<PAGE>   1
                                                                    EXHIBIT 4.26



                            [R&B FALCON LETTERHEAD]



                                 April 23, 1999




Mr. Richard D. Bullock
Cliffs Drilling Company
1200 Smith Street, Suite 300
Houston, Texas 77002

Dear Mr. Bullock,

     Reference is made to the Employment Agreement dated as of February 16,
1999, between Cliffs Drilling Company ("Cliffs") and you. Pursuant to the
Employment Agreement, Cliffs agreed to grant you options to purchase shares of
common stock of R&B Falcon Corporation ("Parent") under the R&B Falcon
Corporation 1998 Employee Long-Term Incentive Plan (the "Plan"). There are
insufficient authorized shares remaining out of the Plan to grant you this
option. Parent proposes to grant you options outside of the Plan, but which
incorporate the terms of the Plan. Please execute and return a copy of this
letter to evidence your agreement that the provisions of Section 2.2 of the
Employment Agreement are satisfied by granting you options that incorporate the
terms of the Plan, not withstanding that these shares are not formally being
granted under the Plan.

Sincerely,

R&B FALCON CORPORATION


By:  /s/ LEIGHTON E. MOSS
  --------------------------------
  Leighton E. Moss


CLIFFS DRILLING COMPANY

By:  /s/ DOUGLAS E. SWANSON
  --------------------------------
  Douglas E. Swanson


ACCEPTED AND AGREED:

  /s/ RICHARD D. BULLOCK
----------------------------------
Richard D. Bullock
<PAGE>   2
                             R&B FALCON CORPORATION
                             STOCK OPTION AGREEMENT


          This Stock Option Agreement ("Agreement") is made between R&B Falcon
Corporation, a Delaware corporation ("Company"), and Richard D. Bullock
("Optionee") as of February 16, 1999 (the "Effective Date").

                                  WITNESSETH:

          WHEREAS, the Optionee has agreed to accept employment with Cliffs
Drilling Company ("Cliffs"), a subsidiary of the Company, pursuant to an
Employment Agreement dated February 16, 1999 between Cliffs and Optionee (the
"Employment Agreement");

          WHEREAS, in order to induce Optionee to accept such employment,
Company agreed to grant Optionee options to acquire shares of common stock of
the Parent;

          NOW THEREFORE, for and in consideration of these premises, it is
hereby agreed as follows:

          1.   As used herein, the terms set forth below shall have the
following respective meanings:

     "Cause" has the meaning given to it in the Employment Agreement.

     "Change of Control" means a Change of Control as defined in Section 18 of
this Agreement.

     "Disability" means Disability as defined in the Company's Personnel
Policies and Procedures, in effect from time to time.

     "Plan" means the R&B Falcon Corporation 1998 Employee Long-Term Incentive
Plan.

          2.   The Option awarded hereunder is not granted pursuant to the
Plan; however, the Option is hereby made subject to all of the terms, conditions
and provisions of the Plan and administrative interpretations thereunder, if
any, which have been adopted by the Committee and are in effect on the date
hereof. Capitalized terms used but not defined herein shall have the meanings
assigned to such terms in the Plan.

          3.   On the terms and subject to the conditions contained herein, the
Company hereby grants to the Optionee an option (the "Option") for the term of
ten years ending on February 16, 2009 ("Option Period") to purchase from the
Company 30,000 shares ("Option Shares") of the Company's Common Stock, at a
price equal to $6.15625 per share.

          4.   This Option shall be exercisable as follows:

          (a)  After one year following the Effective Date, this Option shall
be exercisable for any number of shares up to and including, but not in excess
of, 33-1/3% of the aggregate number of shares subject to this Option;

          (b)  After two years following the Effective Date, this Option shall
be exercisable for any number of shares up to and including, but not in excess
of, 66-2/3% of the aggregate number of shares subject to this Option; and
<PAGE>   3
                                                          R&B Falcon Corporation
                                                          Stock Option Agreement
                                                                     Page 2 of 5


     (c)  After three years following the Effective Date, this Option shall be
exercisable for any number of shares of Common Stock up to and including, but
not in excess of, 100% of the aggregate number of shares subject to this Option;

provided the number of shares as to which this Option becomes exercisable
shall, in each case, be reduced by the number of shares theretofore purchased
pursuant to the terms hereof. Notwithstanding the foregoing, this Option shall
not be exercisable unless and until the Optionee has been continuously employed
by the Company and/or its Affiliates for a period of one year.

     5.   The Option may be exercised by the Optionee, in whole or in part, by
giving written notice to the Compensation and Benefits Department of the
Company setting forth the number of Option Shares with respect to which the
option is to be exercised, accompanied by payment for the shares to be
purchased and any appropriate withholding taxes, and specifying the address to
which the certificate for such shares is to be mailed (or to the extent
permitted by the Company, the written instructions referred to in the last
sentence of this section). Payment shall be by means of cash, certified check,
bank draft or postal money order payable to the order of the Company. As
promptly as practicable after receipt of such written notification and payment,
the Company shall deliver, or cause to be delivered, to the Optionee
certificates for the number of Option Shares with respect to which the Option
has been so exercised (or to the extent permitted by the Company from time to
time, to have such number of Option Shares electronically transferred to
Optionee's account at Optionee's broker in accordance with Optionee's written
instructions).

