Leishmania, the causative agent of kala-azar, reproduces asexually by binary fission.

Binary fission is a method of cell division in which a single parent cell separates into two identical daughter cells. During binary fission, the genetic material of the parent cell is duplicated, and the cell elongates until it eventually divides into two separate cells.

Budding, the first option, is a method of asexual reproduction where a smaller outgrowth, called a bud, develops on the parent organism. The bud later detaches from the parent and becomes a separate organism. Leishmania does not reproduce through budding.

Multiple fission, the third option, is a method of reproduction where a single parent cell divides into multiple daughter cells simultaneously. This process is commonly found in certain protozoans, but it is not the method used by Leishmania.

Sporogony, the fourth option, is a type of sexual reproduction that involves the production of spores. This process is not applicable to Leishmania, as it reproduces asexually through binary fission.

Therefore, the correct answer is option 2 - Leishmania, the causative agent of kala-azar, multiplies asexually by binary fission