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                                                                    Exhibit 10.4

                                 ZIX CORPORATION
                                OUTSIDE DIRECTOR
                             STOCK OPTION AGREEMENT
                       (2004 DIRECTORS' STOCK OPTION PLAN)

Effective Date of Grant                                          Expiration Date

May 6, 2004                                                          May 5, 2014

TO:   Antonio R. Sanchez III ("Optionee")

      WHEREAS, Zix Corporation (the "Company") wishes to recognize the
contributions of the Optionee to the Company and to encourage the Optionee's
sense of proprietorship in the Company by owning the common stock, par value
$.01 per share (the "Common Stock"), of the Company;

      NOW, THEREFORE, in consideration of the mutual agreements and covenants
contained herein, the Company hereby grants to the Optionee a nonqualified stock
option ("Option") to purchase up to a total of 50,000 shares of the Common Stock
at a price per share of $8.89 (the "Option Price") on the terms and conditions
and subject to the restrictions as set forth in this Agreement and the
provisions in the Zix Corporation 2004 Directors' Stock Option Plan (which is
incorporated herein by reference) (the "Plan"). All defined terms contained
herein shall have the meanings ascribed to them in the Plan, except as otherwise
provided herein.

1.    DEFINITIONS.

      a.    Acquiring Person. An "Acquiring Person" shall mean any person
(including any "person" as such term is used in Sections 13(d)(3) or 14(d)(2) of
the Exchange Act that, together with all Affiliates and Associates of such
person, is the beneficial owner (as the term "beneficial owner" is defined under
Rule 13d-3 or any successor rule or regulation promulgated under the Exchange
Act) of 10% or more of the outstanding Common Stock. The term "Acquiring Person"
shall not include the Company, any majority-owned subsidiary of the Company, any
employee benefit plan of the Company or a majority-owned subsidiary of the
Company, or any person to the extent such person is holding Common Stock for or
pursuant to the terms of any such plan. For the purposes of this Agreement, a
person who becomes an Acquiring Person by acquiring beneficial ownership of 10%
or more of the Common Stock at any time after the date of this Agreement shall
continue to be an Acquiring Person whether or not such person continues to be
the beneficial owner of 10% or more of the outstanding Common Stock.

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      b.    Affiliate and Associate. "Affiliate" and "Associate" shall have the
respective meanings ascribed to such terms in Rule 12b-2 of the General Rules
and Regulations under the Exchange Act in effect on the date of this Agreement.

      c.    Cause. "Cause" shall mean the willful engaging by the Optionee in
illegal conduct or gross misconduct that is materially and demonstrably
injurious to the Company or its subsidiaries. For purposes of this definition,
no act or failure to act on the part of the Optionee shall be considered willful
unless it is done, or omitted to be done, by the Optionee in bad faith or
without reasonable belief that the Optionee's action or omission was in the best
interest of the Company.

      d.    Change in Control. A "Change in Control" of the Company shall have
occurred if during the term of this Agreement, any of the following events shall
occur:

            (i)   The Company is merged, consolidated or reorganized into or
      with another corporation or other legal person and as a result of such
      merger, consolidation or reorganization, the Company or its shareholders
      or Affiliates immediately before such transaction beneficially own,
      immediately after or as a result of such transaction, equity securities of
      the surviving or acquiring corporation or such corporation's parent
      corporation possessing less than fifty-one percent (51%) of the voting
      power of the surviving or acquiring person or such person's parent
      corporation;

            (ii)  The Company sells all or substantially all of its assets to
      any other corporation or other legal person and as a result of such sale,
      the Company or its shareholders or Affiliates immediately before such
      transaction beneficially own, immediately after or as a result of such
      transaction, equity securities of the surviving or acquiring corporation
      or such corporation's parent corporation possessing less than fifty-one
      percent (51%) of the voting power of the surviving or acquiring person or
      such person's parent corporation (provided that this provision shall not
      apply to a registered public offering of securities of a subsidiary of the
      Company, which offering is not part of a transaction otherwise a part of
      or related to a Change in Control);

