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A classical no-arbitrage argument shows that for a European put with strike $K$ and time to maturity $T$, the price $p$ satisfies $$p \geq \max(0,Ke^{-rT} - S).$$ Is Black-Scholes in contradiction with this result? I've attached a picture of a Mathematica plot of the price of a long-dated put option against stock prices, which shows a violation of the lower bound. enter image description here

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  • $\begingroup$ Why is this a violation? The lower bound for $S_0 = 0$ is $25 e^{-0.05 \cdot 5} = 19.47$ which is pretty much what I see on your plot. $\endgroup$ – LocalVolatility May 2 '18 at 8:47
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The discounted intrinsic value in your plot is incorrect because you are not discounting up to 5 years maturity. Correct code should be $\max(0,25 \exp(-0.05 \color{red}{\times 5}) - S)$.

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