<PAGE>

                                                                     EXHIBIT 4.5

                              SECOND AMENDMENT TO
                           ATP OIL & GAS CORPORATION
                             1998 STOCK OPTION PLAN


     WHEREAS, ATP OIL & GAS CORPORATION, a Texas corporation (the "Company") has
heretofore adopted the ATP OIL & GAS CORPORATION 1998 STOCK OPTION PLAN (the
"Plan"); and

     WHEREAS, the Company desires to amend the Plan in certain respects;

     NOW, THEREFORE, the Plan shall be amended as follows, effective upon the
date this amendment is adopted by the Board of Directors of the Company:

     1.   Section 6(g) of the Plan shall be deleted and the following shall be
substituted therefor:

          "(g)  Assignability.  An ISO shall not be assignable or otherwise
     transferable except by will or the laws of descent and distribution, and
     during the lifetime of an Optionee, an option shall be exercisable only by
     him or his guardian or legal representative.  A Nonqualified Option shall
     not be assignable or otherwise transferable except (i) by will or the laws
     of descent and distribution, (ii) pursuant to a qualified domestic
     relations order as defined by the Code or Title I of the Employee
     Retirement Income Security Act of 1974, as amended, or the rules
     thereunder, or (iii) with the consent of the Committee."

     2.   Section 6(l) of the Plan shall be deleted and the following shall be
substituted therefor:

          "(l)  Change in Control.  If the Company shall be a party to a merger
     or consolidation that results in at least 40% of the total voting power
     represented by the voting stock of the Company (or the successor to the
     Company) outstanding immediately after such merger or consolidation being
     owned or controlled (including, without limitation, the power to vote) by
     persons or entities other than the shareholders of the Company immediately
     prior to such merger or consolidation (a `Transaction'), then, except as
     provided in any option agreement, upon the consummation of such
     Transaction, all outstanding options shall immediately vest and become
     exercisable and such options shall continue to be exercisable for the
     remainder of the applicable option term.  Further, if (i) the Company shall
     not be the surviving entity in a merger or consolidation (or survives only
     as a subsidiary of an entity) other than a Transaction as described in the
     preceding sentence, (ii) the Company sells, leases or exchanges or agrees
     to sell, lease or exchange all or substantially all of its assets to any
     other person or entity, (iii) the Company is to be dissolved and
     liquidated, (iv) any person or entity, including a `group' as contemplated
     by Section 13(d)(3) of the 1934 Act, acquires or gains ownership or control
     (including, without limitation, power to vote) of more than 50% of the
<PAGE>

     outstanding shares of the Company's voting stock (based upon voting power)
     or (v) as a result of or in connection with a contested election of
     Directors, the persons who were Directors of the Company before such
     election shall cease to constitute a majority of the Board (each such event
     is referred to in clauses (i), (ii), (iii) (iv) and (v) herein as a
     `Corporate Change'), no later than (x) ten days after the approval by the
     shareholders of the Company of such merger, consolidation, reorganization,
     sale, lease or exchange of assets or dissolution or such election of
     Directors or (y) thirty days after a Corporate Change of the type described
     in clause (iv), the Committee, acting in its sole discretion without the
     consent or approval of any Optionee, shall effect one or more of the
     following alternatives, which alternatives may vary among individual
     Optionee and which may vary among options held by any individual Optionee
     (1) accelerate the time at which options then outstanding may be exercised
     so that such options may be exercised in full for a limited period of time
     on or before a specified date (before or after such Corporate Change) fixed
     by the Committee, after which specified date all unexercised options and
     all rights of Optionees thereunder shall terminate, (2) require the
     mandatory surrender to the Company by selected Optionees of some or all of
     the outstanding options held by such Optionees (irrespective of whether
     such options are then exercisable under the provisions of the Plan) as of a
     date, before or after such Corporate Change, specified by the Committee, in
     which event the Committee shall thereupon cancel such options and cause the
     Company to pay to each Optionee an amount of cash per share equal to the
     excess, if any, of the `Change of Control Value' (as hereinafter defined)
     of the shares subject to such Option over the exercise price(s) under such
     options for such shares, (3) make such adjustments to options then
     outstanding as the Committee deems appropriate to reflect such Corporate
     Change (provided, however, that the Committee may determine in its sole
     discretion that no adjustment is necessary to Options then outstanding), or
     (4) provide that the number and class of shares of Common Stock covered by
     an option theretofore granted shall be adjusted so that such option shall
     thereafter cover the number and class of shares of stock or other
     securities or property (including, without limitation, cash) to which the
     Optionee would have been entitled pursuant to the terms of the agreement of
     merger, consolidation or sale of assets and dissolution if, immediately
     prior to such merger, consolidation or sale of assets and dissolution, the
     Optionee had been the holder of record of the number of shares of Common
     Stock then covered by such option.  For the purposes of clause (2) of the
     preceding sentence, the `Change of Control Value' shall equal the amount
     determined in clause (A), (B) or (C), whichever is applicable, as follows:
     (A) the per share price offered to shareholders of the Company in any such
     merger, consolidation, sale of assets or dissolution transaction, (B) the
     price per share offered to shareholders of the Company in any tender offer
     or exchange offer whereby a Corporate Change takes place, or (C) if such
     Corporate Change occurs other than pursuant to a tender or exchange offer,
     the fair market value per share of the shares into which such options being
     surrendered are exercisable, as determined by the Committee as of the date
     determined by the Committee to be the date of cancellation and surrender of
     such options.  In the event that the consideration offered to shareholders
     of the Company in any transaction described in this paragraph consists of
     anything other than cash, the Committee shall determine the fair cash
     equivalent of the portion of the consideration offered which is other than
     cash. The provisions contained in this paragraph shall not terminate any
     rights of the Optionee to further payments pursuant to any other agreement
     with the Company following a Transaction or Corporate Change."

     3.  As amended hereby, the Plan is specifically ratified and reaffirmed.

     EXECUTED this 3rd day of January, 2001.


                              ATP OIL & GAS CORPORATION



                              By: /s/ T. Paul Bulmahn
                                  ------------------------------------------

                                      -2-