We consider 2 European call options with the same underlying asset, the same maturity date $T$ and with 2 different strikes $K_1$ and $K_2$ such that $K_1\leq K_2$. We denote $C^1_{0}$ and $C^{2}_{0}$ their respective prices. Show that under no arbitrage assumption:

$$ C^1_{0} - C^{2}_{0}\leq e^{-rT}(K_2-K_1) $$

with $r$ is the risk-free rate.


The stated equation claims that: Value of call spread <= present value of its maximum payoff. This is almost self evident. For a formal proof, suppose untrue. Then sell the call spread, invest proceeds at r, giving a certain profit at T.


Write your equation as follows:

$$ C_0^1 +e^{-rT}K_1 \leq C_0^2 +e^{-rT}K_2 \quad (1)$$

Both sides consist on the price at $0$ of a portfolio $i$, $i \in \{1,2\}$, containing one option of strike $K_i$ and one risk-free zero-coupon bond of maturity $T$ and principal $K_i$. The payoff $P_i$ at $T$ of each portfolio is $\max(S_T,K_i)$ where $S_T$ is the price of the underlying stock. It comes:

$$ \begin{align} P_2 - P_1 & = \max(S_T,K_2) - \max(S_T,K_1) \\[12pt] & = (K_2-S_T)1_{\{K_2 > S_T > K_1\}} + (K_2-K_1)1_{\{K_1 \geq S_T \}} \end{align}$$

Under all scenarios payoff $P_2$ is greater than payoff $P_1$, hence by no arbitrage portfolio $2$ must have a greater price than portfolio $1$ at any time $0 \leq t \leq T$. The price of each portfolio is equal to the sum of the option price and the zero-coupon bond thus $(1)$ must be enforced, which proves your original inequality.

  • $\begingroup$ Thank so much for your answer. I still have a question, that is the relationship with the payoff and call price, which can prove the inequality. Could you tell me? $\endgroup$ – Son Tran Hoang Jun 11 '17 at 2:41
  • $\begingroup$ I have detailed a bit more the answer. $\endgroup$ – Daneel Olivaw Jun 11 '17 at 11:33

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