Document:

<PAGE>

                                                                 EXHIBIT 10.7.9

                      AMENDMENTS TO STOCK OPTION AGREEMENTS
                               (DATED MAY 5, 2000)

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Michael P. Esposito (the "Option Holder") the
options to purchase common stock, $0.01 par value per share, of the Company set
forth on SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999
Long Term Incentive and Share Award Plans; and this amendment is being entered
into in order to induce the Option Holder to continue to provide his or her
services to the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                       ARCH CAPITAL GROUP LTD.

                                         By: /s/        PETER A. APPEL
                                             ---------------------------------
                                             Peter A. Appel
                                             President & Chief Executive Officer

                                          /s/  MICHAEL P. ESPOSITO, JR.
                                          ------------------------------------
                                          Michael P. Esposito, Jr.

<PAGE>

                                                                     SCHEDULE I

                                     OPTIONS

<TABLE>
<CAPTION>

            ------------------------------------------------------------
            DATE OF GRANT                  NUMBER OF OPTION SHARES
            -------------                  -----------------------
<S>                                        <C>
            ------------------------------------------------------------
            9/13/95                        300
            ------------------------------------------------------------
            1/1/96                         250
            ------------------------------------------------------------
            1/1/97                         250
            ------------------------------------------------------------
</TABLE>

<PAGE>

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Ian R. Heap (the "Option Holder") the options to
purchase common stock, $0.01 par value per share, of the Company set forth on
SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999 Long Term
Incentive and Share Award Plans; and this amendment is being entered into in
order to induce the Option Holder to continue to provide his or her services to
the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                       ARCH CAPITAL GROUP LTD.

                                          By:   /s/ PETER A. APPEL
                                             ---------------------------------
                                             Peter A. Appel
                                             President & Chief Executive Officer

                                                /s/ IAN R. HEAP
                                               -------------------------------
                                               Ian R. Heap

<PAGE>

                                                                    SCHEDULE I

                                     OPTIONS

<TABLE>
<CAPTION>

            ------------------------------------------------------------
            DATE OF GRANT                  NUMBER OF OPTION SHARES
            -------------                  -----------------------
<S>                                        <C>
            ------------------------------------------------------------
            9/13/95                        300
            ------------------------------------------------------------
            1/1/96                         250
            ------------------------------------------------------------
            1/1/97                         250
            ------------------------------------------------------------
</TABLE>

<PAGE>

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Philip L. Wroughton (the "Option Holder") the
options to purchase common stock, $0.01 par value per share, of the Company set
forth on SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999
Long Term Incentive and Share Award Plans; and this amendment is being entered
into in order to induce the Option Holder to continue to provide his or her
services to the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                     ARCH CAPITAL GROUP LTD.

                                        By:   /s/ PETER A. APPEL
                                            -----------------------------------
                                            Peter A. Appel
                                            President & Chief Executive Officer

                                              /s/  PHILIP L. WROUGHTON
                                             ----------------------------------
                                             Philip L. Wroughton

<PAGE>

                                                                     SCHEDULE I

                                     OPTIONS

<TABLE>
<CAPTION>

            ------------------------------------------------------------
            DATE OF GRANT                  NUMBER OF OPTION SHARES
            -------------                  -----------------------
<S>                                        <C>
            ------------------------------------------------------------
            9/13/95                        300
            ------------------------------------------------------------
            1/1/96                         250
            ------------------------------------------------------------
            1/1/97                         250
            ------------------------------------------------------------
</TABLE>

<PAGE>

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Lewis L. Glucksman (the "Option Holder") the options
to purchase common stock, $0.01 par value per share, of the Company set forth on
SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999 Long Term
Incentive and Share Award Plans; and this amendment is being entered into in
order to induce the Option Holder to continue to provide his or her services to
the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                     ARCH CAPITAL GROUP LTD.