     6.   Subject to approval of the Committee, which shall not be unreasonably
withheld, the Optionee may pay for any Option Shares with respect to which the
Option is exercised by tendering to the Company other shares of Common Stock at
the time of the exercise or partial exercise hereof. The certificates
representing such other shares of Common Stock must be accompanied by a stock
power duly executed with signature guaranteed in accordance with market
practice. The value of the Common Stock so tendered shall be its Fair Market
Value.

     7.

     (a)  Upon the termination of the Optionee's employment due to death or
Disability, if Employee has at the date of such termination been continuously
employed by the Company and/or its Affiliates for a period of one year, the
restrictions on exercise set out in Section 4 above shall terminate and the
Optionee's right to exercise this Option thereafter shall no longer be subject
to such restrictions on exercise.

     (b)  Upon termination of Employee's employment by Cliffs without Cause,
the restrictions set forth in Section 4 shall terminate, and the Option shall
be exercisable in full.

     (c)  Upon termination of the Optionee's employment with the Company and
all Affiliates, other than for one of the reasons expressly set out in
paragraphs (a) and (b) preceding, then the Option granted herein shall not
become exercisable as to any shares for which it was not exercisable as of the
date of such termination of employment.

     8.   The Option shall not be transferable by the Optionee otherwise than
as expressly permitted by the Plan. During the lifetime of the Optionee, the
Option shall be exercisable only by her or him. No transfer of the Option shall
be effective to bind the Company unless the Company shall have been furnished
with written notice thereof and a copy of such
<PAGE>   4
                                                          R&B Falcon Corporation
                                                          Stock Option Agreement
                                                                     Page 3 of 5


evidence as the Committee may deem necessary to establish the validity of the
transfer and the acceptance by the transferee or transferees of the terms and
conditions hereof.

     9.   The Optionee shall have no rights as a stockholder with respect to
any Option Shares until the date of issuance of a certificate for Option Shares
purchased pursuant to this Agreement (or to the extent permitted by the
Company, from time to time, the number of such Option Shares has been
electronically transferred to Optionee's account at Optionee's broker). Until
such time, the Optionee shall not be entitled to dividends or to vote at
meetings of the stockholders of the Company.

     10.  The Company may make such provisions as it may deem appropriate for
the withholding of any taxes which it determines is required in connection with
the option herein granted. The Optionee may pay all or any portion of the taxes
required to be withheld by the Company or paid by the Optionee in connection
with the exercise of all or any portion of the option herein granted by
electing to have the Company withhold shares of Common Stock, or by delivering
previously owned shares of Common Stock, having a Fair Market Value equal to
the amount required to be withheld or paid. The Optionee must make the
foregoing election on or before the date that the amount of tax to be withheld
is determined ("Tax Date"). Any such election is irrevocable and subject to
disapproval by the Committee. If the Optionee is subject to the short-swing
profits recapture provisions of Section 16(b) of the Exchange Act, any such
election shall be subject to the following additional restrictions;

     (a)  Such election may not be made within six months of the grant of this
option, provided that this limitation shall not apply in the event of death or
Disability.

     (b)  Such election must be made either in an Election Window (as
hereinafter defined) or at such other time as may be consistent with Section
16(b) of the Exchange Act and the rules promulgated thereunder. Where the Tax
Date in respect of the exercise of all or any portion of this Option is
deferred until after such exercise and the Optionee elects stock withholding,
the full amount of shares of Common Stock will be issued or transferred to the
Optionee upon exercise of this Option, but the Optionee shall be
unconditionally obligated to tender back to the Company on the Tax Date the
number of shares necessary to discharge with respect to such Option exercise
the greater of (i) the Company's withholding obligation and (ii) all or any
portion of the holder's federal and state tax obligation attributable to the
Option exercise. An Election Window is any period commencing on the third
business day following the Company's release of a quarterly or annual summary
statement of sales and earnings and ending on the twelfth business day
following such release.

     11.  Upon the acquisition of any shares pursuant to the exercise of the
Option, the Optionee will enter into such written representations, warranties
and agreements as the Company may reasonably request in order to comply with
applicable securities laws or with this Agreement.

     12.  The certificates representing the Option Shares purchased by exercise
of an option will be stamped or otherwise imprinted with a legend in such form
as the Company or its counsel may require with respect to any applicable
restrictions on sale or transfer, and the stock transfer records of the Company
will reflect stop-transfer instructions, as appropriate, with respect to such
shares.