            (iii) Any Acquiring Person has become the beneficial owner (as the
      term "beneficial owner" is defined under Rule 13d-3 or any successor rule
      or regulation promulgated under the Exchange Act) of securities which,
      when added to any securities already owned by such person, would represent
      in the aggregate 35% or more of the then outstanding securities of the
      Company which are entitled to vote to elect any class of directors;

            (iv)  If, at any time, the Continuing Directors then serving on the
      Board of Directors of the Company cease for any reason to constitute at
      least a majority thereof; or

            (v)   Any occurrence that would be required to be reported in
      response to Item 6(e) of Schedule 14A of Regulation 14A or any successor
      rule or regulation promulgated under the Exchange Act.

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      e.    Continuing Director. A "Continuing Director" shall mean a director
of the Company who (i) is not an Acquiring Person or an Affiliate or Associate
thereof, or a representative of an Acquiring Person or nominated for election by
an Acquiring Person, and (ii) was either a member of the Board of Directors of
the Company on the date of this Agreement or subsequently became a director of
the Company and whose initial election or initial nomination for election by the
Company's shareholders was approved by a majority of the Continuing Directors
then on the Board of Directors of the Company.

      f.    Exchange Act. "Exchange Act" shall mean the Securities Exchange Act
of 1934, as amended.

      g.    Option Shares. "Option Shares" shall mean the Common Stock shares
received upon exercise of the Option.

      h.    Transfer. "Transfer" (or any derivative thereof) means a direct or
indirect assignment, sale, transfer, license, lease, pledge, encumbrance,
hypothecation or execution, attachment or similar process.

2.    TERM OF OPTION. The term of the Option shall expire at 12:00 midnight on
the date set forth in the upper right hand corner on page 1 of this Agreement
(the "Expiration Date" or "stated term"), except as such term may be otherwise
shortened by the other provisions of the Plan or this Agreement.

3.    EXERCISE OF OPTION.

      a.    Exercise. The Option shall become exercisable in increments as
follows:

<TABLE>
<CAPTION>
                            Date Upon Which Right
Number of Shares             To Purchase Accrues
----------------             -------------------
<S>                         <C>
     4,166                        August 6, 2004
     4,167                        November 6, 2004
     4,167                        February 6, 2005
     4,166                        May 6, 2005
     4,167                        August 6, 2005
     4,167                        November 6, 2005
     4,166                        February 6, 2006
     4,167                        May 6, 2006
     4,167                        August 6, 2006
     4,166                        November 6, 2006
     4,167                        February 6, 2007
     4,167                        May 6, 2007
</TABLE>

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            Except as provided in the Plan, the Option shall not be exercisable
unless the Optionee is, at the time of exercise, a director of the Company, and
once the Option has become exercisable with respect to a certain number of
shares as provided above, it shall thereafter be exercisable as to all of that
number of shares, or as to any part thereof, until the expiration or termination
of the Option. However, the Option may not be exercised as to less than 100
shares at any one time (or the remaining shares then purchasable under the
Option, if less than 100 shares).

      b.    Adjustment. In the event there is any adjustment to the Common Stock
pursuant to Section 9 of the Plan, the Board of Directors or Committee shall
make such adjustment as it deems appropriate to the number of shares subject to
the Option or to the exercise price listed above, or both. If a merger,
consolidation, sale of shares, or similar transaction involving the Company, on
the one hand, and one or more persons, on the other hand, with respect to the
Company occurs, and, as a part of such transaction, shares of stock, other
securities, cash or property shall be issuable or deliverable in exchange for
Common Stock, then the Optionee shall be entitled to purchase or receive (in
lieu of the Option Shares that the Optionee would otherwise be entitled to
purchase or receive hereunder), the number of shares of stock, other securities,
cash or property to which that number of shares of Common Stock would have been
entitled in connection with such transaction (and, at an aggregate exercise
price equal to the aggregate exercise price hereunder that would have been
payable if that number of shares of Common Stock had been purchased on the
exercise of the Option immediately before the consummation of the transaction).