                                        By:   /s/  PETER A. APPEL
                                           -----------------------------------
                                           Peter A. Appel
                                           President & Chief Executive Officer

                                              /s/  LEWIS L. GLUCKSMAN
                                             ---------------------------------
                                              Lewis L. Glucksman

<PAGE>

                                                                    SCHEDULE I

                                     OPTIONS

<TABLE>
<CAPTION>

            ------------------------------------------------------------
            DATE OF GRANT                  NUMBER OF OPTION SHARES
            -------------                  -----------------------
<S>                                        <C>
            ------------------------------------------------------------
            11/14/95                       300
            ------------------------------------------------------------
            1/1/96                         250
            ------------------------------------------------------------
            1/1/97                         250
            ------------------------------------------------------------
</TABLE>

<PAGE>

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Thomas V. A. Kelsey (the "Option Holder") the
options to purchase common stock, $0.01 par value per share, of the Company set
forth on SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999
Long Term Incentive and Share Award Plans; and this amendment is being entered
into in order to induce the Option Holder to continue to provide his or her
services to the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                      ARCH CAPITAL GROUP LTD.

                                         By:   /s/  PETER A. APPEL
                                            ------------------------------------
                                            Peter A. Appel
                                            President & Chief Executive Officer

                                               /s/  THOMAS V. A. KELSEY
                                              ---------------------------------
                                              Thomas V. A. Kelsey

<PAGE>

                                                                  SCHEDULE I

                                     OPTIONS

<TABLE>
<CAPTION>

            ------------------------------------------------------------
            DATE OF GRANT                  NUMBER OF OPTION SHARES
            -------------                  -----------------------
<S>                                        <C>
            ------------------------------------------------------------
            9/16/96                        300
            ------------------------------------------------------------
            1/1/97                         250
            ------------------------------------------------------------
</TABLE>

<PAGE>

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Debra M. O'Connor (the "Option Holder") the options
to purchase common stock, $0.01 par value per share, of the Company set forth on
SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999 Long Term
Incentive and Share Award Plans; and this amendment is being entered into in
order to induce the Option Holder to continue to provide his or her services to
the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                      ARCH CAPITAL GROUP LTD.

                                        By:   /s/  PETER A. APPEL
                                           ------------------------------------
                                           Peter A. Appel
                                           President and Chief Executive Officer

                                              /s/  DEBRA M. O'CONNOR
                                             ----------------------------------
                                             Debra M. O'Connor

<PAGE>

                                                                     SCHEDULE I
                                     OPTIONS
<TABLE>
<CAPTION>

            ------------------------------------------------------------
            DATE OF GRANT                  NUMBER OF OPTION SHARES
            -------------                  -----------------------
<S>                                        <C>
            ------------------------------------------------------------
            10/11/95                       7,500
            ------------------------------------------------------------
            11/19/96                       3,800
            ------------------------------------------------------------
            11/18/97                       6,353
            ------------------------------------------------------------
            11/17/98                       7,544
            ------------------------------------------------------------
</TABLE>

<PAGE>

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Louis T. Petrillo (the "Option Holder") the options
to purchase common stock, $0.01 par value per share, of the Company set forth on
SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999 Long Term
Incentive and Share Award Plans; and this amendment is being entered into in
order to induce the Option Holder to continue to provide his or her services to
the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                    ARCH CAPITAL GROUP LTD.

                                       By:   /s/  PETER A. APPEL
                                          -------------------------------------
                                          Peter A. Appel
                                          President and Chief Executive Officer

                                             /s/  LOUIS T. PETRILLO
                                            -----------------------------------
                                            Louis T. Petrillo

<PAGE>

                                                                    SCHEDULE I

                                     OPTIONS

<TABLE>
<CAPTION>

            ------------------------------- ----------------------------
            DATE OF GRANT                   NUMBER OF OPTION SHARES
            -------------                   -----------------------
<S>                                         <C>
            ------------------------------- ----------------------------
            1/2/96                          5,000
            ------------------------------- ----------------------------
            11/19/96                        6,200
            ------------------------------- ----------------------------
            11/18/97                        2,653
            ------------------------------- ----------------------------
            11/17/98                        5,144
            ------------------------------- ----------------------------
</TABLE>

<PAGE>

                             ARCH CAPITAL GROUP LTD.

