Put Options and Arbitrage I came across the following problem on put options:

A European put with strike price $100$ expiring in $1$ year has premium $ \$ 1$ and a European put with strike price $K$ expiring in $1$ year has premium $ \$ 2$. The continuously compounded risk free interest rate is $r>1$. What is the full range of values of $K$ that results in an arbitrage opportunity. 

Why do we assume that we buy the $ \$100$ put option and sell the $K$ put option? In other words our position is the following: $$(\max(0, 100-S_1)-1e^{r})- (\max(0, K-S_1)-2e^{r}) >0$$ which means that $K < 100+e^r$ for arbitrage. 
 A: As the premium of the $K$ option is higher than the premium of the $100$ option, $K \gt 100$.  If the price stays above $K$, both options expire worthless and we keep the dollar, worth $e^r$ at the end of the year.  If the price falls below $100$, both are exercised and we have the $e^r+100-K$.  If the price is between $100$ and $K$, say $P$, the $K$ option is exercised and we sell in the market, ending with $e^r+P-K$.If you were to buy the $K$ and sell the $100$, you would be out if the price stayed high.
A: You raise a good point. Suppose $K=300$, for example. Then we could sell three $100$-put options and buy one $300$-put option, netting $1$ unit at time $t=0$. Our wealth at time $t=1$ will be
  $$
  e^r + \max(0,300-S_1) - 3\max(0,100-S_1).
  $$
If $S_1\le100$, then this reduces to
  $$
  e^r + 300 - S_1 - 300 + 3S_1 = e^r + 2S_1 \ge e^r.
  $$
If $100<S_1\le 300$, then this reduces to
  $$
  e^r + 300 - S_1 \ge e^r.
  $$
And if $S_1 > 300$, then this reduces to $e^r$. So it appears we have arbitrage at $K=300$, and the answer $K<100+e^r$ is incomplete.