      c.    Accelerated Vesting. The Option shall become fully exercisable (i)
upon the occurrence of a Change of Control, if the Optionee is still a director
of the Company on the date of the occurrence of the Change of Control or (ii) if
the Optionee is removed from the Board of Directors of the Company by a vote of
the shareholders other than for Cause. If either of such events occurs, the
Option may be exercised at any time or times thereafter until the expiration or
termination of the Option.

      d.    Method of Exercise. To exercise the Option with respect to any
vested shares of Common Stock hereunder, the Optionee shall provide written
notice (the "Exercise Notice") to the Company at its principal executive office.
The Exercise Notice shall be deemed given when deposited in the U. S. mails,
postage prepaid, addressed to the Company at its principal executive office, or
if given other than by deposit in the U.S. mails, when delivered in person to an
officer of the Company at that office. The date of exercise of the Option (the
"Exercise Date") shall be the date of the postmark if the notice is mailed or
the date received if the notice is delivered other than by mail. The Exercise
Notice shall state the number of shares in respect of which the Option is being
exercised and, if the shares for which the Option is being exercised are to be
evidenced by more than one stock certificate, the denominations in which the
stock certificates are to be issued. The Exercise Notice shall be signed by the
Optionee and shall include the complete address of such person, together with
such person's social security number. If the Option is exercised in full, the
Optionee shall surrender this Agreement to the Company for cancellation. If the
Option is exercised in part, the Optionee shall surrender this Agreement to the
Company so that the Company may make appropriate notation hereon or cancel this
Agreement and issue a new agreement representing the unexercised portion of the
Option.

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      At the time of exercise, the Optionee shall pay to the Company the Option
Price times the number of vested shares as to which the Option is being
exercised. The Optionee shall make such payment by delivering (a) cash, (b) a
certified cashier's check or (c) at the Committee's election, any other
consideration that the Committee determines is consistent with the Plan and
applicable law. If the shares to be purchased are covered by an effective
registration statement under the Securities Act of 1933, as amended, the Option
may be exercised by a broker-dealer acting on behalf of the Optionee if (a) the
broker-dealer has received from the Optionee or the Company a fully- and
duly-endorsed agreement evidencing such Option, together with instructions
signed by the Optionee requesting the Company to deliver the shares of Common
Stock subject to such Option to the broker-dealer on behalf of the Optionee and
specifying the account into which such shares should be deposited, (b) adequate
provision has been made with respect to the payment of any withholding taxes due
upon such exercise, and (c) the broker-dealer and the Optionee have otherwise
complied with Section 220.3(e)(4) of Regulation T, 12 CFR Part 220, or any
successor provision.

      The certificates for shares of Common Stock as to which the Option shall
have been so exercised shall be registered in the name of the Optionee and shall
be delivered to the Optionee at the address specified in the Exercise Notice. If
applicable, the stock certificates shall contain an appropriate legend
referencing the transfer restrictions noted in Subparagraph 3.d. An Option
exercise shall be valid only if the Optionee makes payment or other arrangements
relating to the withholding tax obligations discussed in Paragraph 11. In the
event the person exercising the Option is a transferee of the Optionee by will
or under the laws of descent and distribution, the Exercise Notice shall be
accompanied by appropriate proof of the right of such transferee to exercise the
Option.

4.    WHO MAY EXERCISE OPTION. The Option shall be exercisable during the
lifetime of the Optionee only by the Optionee. To the extent exercisable after
the Optionee's death, the Option shall be exercised only by a person who has
obtained the Optionee's rights under the Option by will or under the laws of
descent and distribution.

5.    TERMINATION OF OPTION. If the Optionee's directorship is terminated by a
vote of the shareholders or directors for Cause, the Option shall automatically
expire (and shall not thereafter be exercisable) simultaneously with such
termination. If the Optionee's directorship terminates for any other reason, the
Option shall be exercisable with respect to the shares that were vested as of
the termination date until the first anniversary of the date that the Optionee
ceased to be a director of the Company (or, if the remaining stated term of the
Option is shorter, until 12:00 midnight on the Expiration Date). The Option
shall not thereafter be exercisable.

