<PAGE>   1
                       SECURITIES AND EXCHANGE COMMISSION
                             Washington, D.C. 20549

                                    FORM 8-K


                                 CURRENT REPORT


                     Pursuant to Section 13 or 15(d) of the
                         Securities Exchange Act of 1934

                        Date of Report: October 14, 1998
                        (Date of earliest event reported)


                         CMC SECURITIES CORPORATION III
             (Exact name of Registrant as specified in its charter)


        Delaware                      33-47913                  75-2431913
(State of Incorporation)        (Commission File No.)        (I.R.S. Employer
                                                             Identification No.)

         2711 N. Haskell Avenue
               Suite 900
             Dallas, Texas                                         75204
(Address of Principal executive offices)                         (Zip Code)


       Registrant's Telephone Number, Including Area Code: (214) 874-2323






<PAGE>   2



Item 5.  Other Events.


         Reference is hereby made to the Registrant's Registration Statement on
Form S-3 (File No. 33-47913) filed with the Securities and Exchange Commission
(the "Commission") on May 14, 1992, as amended by Amendment No. 1 thereto filed
with the Commission on August 7, 1992, as further amended by Post Effective
Amendment No. 1 thereto filed with the Commission on September 16, 1993, as
further amended by Post Effective Amendment No. 2 thereto filed with the
Commission on December 13, 1993, and as further amended by Post Effective
Amendment No. 3 thereto filed with the Commission on February 14, 1994
(collectively, the "Registration Statement"), pursuant to which the Registrant
registered $4,000,000,000 aggregate principal amount of its collateralized
mortgage obligations, issuable in various series, for sale in accordance with
the provisions of the Securities Act of 1933, as amended (the "Act"). Reference
is also hereby made to the Prospectus dated September 25, 1998 and the related
Prospectus Supplement, dated September 25, 1998 (collectively, the
"Prospectus"), which have been filed with the Commission pursuant to Rule
424(b)(5), with respect to the Registrant's Collateralized Mortgage Obligations,
Series 1998-2 (the "Bonds").

         On September 30, 1998, the Registrant caused the issuance and sale of
approximately $538,716,245 aggregate initial principal amount of Bonds. The
Bonds are collateralized by mortgage pass-through certificates (the
"Conventional Certificates") evidencing the beneficial ownership interest in
entire pools of certain conventional, fixed-rate, fully-amortizing, one- to
four-family, first lien mortgage loans (the "Mortgage Loans") originated or
acquired by (i) Cendant Mortgage Corporation ("Cendant"), Cendant Residential
Mortgage Trust ("CRMT"), National City Mortgage Co. ("NCM") and (iv) Washington
Mutual Bank, F.A. ("WMBFA"). Cendant, CRMT, NCM and WMBFA are referred to herein
collectively as the "Loan Sellers". The Conventional Certificates were created
pursuant to a Pooling and Servicing Agreement dated as of September 1, 1998 (the
"Pooling and Servicing Agreement") by and among the Registrant, as Seller,
Norwest Bank Minnesota, National Association ("NBMNA"), as Master Servicer and
Custodian, and The First National Bank of Chicago ("FNBC"), as Certificate
Trustee. A copy of the Pooling and Servicing Agreement is filed herewith as
Exhibit 10.1. The Bonds were issued pursuant to an Indenture dated as of March
1, 1998 (the "Indenture"), by and between the Registrant and FNBC, as Indenture
Trustee (the "Indenture Trustee"), as supplemented by the Series 1998-2
Supplement thereto dated as of September 30, 1998 (the "Series Supplement"), by
and among the Registrant, the Indenture Trustee and NBMNA, as Bond
Administrator. A copy of the Indenture was filed as Exhibit 4.1 to the
Registrant's Current Report on Form 8-K dated April 7, 1998, as filed with the
Commission on April 7, 1998. A copy of the Series Supplement is filed herewith
as Exhibit 4.2.

         The Offered Bonds (as defined in the Prospectus), having an aggregate
principal balance of $534,277,116, have been sold by the Registrant to Bear,
Stearns & Co. Inc. ("Bear, Stearns") pursuant to an Underwriting Agreement dated
as of March 27, 1998 (the "Underwriting Agreement"), as supplemented by a Terms
Agreement dated as of September 25, 1998, each among Bear, Stearns, the
Registrant and Capstead Mortgage Corporation ("CMC"). A copy of the Underwriting
Agreement was filed as Exhibit 1.1 to the Registrant's Current Report on Form
8-K dated April 7, 1998, as filed with the Commission on April 7, 1998.



                                       -2-


<PAGE>   3







Item 7.  Financial Statements and Exhibits.

         (c)      Exhibits

                  Exhibit No.       Description

                     1.1            Underwriting Agreement dated as of March 27,
                                    1998 by and among Bear, Stearns, the
                                    Registrant and CMC(1)

                     4.1            Indenture dated as of March 1, 1998 by and
                                    between the Registrant, as Issuer and FNBC,
                                    as Indenture Trustee(1)

                     4.2            Series 1998-2 Supplement dated as of
                                    September 30, 1998 by and among the
                                    Registrant, as Issuer, FNBC, as Indenture
                                    Trustee, and NBMNA, as Bond 
                                    Administrator(2)

                    10.1            Pooling and Servicing Agreement dated as of
                                    September 1, 1998 by and among the
                                    Registrant, as Seller, NBMNA, as Master
                                    Servicer and Custodian, and FNBC, as
                                    Certificate Trustee(2)


--------------------

(1)      Previously filed as an Exhibit to the Registrant's Current Report on
         Form 8-K, filed with the Commission on April 7, 1998, and incorporated
         by reference therein.

(2)      Filed herewith.








                                       -3-


<PAGE>   4





                                    Signature

         Pursuant to the requirements of the Securities Exchange Act of 1934,
the Registrant has duly caused this Report to be signed on its behalf by the
undersigned hereunto duly authorized.


                                         CMC SECURITIES CORPORATION III



October 14, 1998                         By:/s/ WADE WALKER
                                            ------------------------------------
                                            Wade Walker,
                                            Vice President - Asset and Liability
                                                             Management






                                       -4-

<PAGE>   5



                                INDEX TO EXHIBITS


<TABLE>
<CAPTION>
Exhibit No. 
----------- 
<S>      <C>
1.1      Underwriting Agreement dated as of March 27, 1998 by and among Bear,
         Stearns, the Registrant and CMC(1)

4.1      Indenture dated as of March 1, 1998 by and between the Registrant, as
         Issuer, and FNBC, as Indenture Trustee(1)

4.2      Series 1998-2 Supplement dated as of September 30, 1998 by and among
         the Registrant, as Issuer, FNBC, as Indenture Trustee and NBMNA, as
         Bond Administrator(2)

10.1     Pooling and Servicing Agreement dated as of September 1, 1998 by and
         among the Registrant, as Seller, NBMNA, as Master Servicer and
         Custodian, and FNBC, as Certificate Trustee(2)
</TABLE>

---------------

(1)      Previously filed as an Exhibit to the Registrant's Current Report on
         Form 8-K, filed with the Commission on April 7, 1998, and incorporated
         by reference therein.

(2)      Filed herewith.