QuickLinks -- Click here to rapidly navigate through this document

Exhibit 10.4

APEX SILVER MINES LIMITED

2004 EQUITY INCENTIVE PLAN

INCENTIVE SHARE OPTION AGREEMENT

        This Incentive Share Option Agreement (the "Agreement"), made as of
the            day of                        , by and between Apex Silver Mines
Limited, an exempted limited liability company duly formed and existing under
the laws of the Cayman Islands (the "Company"), and                        (the
"Participant").

        WHEREAS, the Company desires to encourage and enable the Participant to
acquire a proprietary interest in the Company through the ownership of the
Company's ordinary shares, par value US$0.01 per share (the "Ordinary Shares")
pursuant to the terms and conditions of the Apex Silver Mines Limited 2004
Equity Incentive Plan (the "Plan") and this Agreement. Such ownership will
provide the Participant with a more direct stake in the future of the Company
and encourage the Participant to remain with the Company and/or its Affiliates,
as applicable.

        NOW, THEREFORE, in consideration of the mutual covenants hereinafter set
forth and for other good and valuable consideration, the parties agree as
follows:

        1.     DEFINITIONS. For purposes of this Agreement, all capitalized
terms used herein and not otherwise defined herein shall have the meanings
ascribed to them in the Plan.

        2.     GRANT OF OPTION. The Company hereby grants to the Participant an
incentive share option (the "Option") to
purchase                        Ordinary Shares at the exercise price (the
"Exercise Price") of $            per Ordinary Share, subject to the terms and
conditions of this Agreement and the Plan.

        3.     OPTION TERM. The Option granted hereby shall expire
on                        (the "Expiration Date"), unless sooner terminated or
modified under the provisions of this Agreement or the Plan. Except as otherwise
set forth herein, the Option may not be exercised after the Expiration Date.

        4.     WHEN VESTED AND EXERCISABLE. The Option shall vest and be
exercisable by the Participant in accordance with the following schedule:

Date

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

  Number of Options Vested

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

 
 
 

        5.     EMPLOYMENT TERMINATION: DEATH; DISABILITY; RETIREMENT; CAUSE.

        (a)   If the services of the Participant are terminated for any reason
other than death, Disability, Retirement, Change of Control, or Cause, the
portion of this Option to purchase Ordinary Shares that is not vested on the
date of such termination of service shall terminate and be forfeited on such
date of termination; however, the vested portion of this Option shall be
exercisable by the Participant at any time on or prior to the earlier of (i) the
Expiration Date or (ii) the 180 day anniversary of the date of such termination
of service. Any portion of this Option not exercised within the period described
in the preceding sentence, for whatever reason, shall terminate.

        (b)   In the event of the death or Disability of the Participant, the
unvested portion of this Option shall immediately terminate and be forfeited,
and the vested portion of the Option on such date shall be exercisable at any
time on or prior to the 12 month anniversary of such date by the

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

beneficiary designated by the Participant for such purpose (the "Designated
Beneficiary") or if no Designated Beneficiary shall be appointed or if the
Designated Beneficiary shall predecease the Participant, by the Participant's
personal representatives, heirs or legatees. Any portion of the Option not
exercised within the period described in the preceding sentence, for whatever
reason, shall terminate.

        (c)   In the event of the Retirement of the Participant, any unvested
portion of this Option shall immediately vest on the date of Retirement and this
Option shall be exercisable by such Participant at any time on or prior to the
earlier of (i) the Expiration Date, or (ii) the 24 month anniversary of the date
of such retirement.

        (d)   In the event the service of the Participant is terminated for
Cause, this Option (including any vested portion) shall be forfeited as of the
date of termination.

        6.     CHANGE OF CONTROL. In the event of a Change in Control, the
Company shall give the Participant notice thereof and this Option, whether or
not currently vested and exercisable, shall become immediately vested and
exercisable immediately prior to the effective date of the Change of Control,
and the Board shall have the power and discretion to provide alternatives
regarding the terms and conditions for the exercise of, or modification of, this
Option in accordance with the Plan.

        7.     INCENTIVE STOCK OPTION TAX MATTERS.

        (a)   This Option is intended to qualify as an incentive stock option
within the meaning of Section 422 of the Code. The Board may take all
appropriate action to achieve this result. The Exercise Price has been
determined to be equal to or greater than the Fair Market Value per Ordinary
Share at the time of grant.

        (b)   To the extent the aggregate Fair Market Value (determined at the
time of grant) of the Ordinary Shares with respect to which the Option plus all
other incentive stock options Participant holds that are exercisable for the
first time by Participant during any calendar year exceeds one hundred thousand
dollars ($100,000), Participant's option(s) or portions thereof that exceed such
limit (according to the order in which they were granted) shall be treated as
Non-Qualified Share Options.

