A sublease agreement is considered a written document that offers the lessee a chance to pass on the responsibility and rights of tenancy to another person. A sublet, or sublease agreement, adds someone new to an existing lease.Usually the new person (subletter) replaces someone who is moving out. You can lease your apartment to another person. If you will be returning to live in your apartment after the other person leaves, this is called a sublease. In New York State, tenants have a right to sublease their apartments. Most leases require the landlord's written permission. Subleasing is when the current tenant of a property rents out their unit to a third party for a portion of their existing lease. A sublease is an agreement between the primary tenant on the lease and a secondary tenant or subtenant. The sub-tenant pays the tenant their rent every month.