# Price of european call option for different strike prices

Consider two european put options with strike prices $$K, J$$ with $$K and maturity $$T$$. Then the no arbitrage assumption implies $$P_{K}(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.

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

• 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 $K$. – Kevin Apr 5 at 12:52
• @Kevin Ok so there is no other possibility apart from negative interest rates? The proof for the normal case is right isn' it? – Sarah Apr 5 at 13:08
• 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. – Kevin Apr 5 at 13:13
• @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. – Sarah Apr 5 at 13:16
• 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? – Sarah Apr 5 at 13:21