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.

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 $100 300$, then this reduces to $e^r$. So it appears we have arbitrage at $K=300$, and the answer $K<100+e^r$ is incomplete.