# European put options

Why is it that for European Puts on Non-Dividend-Paying Stocks, the lower-bound for price is $$p=Ke^{-rT}-S_0?$$

• It is a consequence of a no arbitrage argument which you can easily workout. Hint: compare a strategy where you initially hold a put with a strategy where you initially hold -1 share of stock and an amount $Ke^{-rT}$ of cash. – Antoine Conze Jan 23 '19 at 15:43

Portfolio B: a zero coupon bond paying off $$K$$ at time $$T$$.
If $$S_T then the option in portfolio A is exercised at $$T$$ and the portfolio is worth $$K$$.
If $$S_T>K$$, then the put option expires worthless and the portfolio is worth $$S_T$$ at this time. Hence, portfolio A is worth $$max(S_T, K)$$ at $$T$$. Portfolio B is worth $$K$$ in time $$T$$. Hence, portfolio A is always worth as much as, and can sometimes be worth more than more than B at $$T$$. It follows then that in the absence of arbitrage opportunities portfolio A must be worth at least as much as B today. Hence: $$p+S_0>= Ke^{-rT}$$ or $$p>=Ke^{-rT}-S_0$$