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I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, but that would only leave the stock price, $S_t$, on the right-hand side, if I am not mistaken. This seems a bit too simplistic in my opinion, especially since questions 2 and 3 are asking to elaborate on the answer from part 1. Any help/hints would be much appreciated. Thanks!

A stock whose price S follows geometric Brownian motion, $\frac{dS}{S} = \mu dt+ \sigma dB$, has options that never expire.

  1. What differential equation do the option values satisfy?
  2. What is the most general solution of the differential equation?
  3. Consider the cases of American-style call and put options of strike K that are initially out of the money. Does the general solution hold in each case? Does it hold for all values of S and t?
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