                      AMENDMENT TO STOCK OPTION AGREEMENTS

           WHEREAS, Arch Capital Group Ltd. (the "Company"), a Delaware
corporation, has granted to Peter A. Appel (the "Option Holder") the options to
purchase common stock, $0.01 par value per share, of the Company set forth on
SCHEDULE I hereto (the "Options") under the Company's 1995 and/or 1999 Long Term
Incentive and Share Award Plans; and this amendment is being entered into in
order to induce the Option Holder to continue to provide his or her services to
the Company;

           NOW, THEREFORE, the parties have agreed to amend the Options as
follows:

           1. Paragraph (d) of each Option shall be amended and restated in its
entirety as follows:

                "(d) The Option may be exercised only during the period (the
                "Option Period") ending on May 5, 2007; thereafter the Option
                Holder shall cease to have any rights in respect thereof. The
                right to exercise the Option may be subject to sooner
                termination in the event service with the Company is terminated,
                as provided in paragraph (j) below."

           2. All other provisions of the Options shall remain in full force and
effect. This amendment shall be governed by and construed in accordance with the
laws of the State of New York, without giving effect to principles of conflict
of laws, and may be executed in two counterparts, each of which shall constitute
one and the same instrument.

           IN WITNESS WHEREOF, the undersigned have executed this agreement as
of May 5, 2000.

                                  ARCH CAPITAL GROUP LTD.

                                     By:   /s/  LOUIS T. PETRILLO
                                        ---------------------------------------
                                        Senior Vice President, General Counsel
                                        and Secretary

                                           /s/  PETER A. APPEL
                                          -------------------------------------
                                          Peter A. Appel

<PAGE>

                                                                  SCHEDULE I

                                     OPTIONS

<TABLE>
<CAPTION>

         ---------------------------------- ------------------------------
         DATE OF GRANT                      NUMBER OF OPTION SHARES
         -------------                      -----------------------
<S>                                         <C>
         ---------------------------------- ------------------------------
         11/13/95                           25,000
         ---------------------------------- ------------------------------
         11/19/96                           39,500
         ---------------------------------- ------------------------------
         11/18/97                           34,153
         ---------------------------------- ------------------------------
         11/17/98                           58,344
         ---------------------------------- ------------------------------
</TABLE><PAGE>

                                                                 Exhibit 10.7.10

                STOCK OPTION AGREEMENTS -- NON EMPLOYEE DIRECTORS
                              (2001 ANNUAL GRANTS)

         Each of the non-employee directors of Arch Capital Group Ltd. ("ACGL")
listed below has entered into Stock Option Agreements with ACGL that are
substantially identical in all material respects to the agreement, dated as of
January 1, 2001, between ACGL and Michael P. Esposito, Jr., a copy of which is
included as part of this Exhibit 10.7.10.

Lewis L. Glucksman
Ian R. Heap
Thomas V.A. Kelsey
Mark D. Mosca
Robert F. Works
Philip L. Wroughton

                             ARCH CAPITAL GROUP LTD

                            Director Option Agreement

                  FOR GOOD AND VALUABLE CONSIDERATION, receipt of which is
hereby acknowledged, Arch Capital Group Ltd. (the "Company"), a Bermuda company,
hereby grants to Michael P. Esposito, Jr., a director of the Company on the date
hereof (the "Option Holder"), the option to purchase common shares, $.01 par
value per share, of the Company ("Shares"), upon the following terms:

                  WHEREAS, the following terms reflect the Company's 1999 Long
Term Incentive and Share Award Plan (the "Plan");

                  (a) GRANT. The Option Holder is hereby granted an option (the
"Option") to purchase 1,500 Shares (the "Option Shares") pursuant to the Plan,
the terms of which are incorporated herein by reference. The Option is granted
as of January 1, 2001 (the "Date of Grant") and such grant is subject to the
terms and conditions herein and the terms and conditions of the applicable
provisions of the Plan. Such Option shall not be treated as an incentive stock
option under Section 422 of the Internal Revenue Code of 1986, as amended.

                  (b) STATUS OF OPTION SHARES. The Option Shares shall upon
issue rank equally in all respects with the other Shares.

                  (c) OPTION PRICE. The purchase price for the Option Shares
shall be, except as herein provided, $15.06 per Option Share, hereinafter
sometimes referred to as the "Option Price," payable immediately in full upon
the exercise of the Option.

                  (d) TERM OF OPTION. The Option may be exercised only during
the period (the "Option Period") commencing in accordance with paragraph (f)
below and shall continue until January 1, 2011; thereafter the Option Holder
shall cease to have any rights in respect thereof. The right to exercise the
Option may be subject to sooner termination as provided in paragraph (j) below.

