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Options are said to constitute in essence a legal contract for one party to buy or sell an assets at a set price in a future time. In modern trading, is there also an actual contract in fact, that is being written and said? How is it phrased, exactly? Where can I see a realistic example of such a contract?

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There is a document titled Characteristics and Risks of Standardized Options. It is 204 pages and not easily summarized.

The concept of an option is simple. I see Apple is trading at $200, and think it will go up, big time. I see that I can buy a $250 Jan '20 call for $20. I pay $2000 for the right, but not the obligation, to buy 100 shares at $250 ea until the day of expiration. My purchase is done via voice request or clicks on a web page. I don't have a written 'contract' for each purchase. The word 'contract' is what the terms of the purchase are called. (i.e. to me a 'contract' is an option to buy/sell 100 shares of XYZ stock at $ZZ)

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A financial option is a contract between two counterparties. It has specified terms (strike price and expiration date).

For OTC contracts, there is a written agreement. For exchange traded options, there is simply a book entry in your account when you buy or sell to open an option contract. Such contracts are standardized and guaranteed by the Options Clearing Corporation (OCC) and they are settled through a clearing house.

Examples of standardized contracts can be found in the quote chains for any underlying that offers options.

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For listed options, you can usually find the features of said option on the exchange that sells this product or perhaps on your broker's information page. For example, here are the contract specs for CME crude oil options:

https://www.cmegroup.com/trading/energy/crude-oil/light-sweet-crude_contractSpecs_options.html?optionProductId=190#optionProductId=190

Notice that some terms contain links to further information. Furthermore, exchanges must create and enforce rulebooks that govern all non-product specific aspects of trading on their platform. Here you can find CME's rulebook:

https://www.cmegroup.com/rulebook/CME/

I do not know that there is a single large contract that contains all of those details.

For OTC options, the contract can take a lot of different forms. As per the nature of OTC derivatives, the format is not standardized. However, to reduce the legal burden of all parties to create a detailed contract governing all possible scenarios, parties usually enter into a master agreement that specifies how the two parties will enter into OTC derivatives together. That master agreement will usually specify a specific format of confirmations, which are the documents that contain the terms of an OTC derivative.

One such example of master agreement is the ISDA master agreement, which is itself a standardized document produced by the International Swap Dealers Association (ISDA).

https://en.wikipedia.org/wiki/ISDA_Master_Agreement

Here is what an ISDA master agreement looks like:

https://www.sec.gov/Archives/edgar/data/1065696/000119312511118050/dex101.htm

You can find an example of a legal confirmation here:

https://www.sec.gov/Archives/edgar/data/77159/000119312507259674/dex105.htm

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A legally enforceable contract between two parties usually consists of an offer and an acceptance of that offer. It need not necessarily be in writing, and the terms can also be defined by convention rather than words (either spoken or in writing).

In the case of over the counter (or OTC options) which are contracts between two financially sophisticated organizations, any dispute over the terms will potentially end up in court (or at least arbitration) so for the purposes of certainty, such agreements will often be generated and agreed (sometimes hundreds or thousands of pages long).

Exchange traded options however keep the scope of definition much smaller, to a handful of parameters: Underlying Security (or securities), Expiration Date, Strike Price, the type of Right (whether it is a Call or Put option); and any position limit restrictions.

The buyer or seller generates an order indicating these parameters by specifying a particular security symbol, along with a desired quantity, price constraints and or other conditions. The exchange executes the portions of the submitted orders that match, which results in a trade. The execution report of that trade is effectively the resulting contract (both the buyer and seller receive a similar execution report).

On the days of trading on the trading floor, orders between two exchange participants on the trading floor would be handwritten onto a form specifying the parameters of the trade (an illustration can be found in figure 1.3 on pages 3 and 4 of Natenberg)

Legally speaking however, the contract is not between the buyer and seller, but between the buyer and the Options Clearing Corporation (the OCC) and the seller and the Options Clearing Corporation, who is also the custodian of both the buyer's and the seller's options account.

US Options are listed on an exchange via the OCC's option listing procedures. It involves the Exchange who wishes to list the option submitting a certificate to the OCC describing the pertinent information. The OCC does not appear to make those certificates available to the public (presumably due to restrictions over the publication of CUSIP numbers) but does make the parameters provided in the certificate available through their web site.

Any nuances of Exchange Listed Option contracts are generally covered in the document: Characteristics and Risks of Standardized Options document that @JoeTaxpayer references in his answer. This is required to be signed by the client before a specific option strategy can be discussed with a broker or traded in the client's account.

The OCC will also issue notices in the changes of option contracts through their website - for example if a company spins off a subsidiary prior to option expiration, the option symbol will change to reflect that it is now irregular and the OCC notice will describe details of the basket (as the underlying is now a collection of more than one security and or cash)

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