Proof of Calendar-Spread-Inequality

Question

The Calendar-Spread-Inequality compares the prices of two European Call Options on the same underlying non-dividend-paying stock, but with different maturities $T_1<T_2$. Denote the value of a call option with strike $K$ and maturity $T$ at time $t\leq T$ as $C_K(t,T)$ (a put will be denoted by $P_K(t,T)$). The calendar-spread-inequality then states:

$$C_{K'}(t,T_1)\leq C_K(t,T_2),$$

where $K'=Ke^{-r(T_2-T_1)}$.

To prove this we first consider the Monotonicity Lemma (see "An Introduction to Quantitative Finance" by Stephen Blyth), which states that, under the no-arbitrage assumption, if two portfolios, $A$ and $B$, have values $V^A(T')\leq V^B(T')$ at time $T'$, their values must obey $V^A(t)\leq V^B(t)$ at any time $t\leq T'$. (A proof will not be given here.)

Now, consider two portfolios, $A$ and $B$, with $A=\{\text{own one call with strike $K'$ and maturity $T_1$}\}$ and $B=\{\text{own one call with strike $K$ and maturity $T_2$}\}$. At time $T_1$ we have $V^A(T_1)=\text{max}\{0,S_{T_1}-K'\}$.

According to Put-Call-Parity for European options: $C_K(t,T)-P_K(t,T)=S_t-Ke^{-r(T-t)}$, which means that \begin{align} C_{K'}(T_1,T_1)=V^A(T_1)&=\text{max}\{0, S_{T_1}-Ke^{-r(T_2-T_1)}\}\\ &=\text{max}\{0,C_K(T_1,T_2)-P_K(T_1,T_2)\}\\ &\leq C_K(T_1,T_2)=V^B(T_1), \end{align} since $P_K\geq0$. By the monotonicity lemma we find that $C_{K'}(t,T_1)\leq C_K(t,T_2)$ for $t\leq T_1$.

Is this proof correct or am I missing something?

Answer 1

I have not checked all the details of your proof although I believe the general idea is fine.

Here is a shorter proof with $r=0$ which I hope is helpful for you as well. I'll leave $r\neq0$ for you to generalise. Let $T > t$, then:

\begin{align} E_0 \left[ (S_T - K)_+ \right] &= E_0 \left[ E_t \left[ (S_T - K)_+ \right] \right] \\ &\geq E_0 \left[ ( E_t (S_T) - K)_+ \right] \\ &= E_0 \left[ ( S_t - K)_+ \right]. \end{align}