In this exercise we use the Girsanov theorem to price the chooser option. The chooser option is

Question:

In this exercise we use the Girsanov theorem to price the chooser option. The chooser option is an exotic option that gives the holder the right to choose, at some future date, between a call and a put written on the same underlying asset. Let the T be the expiration date, St be the stock price, K the strike price. If we buy the chooser option at time t, we can choose between call or put with strike K, written on St. At time t the value of the call is

whereas the value of the put is:

and thus, at time t, the chooser option is worth:
H (St, t) = max [C (St, t), P (St, t)]
(a) Using these, show that:
C (St, t) ˆ’ P (St, t) = St ˆ’ Keˆ’r(Tˆ’t)
Does this remind you of a well-known parity condition?
(b) Next, show that the value of the chooser option at time t is given by
H (St, t) = max [C (St, t), C (St, t) - St + Keˆ’r(Tˆ’t)]
(c) Consequently, show that the option price at time zero will be given by

where S0 is the underlying price observed at time zero.
(d) Now comes the point where you use the Girsanov theorem. How can you exploit the Girsanov theorem and evaluate the expectation in the above formula easily?
(e) Write the final formula for the chooser option.