a)    The option to expand the capacity of a project can be viewed as owning what kind of option written on the underlying project?  Explain

b)    The option to shutdown a project can be viewed as owning what kind of option written on the underlying project?  Explain

c)    Gotham city faces the risk of attack based on threats it has received from the Joker, who is threatening to destroy a number of important pieces of infrastructure.  The police commissioner has estimated that with an expenditure of $10 million per week (the city does not pay taxes), they can reduce the risk of an attack by half.  Assuming an initial probability of 30% that the attack will succeed and no one is actually injured in the attack, what is the minimum amount of economic damage the Joker would have to inflict in an attack to justify this expenditure?

d)    Assuming a successful attack would inflict $150 million worth of economic damage (and again, no human harm or further externalities are incurred) and that the probability of attack falls by a factor of 0.9 with each week that passes, after how many weeks should the city stop paying for the additional $10 million in police services? (you cannot pay for fractional weeks of extra protection)

e)    How does the volatility of outcomes (in terms of both probability and damage costs) impact the value of paying for extra protection?