Why Call Option Prices Cannot Fall with Maturity

Summary

The note explains why, with zero interest and dividend rates, a European call’s value cannot decrease as its expiry moves later. It applies the tower property of conditional expectation and the submartingale property of the positive part of the stock price minus strike. Conditional Jensen inequality supplies the submartingale step when the stock price is a martingale under the pricing measure.

The stated arbitrage intuition is a calendar spread: buy the longer-dated call and sell the shorter-dated call. If this position had a negative initial cost, it would contradict the maturity-price inequality. The argument is limited to the assumptions given, especially zero rates and dividends and the specified martingale setup; with carry, the comparison needs adjustment. The note raises implied volatility patterns as context but does not establish that implied volatility itself must rise with maturity.

Key ideas

• Under the stated zero-rate, zero-dividend assumptions, call value is nondecreasing with expiry.
• The tower property relates the later-expiry payoff to its conditional expectation at the earlier expiry.
• Conditional Jensen inequality makes the call payoff a submartingale when the stock price is a martingale.
• A negative-cost long-later, short-earlier calendar spread would violate the price inequality.

Tags

# Martingale proof: Call-prices must be increasing in maturity


# Martingale proof: Call-prices must be increasing in maturity












I have observed that IV is increasing with time to maturity by using market prices and plotting IV (from Black-Scholes) against log-moneyness, $\log(S_t/K)$. $S_t$ being the price of the stock at time $t$ and $K$ being the strike.

Using Martingales we can prove that the call-option's payoff function - i.e. $\max(S_t-K, 0)$ - is a submartingale under the $Q$-measure. Now this article from Columbia says that the call-price as a function of time to expiry, that is $C_t(T)$, must be not-decreasing to avoid arbitrage, which can be shown using standard martingale results - but why is that?

What are the calculations performed by "standard martinale results" which imply that if the call price was decreasing as a function of $T$ then there would be an arbitrage?

The argument that I do not understand is highlighted here:

## Answer by ir7 (score 3, accepted)

https://quant.stackexchange.com/a/65451

For $r=q=0$ and $t\leq T'\leq T$:

$$ C_t(T)=E_{t}[(S_T -K)^+] = E_{t}[E_{T'}[(S_T -K)^+] \geq E_t[(S_{T'} -K)^+]=C_t(T'),$$

where we used the tower property of conditional expectation and the sub-martingality of $(S_{T'}-K)^+$ they mentioned (which is a consequence of Jensen inequality for conditional expectation).

A calendar spread (one long call with expiry $T$ and one short call with expiry $T'$) with negative price would violate the above inequality.

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