6.    NO RIGHTS AS SHAREHOLDER. Neither the Optionee nor any person claiming
under or through the Optionee shall be or have any rights or privileges of a
shareholder of the Company in respect of any of the shares issuable upon the
exercise of the Option, unless and until certificates representing such shares
shall have been issued (as evidenced by the appropriate entry on the books of
the Company or of a duly authorized transfer agent of the Company).

7.    STATE AND FEDERAL SECURITIES REGULATION. No shares shall be issued by the
Company upon the exercise of the Option unless and until any then-applicable
requirements of state and federal laws

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and regulatory agencies shall have been fully complied with to the satisfaction
of the Company and its counsel. The Company may suspend for a reasonable period
or periods the time during which the Option may be exercised if, in the opinion
of the Company, such suspension is required to enable the Company to remain in
compliance with regulatory requirements relating to the issuance of shares of
Common Stock subject to the Option. The Option is subject to the requirement
that, if at any time the Company shall determine, in its discretion, that the
listing, registration or qualification of the shares of Common Stock subject to
the Option upon any securities exchange or under any state or federal law, or
the consent or approval of any government regulatory body, is necessary or
desirable as a condition of, or in connection with, the granting or exercise of
the Option or the issue or purchase of shares under the Option, the Option may
not be exercised in whole or in part until such listing, registration,
qualification, consent or approval shall have been effected or obtained free of
any conditions not acceptable to the Company. The Company shall be under no
obligation to effect or obtain any such listing, registration, qualification,
consent or approval if the Company shall determine, in its discretion, that such
action would not be in the best interest of the Company. The Company shall not
be liable for damages due to a delay in the delivery or issuance of any stock
certificates for any reason whatsoever, including, but not limited to, a delay
caused by listing, registration or qualification of the shares of Common Stock
subject to an option upon any securities exchange or under any federal or state
law or the effecting or obtaining of any consent or approval of any governmental
body with respect to the granting or exercise of the Option or the issue or
purchase of shares under the Option.

8.    MODIFICATION OF OPTIONS. At any time and from time-to-time, the Committee
may execute an instrument providing for modification, extension or renewal of
the Option, provided that no such modification, extension or renewal shall (i)
impair the Option in any respect without the written consent of the holder of
the Option or (ii) conflict with the provisions of Rule 16b-3 under the Exchange
Act. Except as provided in the preceding sentence, no supplement, modification
or amendment of this Agreement or waiver of any provision of this Agreement
shall be binding unless executed in writing by all parties to this Agreement. No
waiver of any of the provisions of this Agreement shall be deemed or shall
constitute a waiver of any other provision of this Agreement (regardless of
whether similar), nor shall any such waiver constitute a continuing waiver
unless otherwise expressly provided.

9.    CONTINUED DIRECTORSHIP NOT PRESUMED. Nothing in this Agreement, the Plan
or any document describing it nor the grant of an option shall give the Optionee
the right to continue as a director of the Company.

10.   OPTION ISSUED PURSUANT TO PLAN. The Optionee accepts the Option herein
subject to all the provisions of the Plan, which are incorporated herein,
including the provisions that authorize the Committee to administer and
interpret the Plan and provide that the Committee's determinations and
interpretations with respect to the Plan are final and conclusive and binding on
all persons affected thereby.

11.   TAX WITHHOLDING. Any provision of this Agreement to the contrary
notwithstanding, the Company may take such steps as it deems necessary or
desirable for the withholding of any taxes that it is required by law or
regulation of any governmental authority, federal, state or local, domestic or

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foreign, to withhold in connection with any of the shares of Common Stock
subject hereto.

12.   NO LIABILITY OF OPTION. The Option is not liable for or subject to, in
whole or in part, the debts, contracts, liabilities or torts of the Optionee nor
shall it be subject to garnishment, attachment, execution, levy or other legal
or equitable process.

