Rocco works for an international arms dealer who has just publicly listed. What with the world being how it is these days, there's money to be made on more than just selling the tools of mass destruction. The price of a Guns'R'U.S. share, S, is $3 today. It will either double or halve depending on whether a secret arms deal with a secret international organisation (represented by someone calling themselves Albert Kaida, or some such. Just call him "Al") goes through. Fearing an American putdown, Rocco must price an American Put option on S, expiring at time 3 using a 3 period Binomial Tree. Assume interest rates are zero. Who pays interest these days? Rocco looks confused. CTF was beyond him. It would be beyond anyone having only a primary school education. He needs your help. The security of the world is in your hands. Or at least a lot of money for Rocco and his associates (and you if you play your cards right).
a) When should someone exercise the option early? Why?
b) Define the Stopping Time in the American Option Pricing algorithm.
c) Show that the discounted American price process is a supermartingale.
d) Show that the discounted Stopped American price process is a martingale.
e) Find a strategy which replicates the payoff of the American Put.
f) Show that this results in additional income if the option is not exercised at the optimal time.