Home Box Office has signed an exclusive agreement for the pay-television rights to Paramount Pictures films released after mid-1988. The deal, which includes 85 films, is expected to last at least five years.

The rights to films from Paramount, currently the nation's most successful movie company, had been aggressively sought by Home Box Office, the nation's largest pay television service, and Showtime, which is No. 2 and which has had an exclusive deal with Paramount.

No price was disclosed, but some industry experts estimated that the deal was at least $500 million. One knowledgeable pay television executive said an actual price was hard to determine because Home Box Office generally buys films at prices based on their box office performance as well as on the number of HBO subscribers.

Nevertheless, the deal appears extremely lucrative for Gulf and Western, Paramount's parent company. Paramount's hits include ''Beverly Hills Cop II,'' ''The Untouchables'' and the ''Star Trek'' series.

One industry expert who declined to be named said he thought the deal could be a programming coup for Home Box Office, which is a unit of Time Inc.

Home Box Office, with 19 million subscribers, already owns the exclusive rights to films from Columbia Pictures and Tri-Star Pictures. Through a complex agreement, it effectively has the exclusive rights to Warner Brothers films as well.

Showtime/The Movie Channel, with eight million subscribers, is owned by Viacom International, which was recently acquired by the theater owner Sumner M. Redstone. Although it is losing its right to Paramount films, Showtime still has an exclusive contract with the Walt Disney Company's Touchstone Films, as well as with Orion Pictures.

Dennis Leibowitz, a media analyst at Donaldson, Lufkin & Jenrette, speculated that despite the defeat, Showtime might ultimately benefit. He said that if cable and movie industry companies believed Showtime had severe problems, they might take equity positions in the service. That would insure its survival and help Mr. Redstone pay down the debt he incurred in the buyout.

Paramount and Home Box Office also entered into an agreement through which they will jointly finance a number of original films made for pay television.