# Option arbitrage with dividends?

If a stock pays a discrete dividend, the stock price falls by the amount of the dividend. There is no arbitrage opportunity from this predictable jump, because the investors receive the same amount of price depreciation back in cash from the dividend.

But how about options? If the underlying jumps down at the dividend payment, the call option holder receives a direct loss without any compensation in cash as the dividend is only paid to the stock owners and the stock price is now worth less for exercise. So isnt there an arbitrage opportunity to short all call before the dividend and buy them back cheaper afterwards?

## 2 Answers

Fact 1: if you are not good at pricing options, of course you can create a lot of arbitrage opportunities for the rest of the market. It does not matter whether the reason is in dividends or anything else.

Fact 2: if you are good in pricing options, you price the dividend effect in advance. Consider the situation of the European calls, and suppose that both the volatility and the drift are zero. The stock is at a 100, and there is gonna be a dividend of 4 before expiry. If you hold an ITM option with a strike 90, you would not price is at 10 = 100 - 90, rather you would price it at 6 = (100 - 4) - 90. They also say that European options expire on the forward rather than on the spot, notice that the forward for that expiry would be exactly 96 = 100 - 4. At the same time, American call would still be 10 worth since you would exercise it just before the dividend drop out time. Essentially, when pricing option around dividend one makes sure that the price of the option along the path of the stock (which does feature the jump) stay continuous: $V(S_{t-},t-) = V(S_{t+},t+)$ - see any book by Wilmott, perhaps also appears in Hull (maybe M. Joshi also has that). That's done exactly to avoid arbitrage. It's like you know in advance that your call would not be worth that much, so why would you buy it for such price?

• Ok but "6 = (100 - 5) - 90"? ; ) – emcor Jun 10 '15 at 15:04
• @emcor: there was an example with dividend of 5, but then I realized that the option price is also 5, so a possible confusion may arise. Decided to change it to 4, but not everywhere apparently, thanks for spotting this. – Ulysses Jun 11 '15 at 10:58

Generally no, because 'dividends' are already 'priced into' the options. Which means, if an ATM call cost 0.50, and stock price drops by 1.00(amount of dividend), the ATM becomes OTM, but it may still cost 0.50, because the initial price of 0.50 already factored in the dividend.

• Can you show a model for this? E.g. Black Scholes with discrete dividends? – emcor Jun 10 '15 at 14:10