I own options on a stock, and it's just announced a split. What happens to my options?
When the underlying stock of your option splits or even begins issuing a stock dividend, the contract undergoes an adjustment that is often referred to as "being made whole," which means the option contact is modified accordingly so that you are neither negatively nor positively affected by the corporate action. While a stock split will adjust the price of the underlying security of an option, the option is adjusted so that any changes in price due to the split do not affect the value of the option. Keep in mind that if your option is purchased post-split (i. e. after the split is announced) then your option will not be adjusted as it will have been created according to the new split price of the underlying security.
How do you calculate what the new option will be worth? Well, you typically don't need to worry about such things because the Options Clearing Corporation will automatically do it for you (for the sake of orderly and smooth markets). But if you are wondering how it's done, the calculation is relatively straightforward. Each option contract is typically in control of 100 shares of an underlying security at a predetermined strike price. To find the new coverage of the option, take the split ratio and multiply by the old coverage (normally 100 shares). To find the new strike price, take the old strike price and divide by the split ratio. Say, for example, you own a call for 100 shares of XYZ with a strike price of $75. Now, if XYZ had a stock split of 2 for 1, then the option would now be for 200 shares with a strike price of $37.50. If, on the other hand, the stock split was 3 for 2, then the option would be for 150 shares with a strike price of $50.

How does stock splits affect options.
February exisitng home sales rose 5. When a stock splits, the OCC or Options Clearing Corporation , automatically adjusts your options holding through your option trading broker to reflect the proportion of the split such that you too will end up with a net position value which is equivalent to before the split. The following chart may help you to see the differences. You then own three times as many calls. Typical split ratios include 2 for 1, 3 for 1, 3 for 2 and 5 for 2. This is what we commonly call a 2 for 1 split and which is the most common form of stock split.
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Stock Market : How Stock Splits Affect Call Options.
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How do stock splits affect call options

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How splits and dividends affect option prices.
When a stock goes through a split or offers a dividend, do the option prices automatically change? If yes, how? A 2 for 1 split will automatically halve the option price? A dividend of 1$ will automatically decrease the option price by 1$.
Does the answer depend on the strike and expiry?
Investopedia explains how a stock split impacts the stock's options:
Each option contract is typically in control of 100 shares of an underlying security at a predetermined strike price. To find the new coverage of the option, take the split ratio and multiply by the old coverage (normally 100 shares). To find the new strike price, take the old strike price and divide by the split ratio. Say, for example, you own a call for 100 shares of XYZ with a strike price of $75. Now, if XYZ had a stock split of 2 for 1, then the option would now be for 200 shares with a strike price of $37.50. If, on the other hand, the stock split was 3 for 2, then the option would be for 150 shares with a strike price of $50.
So, yes, a 2 for 1 stock split would halve the option strike prices. Also, in case the Investopedia article isn't clear, after a split the options still control 100 shares per contract.
Regarding how a dividend affects option prices, I found an article with a good explanation:
As mentioned above, dividends payment could reduce the price of a stock due to reduction of the company's assets. It becomes intuitive to know that if a stock is expected to go down, its call options will drop in extrinsic value while its put options will gain in extrinsic value before it happens. Indeed, dividends deflate the extrinsic value of call options and inflate the extrinsic value of put options weeks or even months before an expected dividend payment.
Extrinsic value of Call Options are deflated due to dividends not only because of an expected reduction in the price of the stock but also due to the fact that call options buyers do not get paid the dividends that the stock buyers do. This makes call options of dividend paying stocks less attractive to own than the stocks itself, thereby depressing its extrinsic value. How much the value of call options drop due to dividends is really a function of its moneyness. In the money call options with high delta would be expected to drop the most on ex-date while out of the money call options with lower delta would be least affected.
If a stock is expected to drop by a certain amount, that drop would already have been priced into the extrinsic value of its put options way beforehand. This is what happens to put options of dividend paying stocks. This effect is again a function of options moneyness but this time, in the money put options raise in extrinsic value more than out of the money put options. This is because in the money put options with delta of close to -1 would gain almost dollar or dollar on the drop of a stock. As such, in the money put options would rise in extrinsic value almost as much as the dividend rate itself while out of the money put options may not experience any changes since the dividend effect may not be strong enough to bring the stock down to take those out of the money put options in the money.
So, no, a dividend of $1 will not necessarily decrease an option's price by $1 on the ex-dividend date. It depends on whether it's a call or put option, and whether the option is "in the money" or "out of the money" and by how much.

What Happens To Options During Stock Splits.
What Happens To Options During Stock Splits - Introduction.
"What Happens To Options During Stock Splits?" This is perhaps one of the first questions beginner option traders ask shortly after option trading for real. This is an extremely important question to answer as stock splits does happen often and not knowing what's going on definitely throws every amateur options trader into disarray and confusion, leading to all the wrong actions.
What Happens To Options During Stock Splits - What Is A Stock Split?
A stock split happens when a company "splits" its shares up into smaller portions while maintaining overall share capital. A company with 10,000 shares trading at $50 can split into 20,000 shares of $25. This is what we commonly call a 2 for 1 split and which is the most common form of stock split. If you are holding shares 100 shares of that company at $50 before the split, you will end up with 200 shares of $25 after a 2 for 1 split. But what happens when you are holding options instead?
What Happens To Options During Stock Splits?
When a stock splits, the OCC or Options Clearing Corporation, automatically adjusts your options holding through your option trading broker to reflect the proportion of the split such that you too will end up with a net position value which is equivalent to before the split. If you own 1 contract of $50 strike price call options on the company mentioned above valued at $2 per contract on the day of a 2 for 1 split, you will end up with 2 contracts of $25 strike price call options valued at $1 per contract. Let's look at the net effect of the split:
This is also the same treatment if you owned employee stock options. Options that underwent such adjustments are known as Adjusted Options.
What Happens To Options During Stock Splits - The Drawback.
While this adjustment to your stock options may seem like a fair deal, it does change some things. First of all, it increases the number of options contracts that you are holding, which may or may not conform to your option trading plan. The more options contracts you are holding, the higher the real dollar loss in the short term should the stock take a ditch. Let's assume that the company's stocks fall from $25 to $24 right after the stock split.