EXHIBIT 32.1

                           CERTIFICATION PURSUANT TO
                             18 U.S.C. SECTION 1350,
                             AS ADOPTED PURSUANT TO
                  SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

         Pursuant to 18 U.S.C. ss. 1350, as created by Section 906 of the
Sarbanes-Oxley Act of 2002, each of the undersigned principal executive officer
of Dearborn Capital Management, L.L.C., the general partner (the "General
Partner") of Grant Park Futures Fund Limited Partnership ("Grant Park"), and
principal financial officer of the General Partner hereby certifies that:

         (1)      the accompanying Quarterly Report on Form 10-Q of Grant Park
                  for the quarterly period ended September 30, 2004 (the
                  "Report") fully complies with the requirements of section
                  13(a) or 15(d), as applicable, of the Securities Exchange Act
                  of 1934, as amended; and

         (2)      the information contained in the Report fairly presents, in
                  all material respects, the financial condition and results of
                  operations of Grant Park.

Dated: November 15, 2004                    /s/ David M. Kavanagh
                                            -----------------------------------
                                            David M. Kavanagh
                                            President
                                            (principal executive officer)
                                            Dearborn Capital Management, L.L.C.
                                            General Partner

                                            /s/ Maureen O'Rourke
                                            -----------------------------------
                                            Maureen O'Rourke
                                            Chief Financial Officer
                                            Dearborn Capital Management, L.L.C.
                                            General Partner


         This certification accompanies this Report pursuant to Section 906 of
the Sarbanes-Oxley Act of 2002 and shall not be deemed filed by Grant Park for
purposes of Section 18 of the Securities Exchange Act of 1934, as amended.