Assignment of Trade (AOT)

In general, the term "assignment of trade" refers to a process where someone in a forward contract (meaning a deal that is set to take place at some point in a future) makes a separate side deal with to assign their part of the agreement to a third party. Remember the game of telephone as a kid? You know, the one where you whisper something to the person next to you and they repeat what they heard to the person next to them and on down the line until "I like your shoes" becomes "Eye lines in York are a snooze."
Assignment of trade is basically the transactional equivalent of telephone (only hopefully nothing gets degraded along the way - you're just passing the deal along to the next person).
The term has its most notable relevance in the market for mortgage-backed securities. An MBS is a tradable asset similar to bonds that derive their income from revenue from home loans. Some mortgage-backed securities are traded on what is called the "to be announced," or TBA, market. This sounds like something a bad business school student would make up after forgetting to do their homework, but it's a real thing in the MBS world.
Individual mortgages differ in a lot of ways. The size or quality of the home, the location, the credit histories of the people who own the house, etc. In the TBA market, companies take pains to make sure the mortgages are as similar as possible, to the extent that they are virtually interchangeable.
Because of this, the exact nature of a particular mortgage becomes irrelevant - any particular mortgage within a certain type is as good as another. Thus, companies trade in the market without having to know specifics of the mortgage pool they are trading - the details are literally TBA.
In this context, companies are able to move pretty fast and can use the assignment of trade technique. By making a deal and immediately assigning the assets to a third party, a company seeks to fine-tune the timing of the transaction. This way, the firm can maximize its return while minimizing its risk. When using this strategy, the company at the center of the trade doesn't want to hold the MBS for any length of time. Instead, it creates a situation where it moves the securities onto a third party, lowering its risk and optimizing its return.