGlyZWN0ZWQuLi5GJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= print(); # input placeholder JCIrJilwZnZTISM1 JSFH
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