Calendar Spread Inequality for European Call Options
Summary
The document states a calendar-spread inequality for European calls on the same non-dividend-paying stock with different maturities. The shorter-dated option uses a lower strike, discounted to account for the interval between expiries; the claim is that its value cannot exceed that of the longer-dated call at the original strike. The proposed proof compares the two portfolios at the earlier expiry, applies put-call parity, and uses the non-negativity of the corresponding put to establish an ordering of their values.
The answer offers a shorter route for zero interest rates: express the later call payoff through conditional expectation, apply Jensen’s inequality to the convex positive-part payoff, then use the martingale property of the stock price. The response says the original argument’s general idea seems sound but does not verify every detail, and leaves extension to nonzero rates to the reader. The setup relies on no arbitrage and the stated Black-Scholes-style assumptions; the excerpt does not discuss dividends, early exercise, or broader market conditions. It is a proof discussion, not an empirical trading result.
Key ideas
- The inequality compares calls with different expiries and appropriately adjusted strikes.
- The proposed proof orders portfolio values at the earlier expiry using put-call parity and nonnegative put value.
- For zero rates, conditional expectation, Jensen’s inequality, and the stock martingale property give a shorter argument.
- The response does not fully verify the original proof or derive the nonzero-rate extension.
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Full text
# Proof of Calendar-Spread-Inequality
# Proof of Calendar-Spread-Inequality
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 by user34971 (score 1, accepted)
https://quant.stackexchange.com/a/69582
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}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.