        8.     NON-ASSIGNABILITY. The Option granted hereby and any right
arising thereunder may not be transferred, assigned, pledged or hypothecated
(whether by operation of law or otherwise), except by will or the applicable
laws of descent and distribution, and the Option and any right arising
thereunder shall not be subject to execution, attachment or similar process. Any
attempted assignment, transfer, pledge, hypothecation or other disposition of an
Option not specifically permitted herein or in the Plan shall be null and void
and without effect. An Option may be exercised solely by the Participant during
his or her lifetime, or following his or her death pursuant to
Section 5(b) hereof.

        9.     MODE OF EXERCISE. The Option may be exercised in whole or in
part. Ordinary Shares purchased upon the exercise of the Option shall be paid
for in full at the time of such purchase. Such payment shall be made in cash or
by wire transfer in immediately available funds in either event denominated
Dollars. Upon receipt of notice of exercise and payment in accordance with
procedures to be established by the Board, the Company or its agent shall
deliver to the person exercising the Option (or his or her designee) a
certificate for such Ordinary Shares.

        10.   RECAPITALIZATION. The number of Ordinary Shares covered by this
Option and the Exercise Price shall be proportionately adjusted for any increase
or decrease in the number of issued Ordinary Shares as set forth in the Plan;
provided, however, that any fractional shares resulting from any such adjustment
shall be eliminated. The Board may also make any other changes, including
changes in the classes of securities available, to the extent it is deemed
necessary or desirable to preserve the intended benefits of the Plan for the
Company and the Participants in the event of any

2

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

other reorganization, recapitalization, merger, consolidation, spin-off,
extraordinary dividend or other distribution or similar transaction.
Notwithstanding any other provision of the Plan or this Agreement, the Board may
cause the Option granted hereunder to be canceled in consideration of a cash
payment or alternative Share Award made to the holder of such canceled Share
Award equal in value to the Fair Market Value of such canceled Share Award.

        11.   PLAN CONTROLLING. This Agreement is intended to conform in all
respects with the requirements of the Plan. Inconsistencies between the
requirements of this Agreement and the Plan shall be resolved according to the
terms of the Plan. The Participant acknowledges receipt of a copy of the Plan.

        12.   RIGHTS PRIOR TO EXERCISE OF OPTION. The Participant shall not have
any rights as a shareholder with respect to any Ordinary Shares subject to the
Option prior to the date on which he is recorded as the holder of such Ordinary
Shares on the records of the Company.

        13.   WITHHOLDING TAXES. The Company shall have the right to require
Participants or their beneficiaries or legal representatives to remit to the
Company an amount sufficient to satisfy any Cayman Islands or United States
federal, state and local withholding tax requirements, including upon the grant,
vesting or exercise of this Option. Whenever payments under the Plan or this
Agreement are to be made to any Participant in cash, such payments shall be net
of any amounts sufficient to satisfy all applicable taxes, including without
limitation, all applicable Cayman Islands or United States federal, state and
local withholding tax requirements to be withheld or submitted by the Company
concerning such payments. The Board may, in its sole discretion, permit a
Participant to satisfy his or her tax withholding obligation either by
(i) surrendering Ordinary Shares owned by the Participant or (ii) having the
Company withhold from Ordinary Shares otherwise deliverable to the Participant.
Ordinary Shares surrendered or withheld shall be valued at their Fair Market
Value as of the date on which income is required to be recognized for income tax
purposes.

        14.   NO LIABILITY OF BOARD COMMITTEE MEMBERS. No member of the Board or
any Committee or their designees shall be personally liable by reason of any
contract or other instrument executed by such member or on his behalf in his
capacity as a member of the Board or a Committee nor for any mistake of judgment
made in good faith.

        15.   GOVERNING LAW. This Agreement and all rights arising hereunder
shall be governed by, and construed and interpreted in accordance with, the laws
of the Delaware.

        NEITHER THE PLAN NOR THIS AGREEMENT SHALL BE CONSTRUED AS GIVING THE
PARTICIPANT THE RIGHT TO BE RETAINED IN THE EMPLOY OR SERVICE OF THE COMPANY OR
ANY AFFILIATE THEREOF, NOR SHALL THEY INTERFERE IN ANY WAY WITH THE RIGHT OF THE
COMPANY OR ANY AFFILIATE THEREOF, AS APPLICABLE, TO TERMINATE THE PARTICIPANT'S
EMPLOYMENT OR SERVICE AT ANY TIME WITH OR WITHOUT CAUSE.

* * * * *

3

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

        Executed as of the day and year first above written.

 
 
APEX SILVER MINES LIMITED
 
 
By:
 

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

Name:
Title:
 
 
PARTICIPANT
 
 
By:
 

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

4

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

QuickLinks

APEX SILVER MINES LIMITED 2004 EQUITY INCENTIVE PLAN INCENTIVE SHARE OPTION
AGREEMENT