# 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:

## 1 Answer

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.

• Thanks for showing that calculations! Would you mind explaining why this then results in the statement "the term structure of implied volatility cannot be too inverted"? – Kenno Jun 9 at 6:36
• If the pricing of the calls is done via standard Black-Scholes functional $BS_t$, then $C_t(T) = BS_t(T; \sigma)$ and $C_t(T') = BS_t(T'; \sigma')$ for some vol parameters $\sigma$ and $\sigma'$. You can experiment with very low $\sigma$ and very high $\sigma'$ to see if the inequality above gets broken. – ir7 Jun 9 at 14:32
• I'll try calculating some different values. What exactly is $B$ in your formula? I've only seen $B$ in the notation of the 'Risk Free Assest' (i.e. the bank account). – Kenno Jun 9 at 19:23
• BS stands for the standard Black-Scholes formula itself. – ir7 Jun 9 at 19:27