In a pure diffusion setting, you can equivalently write no calendar arbitrage constraints:
In terms of implied volatility: total implied variance should be non decreasing in time, and that, for any given forward moneyness level, see Gatheral top of page 4.
In terms of European option prices: see Gatheral end of page 3.
The price-based constraint builds on the following lemma
[Lemma] If $X_t$ is a martingale, $L < \infty$ a real constant and $0 < t_1 < t_2$ two future times, then
$$E[(X_{t_2}-L)^+] \geq E [(X_{t_1}-L)^+] $$
[Proof]
\begin{align}
E[(X_{t_2}-L)^+ \vert \mathcal {F}_0 ] &= E[\ E [(X_{t_2}-L)^+ \ \vert \mathcal {F}_1 \ ]\ \vert \mathcal {F}_0 ] \\
&\geq E[\ \left( E[X_{t_2}-L \ \vert \mathcal {F}_1 \ ] \right)^+ \ \vert \mathcal {F}_0 ] \\
&\geq E[ (X_{t_1}-L)^+ \ \vert \mathcal {F}_0 ]
\end{align}
where we have used, in respective order:
- Tower property of conditional expectation
- Jensen's inequality ($f : x \rightarrow x^+$ is a convex function)
- The fact that $X_t$ is a martingale
In a more general setting, one can still use this lemma to derive price-based constraints. The only question is: what martingale $X_t$ should we consider?
[Proportional dividends]
In a pure diffusion setting, it made sense to use $X_t = S_t/F (0,t) $, because $dS_t/S_t = \mu_t dt + \sigma_t dW_t \Rightarrow S_t = F(0,t)X_t$, where $X_t=\mathcal{E}(\int_0^t \sigma_s dW_s)$ is indeed a martingale (Doléans-Dade exponential). Applying the lemma then gives:
\begin{align*}
& E[(X_{t_2}-L)^+] \geq E [(X_{t_1}-L)^+] \\
\iff & E\left[\left(\frac{S_{t_2}}{F(0,t_2)}-L\right)^+\right] \geq E \left[\left(\frac{S_{t_1}}{F(0,t_1)}-L\right)^+\right] \\
\iff & \frac{1}{F(0,t_2)} E[(S_{t_2}-LF(0,t_2))^+] \geq \frac{1}{F(0,t_1)} E[(S_{t_1}-LF(0,t_1))^+] \\
\iff & \frac{\tilde{C}(K_2,t_2)}{F(0,t_2)} \geq \frac{\tilde{C}(K_1,t_1)}{F(0,t_1)}
\end{align*}
where $\tilde{C}(K,T)$ denotes an undiscounted call price and $K_1=LF(0,t_1)$, $K_2=LF(0,t_2)$. This is precisely Gatheral's result:
$$ \frac{C_2}{K_2} \geq \frac{C_1}{K_1} $$
(in his paper, he always uses undiscounted call prices and he chose $L=e^k$), since
$$ \frac{K_2}{K_1} = \frac{F(0,t_2)}{F(0,t_1)} $$
[Cash & Proportional dividends]
In a more elaborate setting, it will depend on how you model dividends. Buehler for instance suggests a no-arbitrage pricing framework which can accommodate cash dividends, proportional dividends, and/or any mix of the two. In his model, it makes sense to use the martingale $X_t = (S_t-D_t)/(F (0,t) - D_t) $ where $D_t $ is related to the future dividend stream (all divs are assumed to be known in advance). Applying the lemma gives:
\begin{align*}
& E[(X_{t_2}-L)^+] \geq E [(X_{t_1}-L)^+] \\
\iff & E\left[\left(\frac{S_{t_2}-D_{t_2}}{F(0,t_2)-D_{t_2}}-L\right)^+\right] \geq E \left[\left(\frac{S_{t_1}-D_{t_1}}{F(0,t_1)-D_{t_1}}-L\right)^+\right] \\
\iff & \frac{E[(S_{t_2}-(D_{t_2}+L(F(0,t_2)-D_{t_2})))^+] }{F(0,t_2)-D_{t_2}} \geq \frac{E[(S_{t_1}-(D_{t_1}+L(F(0,t_1)-D_{t_1})))^+]}{F(0,t_1)-D_{t_1}} \\
\iff & \frac{\tilde{C}(K_2,t_2)}{F(0,t_2)-D_{t_2}} \geq \frac{\tilde{C}(K_1,t_1)}{F(0,t_1)-D_{t_1}}
\end{align*}
where $\tilde{C}(K,T)$ denotes an undiscounted call price and $K_1=D_{t_1}+L(F(0,t_1)-D_{t_1}))$, $K_2=D_{t_2}+L(F(0,t_2)-D_{t_2}))$.
Observe that when $(D_t)_{t\geq0} = 0$, we fall-back on Gatheral's result. In Buehler, $(D_t)_{t\geq0} = 0$ iff there are no cash dividends (meaning there could be either proportional dividends or no dividends at all). This is completely consistent with what we have said in the pure diffusion case.
[Arbitrage opportunity]
Finally, note that the above inequalities describe the calendar arbitrage opportunities. In the second situation for instance, a PF where you are long $\tilde{C}(K=K_2,T=t_2)$ and short $(F(0,t_2)-D_{t_2})/(F(0,t_1)-D_{t_1}) $ units of $\tilde{C}(K=K_1,T=t_1)$ should always have a positive value (we just showed that). If not, it is an arbitrage opportunity.
-Y*exp(r*T1)*C(t2), do you meanBSwhere you sayC? If not, I don't see why a call's price would always change by the risk-free interest rate (even if the underlying remained at the same price, there would be theta decay). I know you're assuming the same strike price, but if the underlying price drops, wouldn'tC(t2)be considerably lower? $\endgroup$