# Calendar Arbitrage in a Vol Surface

I am trying to determine the condition such that my implied vol surface doesn't have calendar arbitrage. I have done research and found that one such condition is that total variance should increase along the time axis. However, I want to find a different condition using the call option price or forwards, or something to that extent.

Furthermore, I do not want to assume proportional dividends, same forward moneyness, etc. The information I do know is option price and forward prices.

My approach is something as follows. Let X and Y be unknowns. at t=0, I would need to pay (or receive) $XC(t_1)+YC(t_2)$ where $C(T)=\exp(-rT)BS(F_T,K,T,r,\sigma)$. Note that I am assuming that we are working with the same strike $K$. Let $X=1$ for simplicty. At $t=1$, if $S_{t_1}<K$, then my call expiring at $t_1$ would be worth nothing, and closing out the portfolio position, the payoff would be $-Y\exp(rT_1)C(t_2)$. If $S_{t_1}>K$, then my payoff would be $S_{t_1}-K-Y\exp(rT_1)C(t_2)$.

I'm not sure how I would continue my argument from here, though perhaps I want to use the fact that $C(t) \ge \exp(-rt)(F_t-K)$. I know I would first need to find out the quantity of $Y$ first.

Any help would be greatly appreciated. Jim

• I'm missing something. Where you say -Y*exp(r*T1)*C(t2), do you mean BS where you say C? 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't C(t2) be considerably lower? – user59 May 5 '16 at 0:08

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.

• If you don't assume proportional dividends then there is only one way to specify the dynamics of 'the stock' properly, see H. Buehler's work. Anyway, I guess you'll start from the result I described above to find what you need, you just need to find the right martingale now! – Quantuple Mar 14 '16 at 20:07
• I perfectly understand Gatheral's result. My main thing is to create an arbitrage portfolio instead, assuming forward prices that are not based on proportional dividends. – Jim Mar 15 '16 at 9:33
• Ok. All I am saying is that you can use the same result to obtain a calendar arbitrage inequality, even with non prop dividends. For instance if you use the modelling assumptions described in \url{papers.ssrn.com/sol3/papers.cfm?abstract_id=1141877}, you see that $X_t = (S_t - D_t)/(F_t - D_t)$ is a martingale with $F_t$ the forward price and $D_t$ can be computed from the expected div stream. – Quantuple Mar 15 '16 at 10:35
• Note that this is equivalent to applying the classic no arbitrage relationships in a cleansed domain where you have got rid of the effect of future (deterministic) div payments. Maybe chap 4 (esp. Section 4.3.1.) of this thesis will help you understand what I mean? \url {google.be/url?sa=t&source=web&rct=j&url=http://…} – Quantuple Mar 15 '16 at 11:13
• Quantuple, seems like a very interesting solution. In the gatheral paper, the European option price condition is that $\frac{C_2}{K_2}>\frac{C_1}{K_1}$. How would this change if you were using the different martingale $X_t=(S_t-D_t)/(F_t-D_t)$ – user19928 Mar 16 '16 at 23:13