# Calendar arbitrage in implied vol grid with discrete and proportional dividends

I have an implied vol discrete grid, obtained from market data. To obtain prices from these implied vols, a dividend model with discrete and proportional dividends is used.

How can I verify if there are calendar arbitrages in this implied vol grid?

This paper (Gatheral, Jacquier. Arbitrage free SVI volatility surfaces) says, in Lemma 2.1 on page 3, that if dividends are proportional only, then verifying that the total variance is increasing is a necessary and sufficient condition of no-arbitrage.

Though, in my case dividends are not only proportional, there are also cash dividends.

On page 4 of the same paper, definition 2.2 (without proof) says that the increasing total variance is a sufficient condition of non-arbitrage, without specifying any assumption on dividends.

I want to know if I interpreted correctly this definition 2.2, i.e. if there are both cash and proportional dividends, then

• If I demonstrate that total variance is increasing, then I am sure that there is no arbitrage in the vol grid.
• If there are some points where total variance is not increasing, I am not sure that if there is or if there is not arbitrage on that point, i.e. the increasing total variance condition is sufficient for non-arbitrage, but not necessary.
• Mathematically: if dividends are both discrete with cash and proportional, then $$\frac{\partial}{\partial T} \left[ \sigma^{2}(k, T)T \right] >0\Rightarrow \text{No-arbitrage}$$

Are these 3 assertions correct?

## 2 Answers

This is a great reference on discrete fixed/proportional dividends with local volatility. In your case you need to consider the pure stock process with the dividends removed and consider the conditions in section 3.2.

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1141877

There’s also a follow up paper to this one here

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2639318

• Sorry, this does not answer my question. Commented Apr 6, 2022 at 21:01

Proposition 2.1 on this paper, the calendar arbitrage condition is derived from portfolio of "a short position on short tenor, $$T_1$$, call and a long position on long tenor, $$T_2$$, call".

In the case of the cash dividend, the value of portfolio at time $$t = T_1$$ is equal to a put option minus the present value of the cash dividends between 2 tenors due to call-put parity.

Therefore, calendar arbitrage exists if $$C(K_1,T_1) > e^{-\int_{T_1}^{T_2} \delta_tdt} C(K_2,T_2) + PV(Cash Dividends)$$