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                                                                    EXHIBIT 10b
 
                             ROWAN COMPANIES, INC.
 
                  1998 NONEMPLOYEE DIRECTORS STOCK OPTION PLAN
 
                             I. PURPOSE OF THE PLAN
 
     The Rowan Companies, Inc. 1998 Nonemployee Directors Stock Option Plan (the
"Plan") is intended to provide a means whereby directors of Rowan Companies,
Inc., a Delaware corporation (the "Company"), who are not employees of the
Company or its subsidiaries may develop a sense of proprietorship and personal
involvement in the development and financial success of the Company and whereby
the economic interests of such directors may be more closely aligned with those
of the Company's shareholders, thereby advancing the interests of the Company
and its shareholders. Accordingly, the Company may grant to such directors
("Optionees") the option ("Option") to purchase shares of the common stock of
the Company ("Stock"), as hereinafter set forth. Options granted under the Plan
are not intended to be incentive stock options within the meaning of section
422(b) of the Internal Revenue Code of 1986, as amended (the "Code").
 
                               II. ADMINISTRATION
 
     The Plan shall be administered by the full Board of Directors of the
Company (the "Board"). Unless the Board determines otherwise, any and all
actions taken by the Board with respect to the Plan or in exercising its duties,
powers and responsibilities under the Plan shall be effected in a manner which
permits the Plan to comply with Rule 16b-3, as currently in effect or as
hereinafter modified or amended ("Rule 16b-3"), promulgated under the Securities
Exchange Act of 1934, as amended (the "1934 Act"). The Board shall have sole
authority to select the Optionees from among those individuals eligible
hereunder and to establish the number of shares which may be issued under each
Option. In selecting the Optionees from among individuals eligible hereunder and
in establishing the number of shares that may be issued under each Option, the
Board may take into account such factors as the Board in its discretion shall
deem relevant. The Board is authorized to interpret the Plan and may from time
to time adopt such rules and regulations, consistent with the provisions of the
Plan, as it may deem advisable to carry out the Plan. All decisions made by the
Board in selecting the Optionees, in establishing the number of shares which may
be issued under each Option and in construing the provisions of the Plan and any
Options shall be final.
 
                             III. OPTION AGREEMENTS
 
     (a) Each Option shall be evidenced by a written agreement between the
Company and the Optionee ("Option Agreement") which shall contain such terms and
conditions as may be approved by the Board. The terms and conditions of the
respective Option Agreements need not be identical.
 
     (b) For all purposes under the Plan, the fair market value of a share of
Stock on a particular date shall be equal to the mean of the high and low sales
prices of the Stock reported on the New York Stock Exchange on that date; or, if
no prices are reported on that date, on the last preceding date on which such
prices of the Stock are so reported. In the event Stock is not publicly traded
at the time a determination of its value is required to be made hereunder or the
foregoing methodology is not appropriate for determining the fair market value
of Stock as determined by the Board for any reason, the determination of its
fair market value shall be made by the Board in such manner as it deems
appropriate.
 
     (c) Except as provided herein, each Option and all rights granted
thereunder shall not be transferable other than by will or the laws of descent
and distribution be exercisable during the Optionee's lifetime only by the
Optionee or, in the case of the Optionee's death or incapacity, by the
Optionee's guardian or legal
 
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representative. The Board may, in its discretion, provide in an Option agreement
that the Option granted thereby to a director may be transferred in whole or in
part.
 
                          IV. ELIGIBILITY OF OPTIONEE
 
     Options may be granted only to individuals who are directors of the Company
and who are not also employees of the Company or any parent or subsidiary
corporation (as defined in section 425 of the Code) of the Company at the time
the Option is granted. Options may be granted to the same individual on more
than one occasion.
 
                         V. SHARES SUBJECT TO THE PLAN
 
     The aggregate number of shares which may be issued under Options granted
under the Plan shall not exceed 200,000 shares of Stock. Such shares may consist
of authorized but unissued shares of Stock or previously issued shares of Stock
reacquired by the Company. Any of such shares which remain unissued and which
are not subject to outstanding Options at the termination of the Plan shall
cease to be subject to the Plan, but, until termination of the Plan, the Company
shall at all times make available a sufficient number of shares to meet the
requirements of the Plan. Should any Option hereunder expire or terminate prior
to its exercise in full, the shares theretofore subject to such Option may again
be subject to an Option granted. The aggregate number of shares which may be
issued under the Plan shall be subject to adjustment in the same manner as
provided in Paragraph VIII hereof with respect to shares of Stock subject to
Options then outstanding. Exercise of an Option in any manner shall result in a
decrease in the number of shares of Stock which may thereafter be available for
purposes of the Plan by the number of shares as to which the Option is
exercised.
 
                                VI. OPTION PRICE
 
     The purchase price of Stock issued under each Option shall be determined by
the Board, but such purchase price shall not be less than the fair market value
of the Stock subject to the Option on the date the Option is granted.
 
                               VII. TERM OF PLAN
 
     The Plan shall be effective as of April 24, 1998, subject to its approval
by the stockholders of the Company. Except with respect to Options then
outstanding, if not sooner terminated under the provisions of Paragraph IX, the
Plan shall terminate upon and no further Options shall be granted after the
expiration of fifteen years from the effective date of the Plan. The expiration
of the Plan will not affect the validity of Options outstanding at that time.
 