<PAGE>

                  (e) NO RIGHTS OF SHAREHOLDER. The Option Holder shall not, by
virtue hereof, be entitled to any rights of a shareholder in the Company, either
at law or in equity.

                  (f) EXERCISABILITY. The Option shall become exercisable on
January 1, 2002, subject to paragraph (j) below; provided that such Option, to
the extent not already exercisable in full, shall become immediately and fully
exercisable (1) to the extent provided in paragraph (j) below and (2) upon a
Change in Control. Subject to paragraph (j) below, the Option may be exercised
at any time or from time to time during the Option Period in regard to all or
any portion of the Option which is then exercisable, as may be adjusted pursuant
to paragraph (g) below.

                "Change in Control" means any of the following occurring after
the date hereof:

                      a. any person (within the meaning of the Securities
                Exchange Act of 1934, as amended (the "Exchange Act")), other
                than a Permitted Person or Initial Investor, is or becomes the
                "beneficial owner" (as defined in Rule 13d-3 under the Exchange
                Act), directly or indirectly, of Voting Securities representing
                35% or more of the total voting power of all the then
                outstanding Voting Securities; or

                      b. any Initial Investor is or becomes the "beneficial
                owner" (as defined in Rule 13d-3 under the Exchange Act),
                directly or indirectly, of Voting Securities representing 50%
                or more of the total voting power of all the then outstanding
                Voting Securities; or

                      c. the individuals who, as of the Date of Grant,
                constitute the Board of Directors of the Company (the "Board")
                together with those who become directors subsequent to such
                date and whose recommendation, election or nomination for
                election to the Board was approved by a vote of at least a
                majority of the directors then still in office who either were
                directors as of such date or whose recommendation, election or
                nomination for election was previously so approved, cease for
                any reason to constitute a majority of the members of the Board;
                or

                      d. the consummation of a merger, consolidation,
                recapitalization, liquidation, sale or disposition by the
                Company of all or substantially all of the Company's assets, or
                reorganization of the Company, other than any such transaction,
                other than any such transaction which would (x) result in at
                least 60% of the total voting power represented by the voting
                securities of the surviving entity or, in the case of an asset
                sale, the successor entity, outstanding immediately after such
                transaction being beneficially owned, directly or indirectly,
                by the stockholders of the Company immediately preceding the
                transaction and (y) not otherwise be deemed a Change in Control
                under subparagraphs a, b, c or e of this paragraph (f); or

                      e. the Board adopts a resolution to the effect that,
                for purposes hereof, a Change in Control has occurred.

                         (i) "Initial Investors" means (A) The Trident
                    Partnership, L.P.; (B) Marsh & McLennan Risk Capital
                    Holdings, Ltd.; or (C) any majority-owned subsidiary or
                    parent (or equivalent in the case of a non-corporate entity)
                    of the foregoing.

                                       2
<PAGE>

                         (ii) "Permitted Persons" means (A) the Company; (B) any
                    Related Party; or (C) any group (as defined in Rule 13d-3
                    under the Exchange Act) comprised of any or all of the
                    foregoing.

                         (iii) "Related Party" means (A) a majority-owned
                    subsidiary of the Company; (B) a trustee or other fiduciary
                    holding securities under an employee benefit plan of the
                    Company or any majority-owned subsidiary of the Company; or
                    (C) a corporation owned directly or indirectly by the
                    stockholders of the Company in substantially the same
                    proportion as their ownership of Voting Securities.

                         (iv) "Voting Security" means any security of the
                    Company which carries the right to vote generally in the
                    election of directors.

                  (g) ADJUSTMENTS FOR RECAPITALIZATION AND DIVIDENDS. In the
event that, prior to the expiration of the Option, any dividend in Shares,
recapitalization, Share split, reverse split, reorganization, merger,
consolidation, spin-off, combination, repurchase, or share exchange, or other
such change affects the Shares such that they are increased or decreased or
changed into or exchanged for a different number or kind of shares, other
securities of the Company or of another corporation or other consideration, then
in order to maintain the proportionate interest of the Option Holder and
preserve the value of the Option, (i) there shall automatically be substituted
for each Share subject to the unexercised Option the number and kind of shares,
other securities or other consideration into which each outstanding Share shall
be changed or for which each such Share shall be exchanged, and (ii) the
exercise price shall be increased or decreased proportionately so that the
aggregate purchase price for the Shares subject to the unexercised Option shall
remain the same as immediately prior to such event.

