Dividend-Related Early Exercise and Call Option Arbitrage Bounds
Summary
The document examines whether an at-the-money American call on a dividend-paying stock should be exercised before the ex-dividend date, then considers an arbitrage when the quoted call price is below a theoretical lower bound. It explains that the stock price and call value can fall around the dividend date, while early exercise allows the holder to receive the stock and its dividend. The example compares the quoted call price with a bound based on stock value after accounting for the dividend and the present value of the strike.
The proposed trade buys the underpriced call, shorts the stock, and invests the short-sale proceeds net of the option cost. At expiration, the call can close the short if the stock is above the strike; otherwise, the stock can be bought in the market. The document gives numerical figures for the setup, but its calculations and explanations are not fully consistent: it mixes early-exercise reasoning with European-call bounds, and the alternative answer reports a different bound. The argument also abstracts from transaction costs, borrow constraints, and precise dividend timing.
Key ideas
- A dividend can reduce the stock price and affect the value of a call around the ex-dividend date.
- Early exercise of an American call may be worth considering immediately before a dividend, depending on the dividend and remaining time value.
- A call priced below a valid lower bound can motivate a position combining the call, a short stock, and a risk-free investment.
- Arbitrage conclusions depend on consistent option assumptions, dividend timing, and executable market prices.
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Full text
# What is the arbitrage opportunity and strategy here?
# What is the arbitrage opportunity and strategy here?
- Suppose that the current stock price is $€100$, the exercise price is $€100$, the annually compounded interest rate is 5 percent, the stock pays a $€1$ dividend in the next instant, and the quoted call price is $€3.50$ for a one year option. Identify the appropriate arbitrage opportunity and show the appropriate arbitrage strategy.
My answer:
When a company declares a dividend, it specifies that the dividend is payable to all stockholders as of a certain date, called the holder-of-record date. Two business days before the holder-of-record date is the ex-dividend date. To be the stockholder of record by the holder-of-record date, one must buy the stock by the ex-dividend date. The stock price tends to fall by the amount of the dividend on the ex-dividend date.
When a stock goes ex-dividend, the call price drops along with it. The amount by which the call price falls cannot be determined at this point in our understanding of option pricing. Since the call is a means of obtaining the stock, however, its price could never change by more than the stock price change. Thus, the call price will fall by no more than the dividend. An investor could avoid this loss in value by exercising the option immediately before the stock goes ex-dividend. This is the only time the call should be exercised early.
Another way to see that early exercise could occur is to recall that we stated that the lower bound of a European call on a dividend-paying stock is Max $[ 0, S'_0 - X(1 + 0.05)^{-1}]$ where $S'_0$ is the stock price minus the present value of dividends. X is the strike price $€100$. To keep things simple, assume only one dividend of the amount D, and that the stock will go ex-dividend in the next instant. Then $S'_0$ is approximately equal to $S_0 -D= €100-€1= €99 $ (since the present value of D is almost D). Since we would consider exercising only at-the-money call, assume that $S_0= €100 $ equals $X(€100)$. Then it is easy to see that $S_0 - X= €100-€100 = 0$ could not exceed $S'_0 - X(1+0.05)^{-1}= €99 - €100(1+0.05)^{-1}= €3.76 $. By exercising the option, the call holder obtains the value $S_0 -X =€100 -€100= 0 $ Here the quoted market price of the call option is $€3.50$ which is below the lower bound of $€3.76$ of a European call on dividend-paying stock.
Arbitrage Portfolio
If the call price is less than the stock price minus dividend minus the present value of the exercise price, we can construct an arbitrage portfolio. We buy the call and risk-free bonds and sell short the stock. This portfolio has a positive initial cash flow, because the call price plus the bond price $(€3.50 + €100(1+0.05)^{-1}= €98.738)$ is less than the stock price- dividend(€99). At expiration, the payoff is $ X - S_T =€100 - S_T$ if $€100 > S_T$ (Stock price at the time of expiry of option) and zero otherwise.
The portfolio has a positive cash flow today and either a zero or positive cash flow at expiration. Again there is no way to lose money
## Answer by AKdemy (score 3, accepted)
https://quant.stackexchange.com/a/74241
Early exercise of American Call options makes only sense iff $D_n \gt K(1-e^{-r(T-t_n)})$.
The lower bound for American call options is $S_{(t_n)}-D_n - K*exp^{-r(T-t_n)}$.
However, 3.5 < 3.87, in which case the call option is less than the theoretical minimum. An arbitrageur can buy the call and short the stock, to get a cashflow equal to the proceeds of the stock minus the cost of the call. Invested at the prevailing one-year interest rate, you can get a certain payoff at the end of the year, where the option expires. If the stock price is above the strike price, the arbitrageur exercises the option, closes out the short position and makes a profit equal to the difference between investment and strike.
If the stock is less than strike, the stock is bought in the market and the short position is closed out - this will yield an even greater profit.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.