Question
Consider a one-year forward binomial model for the stock-price movement over the following year. The current stock price is $S_0 = 100$, its dividend yield is $5\%$ and its volatility is $30\%$. The continuously compounded risk-free rate is $5\%$. Consider also an American call option on this stock maturing at the end of the year. What is the maximum strike price $K$ for which there is early exercise?
My working
I have calculated the $u$ and $d$ factors as $e^{0.3}$ and $e^{-0.3}$ respectively, meaning that $S_u = 100e^{0.3}$ and $S_d = 100e^{-0.3}$. I also know that $p = \frac {1 - e^{-0.3}} {e^{0.3} - e^{-0.3}}$.
Now, if we want early exercise, then we must have $$\max\{100 - K, 0\} > [p\max\{100e^{0.3} - K, 0\} + (1 - p)\max\{100e^{-0.3} - K, 0\}]e^{-0.3}.$$ However, this is where I am stuck. I am able to narrow $K$ down to $$100 < K < 100e^{0.3},$$ but that is all. In particular, is there any way to solve the above inequality in terms of $K$?
Any thoughts will be greatly appreciated :)