                  (h) TRANSFERABILITY. The Option may not be assigned or
otherwise transferred, disposed of or encumbered by the Option Holder, other
than by will or by the laws of descent and distribution. During the lifetime of
the Option Holder, the Option shall be exercisable only by the Option Holder or
by his or her guardian or legal representative. Notwithstanding the foregoing,
the Option may be transferred by the Option Holder to members of his or her
"immediate family" or to a trust established for the exclusive benefit of solely
one or more members of the Option Holder's "immediate family." Any Option held
by the transferee will continue to be subject to the same terms and conditions
that were applicable to the Option immediately prior to the transfer, except
that the Option will be transferable by the transferee only by will or the laws
of descent and distribution. For purposes hereof, "immediate family" means the
Option Holder's children, stepchildren, grandchildren, parents, stepparents,
grandparents, spouse, siblings (including half brothers and sisters), in-laws,
and relationships arising because of legal adoption.

                  (i) EXERCISE OF OPTION. In order to exercise the Option, the
Option Holder shall submit to the Company an instrument in writing signed by the
Option Holder, specifying the number of Option Shares in respect of which the
Option is being exercised, accompanied by payment of the Option Price for the
Option Shares for which the Option is being exercised in cash or Shares already
owned by the Option Holder (provided that the Option Holder has owned such
Shares for a minimum period of six months) and having a total Fair Market Value
(as defined below) equal to the exercise price, or in a combination of cash and
such Shares. Option Shares will be issued accordingly by the Company within 15
business days, and a share certificate dispatched to the Option Holder within 30
days.

                  The Company shall not be required to issue fractional Shares
upon the exercise of the Option. If any fractional interest in a Share would be
deliverable upon the exercise of the Option in whole

                                       3
<PAGE>

or in part but for the provisions of this paragraph, the Company, in lieu of
delivering any such fractional share therefor, shall pay a cash adjustment
therefor in an amount equal to their Fair Market Value (or if any Shares are not
publicly traded, an amount equal to the book value per share at the end of the
most recent fiscal quarter) multiplied by the fraction of the fractional share
which would otherwise have been issued hereunder. Anything to the contrary
herein notwithstanding, the Company shall not be obligated to issue any Option
Shares hereunder if the issuance of such Option Shares would violate the
provision of any applicable law, in which event the Company shall, as soon as
practicable, take whatever action it reasonably can so that such Option Shares
may be issued without resulting in such violations of law. For purposes hereof,
Fair Market Value shall mean the mean between the high and low selling prices
per Share on the immediately preceding date (or, if the Shares were not traded
on that day, the next preceding day that the Shares were traded) on the
principal exchange on which the Shares are traded, as such prices are officially
quoted on such exchange.

                  (j) TERMINATION OF SERVICE. In the event the Option Holder
ceases to be a director of the Company (i) due to retirement after attainment of
age 65 or (ii) due to death or disability, the Option, to the extent not already
exercisable in full, shall become immediately and fully exercisable at the time
of such termination of service, and the Option may be exercised at any time
during the Option Period. Subject to paragraph (f) above, if the Option Holder
ceases to be a director of the Company for any other reason, the portion of the
Option which is not then exercisable shall be cancelled on the date service
terminates, and the portion of the Option which is then exercisable may be
exercised at any time within six months after the date of such termination, but
not later than termination of the Option Period.

                  (k) OBLIGATIONS AS TO CAPITAL. The Company agrees that it will
at all times maintain authorized and unissued share capital sufficient to
fulfill all of its obligations under the Option.

                  (l) TRANSFER OF SHARES. The Option, the Option Shares, or any
interest in either, may be sold, assigned, pledged, hypothecated, encumbered, or
transferred or disposed of in any other manner, in whole or in part, only in
compliance with the terms, conditions and restrictions as set forth in the
governing instruments of the Company, applicable United States federal and state
securities laws and the terms and conditions hereof. Each certificate for Option
Shares issued upon exercise of the Option, unless at the time of exercise such
Option Shares are registered under the Securities Act of 1933, as amended, shall
bear the following legend or such other legend as the Company deems appropriate:

