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Checking Calendar Arbitrage with Cash and Proportional Dividends

Article Quant Q&A · Author: Joanna
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Summary

The discussion asks whether increasing total implied variance is a valid calendar-arbitrage test when an equity pays both discrete cash dividends and proportional dividends. It contrasts a result attributed to Gatheral and Jacquier, where increasing total variance is necessary and sufficient under proportional dividends, with a definition described as giving a sufficient condition more generally. The question is whether that broader statement applies to the mixed dividend setting and whether a failure of monotonicity proves arbitrage.

The replies point to work on local volatility with discrete fixed and proportional dividends, suggesting that the stock process should be adjusted to remove dividends and that conditions in a cited section should be examined. Another reply frames the test through a portfolio short one shorter-dated call and long one longer-dated call. Using put-call parity, it states a price inequality involving discounting and the present value of intervening cash dividends. The exchange does not fully resolve all three assertions or establish that a variance-grid test alone suffices for every dividend model; its references and abbreviated derivation warrant checking against the cited papers.

Key ideas

  • Increasing total variance is stated as necessary and sufficient for avoiding calendar arbitrage under proportional dividends in the cited result.
  • The discussion raises whether that condition remains sufficient when cash dividends are also present.
  • A non-increasing variance point does not by itself settle whether an arbitrage exists, according to the question's interpretation.
  • One reply recommends analyzing a stock process with dividends removed under a local-volatility framework.
  • Another reply derives a calendar condition from a spread of calls and put-call parity, accounting for intervening cash dividends.

Tags

OptionsVolatilityArbitrageDerivatives pricing
Full text
# Calendar arbitrage in implied vol grid with discrete and proportional dividends


# 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.



Are these 3 assertions correct?

## Answer by BrownianBread (score 1)

https://quant.stackexchange.com/a/70418

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

## Answer by StupidMan (score 0)

https://quant.stackexchange.com/a/74949

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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