                    VIII. RECAPITALIZATION OR REORGANIZATION
 
     (a) The existence of the Plan and the Options granted hereunder shall not
affect in any way the right or power of the Board or the shareholders of the
Company to make or authorize any adjustment, recapitalization, reorganization or
other change in the Company's capital structure or its business, any merger or
consolidation of the Company, any issue of debt or equity securities, the
dissolution or liquidation of the Company or any sale, lease, exchange or other
disposition of all or any part of its assets or business or any other corporate
act or proceeding.
 
     (b) The shares with respect to which Options may be granted are shares of
Stock as presently constituted, but if, and whenever, prior to the expiration of
an Option theretofore granted, the Company shall effect a subdivision or
consolidation of shares of Stock or the payment of a stock dividend on Stock
without receipt of consideration by the Company, the number of shares of Stock
with respect to which such Option may thereafter be exercised (i) in the event
of an increase in the number of outstanding shares shall be
 

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proportionately increased, and the purchase price per share shall be
proportionately reduced, and (ii) in the event of a reduction in the number of
outstanding shares shall be proportionately reduced, and the purchase price per
share shall be proportionately increased.
 
     (c) If the Company recapitalizes, reclassifies its capital stock, or
otherwise changes its capital structure (a "recapitalization"), the number and
class of shares of 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 and securities to which the Optionee would have been entitled
pursuant to the terms of the recapitalization if, immediately prior to the
recapitalization, the Optionee had been the holder of record of the number of
shares of Stock then covered by such Option. If (i) the Company shall not be the
surviving entity in any merger, consolidation or other reorganization (or
survives only as a subsidiary of an entity), (ii) the Company sells, leases or
exchanges 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 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 herein as a "Corporate Change"), no later than (a) 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 (b) thirty days after a change of control of
the type described in Clause (iv), the Board, acting in its sole discretion
without the consent or approval of any Optionee, shall act to effect one or more
of the following alternatives, which may vary among individual Optionees 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 Board, 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 Board, in which event the Board shall thereupon cancel such Options and the
Company shall pay to each Optionee an amount of cash per share equal to the
excess, if any, of the amount calculated in Subparagraph (d) below (the "Change
of Control Value") 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 Board deems appropriate to reflect such
Corporate Change (provided, however, that the Board 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 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 Stock then covered by such Option.
 
     (d) For the purposes of clause (2) in Subparagraph (c) above, the "Change
of Control Value" shall equal the amount determined in clause (i), (ii) or
(iii), whichever is applicable, as follows: (i) the per share price offered to
shareholders of the Company in any such merger, consolidation, reorganization,
sale of assets or dissolution transaction, (ii) the price per share offered to
shareholders of the Company in any tender offer or exchange offer whereby a
Corporate Change takes place, or (iii) 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 Board as of the date determined by the Board 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
Subparagraph (d) or Subparagraph (c) above consists of anything other than cash,
the Board shall determine the fair cash equivalent of the portion of the
consideration offered which is other than cash.
 

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     (e) Any adjustment provided for in Subparagraphs (b) or (c) above shall be
subject to any required shareholder action.
 
     (f) Except as hereinbefore expressly provided, the issuance by the Company
of shares of stock of any class or securities convertible into shares of stock
of any class, for cash, property, labor or services, upon direct sale, upon the
exercise of rights or warrants to subscribe therefor, or upon conversion of
shares or obligations of the Company convertible into such shares or other
securities, and in any case whether or not for fair value, shall not affect, and
no adjustment by reason thereof shall be made with respect to, the number of
shares of Stock subject to Options theretofore granted or the purchase price per
share.
 
                    IX. AMENDMENT OR TERMINATION OF THE PLAN
 
     The Board in its discretion may terminate the Plan at any time with respect
to any shares for which Options have not theretofore been granted. The Board
shall have the right to alter or amend the Plan or any part thereof from time to
time; provided, that no change in any Option theretofore granted may be made
which would materially impair the rights of the Optionee without the consent of
such Optionee; and provided, further, that the Board may not make any alteration
or amendment which would materially increase the benefits accruing to
participants under the Plan, increase the aggregate number of shares which may
be issued pursuant to the provisions of the Plan or extend the term of the Plan,
without the approval of the shareholders of the Company.
 
                               X. SECURITIES LAWS
 
     (a) The Company shall not be obligated to issue any Stock pursuant to any
Option granted under the Plan at any time when the offering of the shares
covered by such Option have not been registered under the Securities Act of 1933
and such other state and federal laws, rules or regulations as the Company or
the Board deems applicable and, in the opinion of legal counsel for the Company,
there is no exemption from the registration requirements of such laws, rules or
regulations available for the offering and sale of such shares.
 
     (b) It is intended that the Plan and any grant of an Option made to a
person subject to Section 16 of the 1934 Act meet all of the requirements of
Rule 16b-3. If any provision of the Plan or any such Option would disqualify the
Plan or such Option under, or would otherwise not comply with, Rule 16b-3, such
provision or Option shall be construed or deemed amended to conform to Rule
16b-3. Nevertheless, the Board shall have discretion to cause the Plan to be
administered or otherwise operated without complying with Rule 16b-3, and
neither the Board nor the Company shall be liable to any Optionee or other
person for failure of the Plan or any award under the Plan to comply with Rule
16b-3.