Adjusting Option Terms for Stock Dividends (FRM Part 1)

1. Context

Here, we take a look at how option contracts are adjusted for stock dividends and stock splits. The details of the reading in which this topic appears are given below:

Area	Financial Markets and Products
Reading	Mechanics of Option Markets
Reference	Hull, John, Ch10. Mechanics of Options Markets. In Options, Futures, and Other Derivatives, 9th Edition, Pearson, New York, 2014.

2. Adjusting for Cash Dividends

Exchanges do not make any adjustment to option terms on account of cash dividends paid by underlying stocks. Stock prices do fall on ex-dividend dates on account of these dividends, and option values are affected. However, the impact of cash dividends is already priced into the premium that was paid for the option at the time of purchase. Some OTC options are dividend-protected (for cash dividends). If the firm declares a cash dividend, the strike price of such a dividend-protected option is reduced on the ex-dividend date by the amount of the dividend paid. This adjustment of strike price offsets the drop in the stock price owing the dividend and the option holder is unharmed.

3. Stock Dividends and Stock Splits

Instead of cash dividends, firms can elect to pay dividends in stock wherein current holder’s of the firm’s stock receive more stock in direct proportion to their current holdings. Further, in a $n$-for-$m$ stock split, the firm replaces every $m$ shares held by an investor by $n$ shares. The impact of both a stock dividend and splits is to increase the number of outstanding shares of the firm (and reduce the price of shares). A $q\%$ stock dividend can be effectively seen as a $(1+q)$-for-$1$ stock split.

4. Adjusting for Stock Splits

If the option’s terms are not accurately adjusted, stock splits alter it’s value radically. Consider a ATM call option on a stock that is currently trading at 200USD. If this stock were to undergo a 2-for-1 split it would bring down the price of the stock to 100USD (in an idealized scenario where factors such as additional interest in the stock / more liquidity do not affect the stock price) and double the number of shares. If the exercise price of the option were not adjusted, the call option would be worthless as it becomes virtually impossible for the stock price to exceed exercise price prior to maturity.

In general, a $n$-for-$m$ stock split takes the stock’s price (at time $t$) from $S_t$ to $\frac{m}{n}S_t$. The exchange makes the following adjustments to account for this split:

Strike	$K \rightarrow \frac{m}{n}K$
#Shares or #Options	$N \rightarrow \frac{n}{m}N$

The convention is increase the number of shares covered by each option (usually, 100 shares) in proportion to the stock dividend (i.e. a 15% stock dividend changes the number of shares from 100 to 115 and adjusts the exercise price by $1/1.15$). If the stock dividend or stock split results in new number shares being a multiple of 100, holders of outstanding option contracts are credited with more contracts. In either case, the effect is to alter the total number of shares received upon exercise. However, holding more number of options does affect transaction costs if you were to trade out of them prior to maturity.

5. Is the investor getting the promised payoff?

Yes, he is. We know that the original payoff to the investor (before the stock split or stock dividend) was given by: $$N\times\max\left( S_T – K, 0 \right)$$ After the split and with the proposed adjustments, the payoff has now become: $$\left(\frac{n}{m}N\right)\times\max\left( \left( \frac{m}{n}S_T \right) – \left( \frac{m}{n}K \right), 0 \right)$$ which is the same as before as algebraically the adjustments cancel out.

6. Does it affect the option’s current value?

No, it doesn’t. As we have seen above, the exercise (intrinsic) value of the option is unaffected by the stock split or stock dividends (thanks to the adjustments made by the exchange). Even the time value is not affected since the parameter which the time value relates to i.e. volatility is unaffected by stock splits. This is because the volatility that we input into a pricing model (such as Black Scholes) is the volatility of stock price returns (and not of stock price, the volatility of which will undoubtedly be affected by the split).

Owing to the $n$-for-$m$ stock split, as the stock price goes from $S_t$ to $\frac{m}{n}S_t$, it’s price volatility will go from $\sigma_S$ to $\frac{m}{n}\sigma_S$, but it’s return volatility $\sigma$ stays the same $\because \mbox{var}\left(\frac{\Delta\left(\frac{mS_t}{n}\right)}{\frac{mS_t}{n}}\right)=\mbox{var}\left(\frac{\Delta S_t}{S_t}\right)$