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.
- What differential equation do the option values satisfy?
- What is the most general solution of the differential equation?
- 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?
