# Is it necessary for $P(K, t) - P(K + s, t) \geq se^{-rt}$ to hold?

Let $$P(K, t)$$ be a put option with strike price $$K$$ and expiration time $$t$$. Let $$s > 0$$. Is it necessarily true that the inequality

$$P(K, t) - P(K + s, t) \geq se^{-rt}$$

holds? I know that for a call option $$C(K, t)$$, it can be shown that the inequality $$C(K, t) - C(K + s, t) \leq se^{-rt}$$ must hold, otherwise there is an arbitrage. In particular, this arbitrage can be obtained by selling a call option with strike price $$K$$ and exercise time $$t$$, buying a $$C(K + s, t)$$ call option, and putting the remaining $$C(K, t) - C(K + s, t) \geq se^{-rt}$$ in a bank.

So, I was wondering whether there is a similar justification for the inequality I mentioned above to hold.

• No, why? If the put is enough out of the money both puts and the difference will be close enough to zero to make the inequality false. – Mats Lind May 20 '19 at 11:13
• Apply the Put Call Parity to the inequality about Calls and see what you get for Puts. – noob2 May 20 '19 at 12:26