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Consider two european put options with strike prices $K, J$ with $K<J$ and maturity $T$. Then the no arbitrage assumption implies $P_{K}(0)<P_J(0)$, where $P_K(0)$ denotes the price of the put with strike $K$ at time $t=0$.

Proof: Assume otherwise, then sell option with strike $K$ and buy option with strike $S$, yielding a instant profit of $P_K(0) - P_S(0) \geq 0$. This can be invested in an risk free asset, i.e. buy $P_K(0) - P_S(0)/ B(0,T)$ zero coupon bonds. $B(0,T)$ denotes the price of such a bond with maturity $T$.

At maturity $T$ there are two cases to distinguish. If the underlying exeeds the strike $K$, there is nothing to do. In the other case you can sell the buyer of the put with strike $K$ the underlying for $K$ which can be financed by exercising the bought put recieving $J>K $.

So in each case you made a profit out of nothing.

I hope that is the right argument. My question is, whether it is possible to have $P_{K}(0)=P_J(0)$ for $K<J$.

Can this be the case if you have negative interest rates?

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  • $\begingroup$ The payoff itself, $\max\{K-S_T,0\}$, is always monotone in the strike price $K$. So, you need a weird discount factor. Suppose time is discrete and interest rates are fixed. Then, $P(K)=\frac{1}{1+rT}\mathbb{E}^\mathbb{Q}[\max\{K-S_T,0\}$. Suppose $T=1$ and $r=-150\%$, then put option prices would monotonically decline in the strike price. Of course, this is more than a pathological example. In continuous time, you need $r>-1$, so this kind of argument wouldn't apply to the Black-Scholes model, where puts are always monotone in $K$. $\endgroup$ – Kevin Apr 5 at 12:52
  • $\begingroup$ @Kevin Ok so there is no other possibility apart from negative interest rates? The proof for the normal case is right isn' it? $\endgroup$ – Sarah Apr 5 at 13:08
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    $\begingroup$ Yes, I think so. The option price is the expected discounted payoff. Because the payoff is monotone in $K$, you need the discount factor to flip monotonicity. This only works if discrete interest rates are super negative. $\endgroup$ – Kevin Apr 5 at 13:13
  • $\begingroup$ @Kevin. Thank you for your answer. This would also imply that that the price of the zero coupon bond $B(0,T) >1 $. So everything should make sense then. $\endgroup$ – Sarah Apr 5 at 13:16
  • $\begingroup$ You mentioned he Black-Scholes-Model. I wonder why the considered asset have to be Ito-processes. The reason might be that you have a larger set of integrands apart from only Brownian motion? $\endgroup$ – Sarah Apr 5 at 13:21

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