     13.  Unless otherwise provided herein, every notice hereunder shall be in
writing and be delivered by hand or by registered or certified mail. All
notices of the exercise by the Optionee of any option hereunder shall be
directed to R&B Falcon Corporation,
<PAGE>   5
                                                          R&B Falcon Corporation
                                                          Stock Option Agreement
                                                                     Page 4 of 5

Attention: Benefits and Compensation Department, at the Company's principal
office address from time to time. Any notice given by the Company to the
Optionee directed to him or her at his or her address on file with the Company
shall be effective to bind any other person who shall acquire rights hereunder.
The Company shall be under no obligation whatsoever to advise the Optionee of
the existence, maturity or termination of any of the Optionee's rights
hereunder and the Optionee shall be deemed to have familiarized himself with
all matters contained herein and in the Plan which may affect any of the
Optionee's rights or privileges hereunder.

     14.  Whenever the term "Optionee" is used herein under circumstances
applicable to any other person or persons to whom this award, in accordance
with the provisions of Paragraph 8, may be transferred, the word "Optionee"
shall be deemed to include such person or persons. References to the masculine
gender herein also include the feminine gender for all purposes.

     15.  Notwithstanding any of the other provisions hereof, the Optionee
agree that he or she will not exercise the Option, and that the Company will
not be obligated to issue any share pursuant to this Agreement, if the exercise
of the Option or the issuance of such shares of Common Stock would constitute a
violation by the Optionee or by the Company of any provision of any law or
regulation of any governmental authority or any national securities exchange.

     16.  Although the Options are not granted pursuant to the Plan, the terms
and provisions of the Plan (including any subsequent amendments thereto) are
incorporated herein by reference. In the event of a conflict between any term
or provision contained herein and a term or provision of the Plan, the
applicable terms and provisions of the Plan will govern and prevail. All
definitions of words and terms contained in the Plan shall be applicable to
this Agreement.

     17.  In the event of a corporate merger or other business combination in
which the Company is not the surviving entity, the economic equivalent number
of the voting shares of common stock of, or participating interests in, the
surviving entity, based on the terms of such merger or other business
combination, shall be substituted for the Option Shares hereunder, and the
price per share set out in Section 3 hereof shall be adjusted to reflect
substantially the same economic equivalent value of the Option Shares to the
Optionee immediately prior to any such merger or other business combination.

     18.  For the purpose of this Agreement, a "Change of Control" shall mean:
(a) any "Person", as such term is used in Section 13(d) and 14(d) of the
Securities Exchange Act of 1934, as amended (the "Exchange Act") (other than
(i) the Optionee, (ii) the Company or any of its subsidiaries or Affiliates
(as that term is defined in the Exchange Act), (iii) any Person subject, as of
the date of this Agreement or at any prior time, to the reporting or filing
requirements of Section 13(d) of the Exchange Act with respect to the
securities of the Company or any Affiliate, (iv) any trustee or other fiduciary
holding or owning securities under an employee benefit plan of the Company, (v)
any underwriter temporarily holding or owning securities of the Company, or
(vi) any corporation owned directly or indirectly by the current stockholders
of the Company in substantially the same proportion as their then ownership of
stock of the Company) becomes, after the date of this Agreement, the
"beneficial owner" (as defined in Rule 13d-3 under the Exchange Act), directly
or indirectly, of securities of the Company representing forty percent (40%) or
more of the combined voting power of the Company's then outstanding securities;
or (b) at any time a majority of the members of the board of directors of the
Company is comprised of other than Continuing Directors (and for this purpose
"Continuing Directors" shall mean members of the board of directors of the
Company who were directors as of the date of this Agreement, or who were
nominated by a majority of the
<PAGE>   6
                                                          R&B Falcon Corporation
                                                          Stock Option Agreement
                                                                     Page 5 of 5


members of the board of directors of the Company and such majority was
comprised only of Continuing Directors at the time of such nomination).

     19.  Adjustments. In the event of a corporate merger or other business
combination in which the Company is not the surviving entity, the economic
equivalent number of the voting shares of common stock of, or participating
interests in, the surviving entity, based on the terms of such merger or other
business combination, shall be substituted for the number of Option Shares held
by the Participant hereunder, and the exercise price per share set out in
Section 3 above shall be likewise adjusted, to reflect substantially the same
economic equivalent value of the Option Shares to the Participant prior to any
such merger or other business combination. In the event of a split-off, spin-off
or creating of a different class of common stock of the Company (including,
without limitation, a tracking stock), the Participant shall receive an option
to purchase an equivalent number of the shares of common stock or voting
interests of such separate entity being split-off or spun-off or of the shares
of the new class of common stock of the Company, as if Participant had owned the
shares underlying the Option Shares on the record date for any such split-off,
spin-off or creation of a new class of common stock of the Company, and the
exercise prices set out in Section 3 hereof and applicable to the options to
purchase shares or the voting interests of the new entity being split-off or
spun-off shall be adjusted to reflect substantially the same economic
equivalent value of the Option Shares to the Optionee prior to any such
split-off, spin-off or creation of a new class of common stock of the Company.

                                        R&B FALCON CORPORATION



                                        By: /s/ LEIGHTON E. MOSS
                                            --------------------------------
                                            Leighton E. Moss, Vice President


                                        OPTIONEE


                                        /s/ RICHARD D. BULLOCK
                                        ------------------------------------
                                        Richard D. Bullock