                  "The securities evidenced hereby have not been registered
                  under the Securities Act of 1933, as amended (the `Act'), and
                  may not be offered, sold or otherwise transferred except (i)
                  in compliance with the provisions of any applicable state
                  securities or `Blue Sky' laws and (ii) (A) pursuant to an
                  effective registration under the Act, (B) in compliance with
                  Rule 144 under the Act, (C) inside the United States to a
                  Qualified Institutional Buyer in compliance with Rule 144A
                  under the Act, (D) outside the United States in compliance
                  with Rule 904 of Regulation S under the Act or (E) inside the
                  United States to an institutional `accredited investor' as
                  defined in Rule 501(a)(1), (2), (3) or (7) under the Act in a
                  transaction which, in the opinion of counsel reasonably
                  satisfactory to the Company, qualifies as an exempt
                  transaction under the Act and the rules and regulations
                  promulgated thereunder."

Any certificate issued at any time in exchange or substitution for any
certificate bearing such legend or such other legend deemed appropriate by the
Company shall also bear such legend unless, in the opinion

                                       4
<PAGE>

of counsel for the Company, the securities represented thereby need no longer be
subject to the restrictions set forth therein. The provisions of this paragraph
(l) shall be binding upon all subsequent holders of certificates bearing the
above legend and all subsequent holders of the Option, if any.

                  (m) EXPENSES OF ISSUANCE OF OPTION SHARES. The issuance of
stock certificates upon the exercise of the Option in whole or in part, shall be
without charge to the Option Holder. The Company shall pay, and indemnify the
Option Holder from and against any issuance, stamp or documentary taxes (other
than transfer taxes) or charges imposed by any governmental body, agency or
official (other than income taxes) by reason of the exercise of the Option in
whole or in part or the resulting issuance of the Option Shares.

                  (n) WITHHOLDING. The Option Holder agrees to make appropriate
arrangements with the Company for satisfaction of any applicable tax withholding
requirements, or similar requirements, arising out of the Option.

                  (o) REFERENCES. References herein to rights and obligations of
the Option Holder shall apply, where appropriate, to the Option Holder's legal
representative or estate without regard to whether specific reference to such
legal representative or estate is contained in a particular provision of this
Option.

                  (p) SETTLEMENT OF DISPUTES. Any dispute between the parties
arising from or relating to the terms of this Option shall be resolved by
arbitration held in the State of Connecticut in accordance with the rules of the
American Arbitration Association. All costs associated with any arbitration,
including all legal expenses, for both parties shall be borne by the Company.

                  (q) NO MITIGATION. To the extent that the vesting of the
Option is accelerated upon a Change in Control or upon a termination of service
as provided herein, neither the Option, nor any Option Shares nor any interest
in either, shall be reduced by any compensation received by the Option Holder in
connection with any other employment.

                  (r) NOTICES. Any notice required or permitted to be given
under this agreement shall be in writing and shall be deemed to have been given
when delivered personally or by courier, or sent by certified or registered
mail, postage prepaid, return receipt requested, duly addressed to the party
concerned at the address indicated below or to such changed address as such
party may subsequently by similar process give notice of:

           If to the Company:

                  Arch Capital Group Ltd.
                  EXECUTIVE OFFICES:
                  20 Horseneck Lane
                  Greenwich, CT  06830
                  Attn:  Secretary

           If to the Option Holder:

                  [Address of Option Holder]

                                       5
<PAGE>

                  (s) GOVERNING LAW. This agreement shall be governed by and
construed in accordance with the laws of the State of New York, without giving
effect to principles of conflict of laws.

                  (t) ENTIRE AGREEMENT. This agreement constitutes the entire
agreement among the parties relating to the subject matter hereof, and any
previous agreement or understanding among the parties with respect thereto is
superseded by this agreement.

                  (u) COUNTERPARTS. This agreement may be executed in two
counterparts, each of which shall constitute one and the same instrument.

         IN WITNESS WHEREOF, the undersigned have duly executed this agreement
as of the Date of Grant.

                                      ARCH CAPITAL GROUP LTD.

                                      By:    /s/  PETER A. APPEL
                                          ------------------------------------
                                          Peter A. Appel
                                          President & Chief Executive Officer

                                             /s/ MICHAEL P. ESPOSITO
                                            -----------------------------------
                                            Michael P. Esposito, Jr.

                                       6

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