Put Option Prices Increase with Strike Under No Arbitrage

Summary

This discussion examines whether European put options with the same maturity can have equal prices when one has a higher strike. It presents a no-arbitrage argument: if the lower-strike put were at least as expensive as the higher-strike put, an investor could sell the former, buy the latter, and invest any initial proceeds. At expiry, the higher-strike put would cover the obligation created by the lower-strike put, leaving a nonnegative payoff in every case.

The argument motivates the usual monotonicity of put prices across strikes, but the post includes an uncertain proof and asks whether equality could occur under negative interest rates. It does not resolve that caveat. Strict inequalities also depend on assumptions about payoff conventions and market frictions; the post offers no numerical example or broader treatment of those conditions.

Key ideas

• For puts with the same expiry, a higher strike gives the holder a payoff at least as large in every underlying-price state.
• A proposed arbitrage sells the lower-strike put, buys the higher-strike put, and invests any initial price difference.
• The discussion raises negative interest rates as a possible caveat but does not answer it.
• The argument assumes trades and financing can be arranged without frictions.

Tags

# Price of european call option for different strike prices


# Price of european call option for different strike prices












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.