13.   NO ASSIGNMENT. The Option is not Transferable otherwise than by will or
the laws of descent and distribution, and is exercisable during the Optionee's
lifetime only by him or her. Without limiting the generality of the foregoing,
the Option may not be Transferred (except as aforesaid), and shall not be
subject to execution, attachment or similar process, without the prior written
consent of the Company. Any attempted Transfer contrary to the provisions hereof
shall be void and ineffective for all purposes.

14.   GOVERNING LAW. This Agreement has been executed in, and shall be deemed to
be performable in, Dallas, Dallas County, Texas. The parties agree that this
Agreement shall be governed by and construed in accordance with the laws of the
State of Texas (excluding its conflict of laws rules). The parties further agree
that the courts of the State of Texas, and any courts whose jurisdiction is
derivative on the jurisdiction of the courts of the State of Texas, shall have
personal jurisdiction over all parties to this Agreement.

15.   ENTIRE AGREEMENT. Except for the Plan, this Agreement constitutes the
entire agreement between the parties pertaining to the subject matter hereof and
supersedes all prior and contemporaneous agreements, representations and
understandings of the parties. No supplement, modification or amendment of this
Agreement shall be binding unless executed in writing by the party to be charged
therewith. No waiver of any of the provisions of this Agreement shall be deemed,
or shall, constitute a waiver of any other provision, whether or not similar,
nor shall any waiver constitute a continuing waiver.

16.   DUPLICATE ORIGINALS. Duplicate originals of this document shall be
executed by both the Company and the Optionee, each of which shall retain one
duplicate original.

17.   NOTICE. Other than any Exercise Notice, any notice required or permitted
to be given under the Plan or this Agreement shall be in writing and delivered
in person or sent by registered or certified mail, return receipt requested,
first-class postage prepaid, (i) if to the Optionee, at the address shown on the
books and records of the Company or at the Optionee's place of employment, or
(ii) if to the Company, at 2711 N. Haskell Avenue, Suite 2300, Dallas, Texas
75204-2960, Attention: Vice President Finance & Administration, or any other
address that may be given by either party to the other party by notice pursuant
to this Paragraph 17. Any notice other than any Exercise Notice, if sent by
registered or certified mail, shall be deemed to have been given when received.

18.   MISCELLANEOUS.

      a.    The Option herein is intended to be a nonqualified stock option
under applicable tax laws, and it is not to be characterized or treated as an
incentive stock option under such laws.

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      b.    Subject to the limitations herein on the Transferability by the
Optionee of the Option and any shares of Common Stock, this Agreement shall be
binding upon and inure to the benefit of the representatives, executors,
successors or beneficiaries of the parties hereto.

      c.    If any provision of this Agreement is declared or found to be
illegal, unenforceable or void, in whole or in part, then the parties shall be
relieved of all obligations arising under such provision, but only to the extent
that it is illegal, unenforceable or void, it being the intent and agreement of
the parties that this Agreement shall be deemed amended by modifying such
provision to the extent necessary to make it legal and enforceable while
preserving its intent or, if that is not possible, by substituting therefor
another provision that is legal and enforceable and achieves the same
objectives.

      d.    All section titles and captions in this Agreement are for
convenience only, shall not be deemed part of this Agreement, and in no way
shall define, limit, extend or describe the scope or intent of any provisions of
this Agreement.

      e.    The parties shall execute all documents, provide all information and
take or refrain from taking all actions as may be necessary or appropriate to
achieve the purposes of this Agreement.

      f.    No failure by any party to insist upon the strict performance of any
covenant, duty, agreement or condition of this Agreement or to exercise any
right or remedy consequent upon a breach thereof shall constitute a waiver of
any such breach or any other covenant, duty, agreement or condition.

      g.    In addition to all other rights or remedies available at law or in
equity, the Company shall be entitled to injunctive and other equitable relief
to prevent or enjoin any violation of the provisions of this Agreement.

                                      ZIX CORPORATION

                                      By: /s/ Bradley C. Almond
                                          -----------------------------
                                          Bradley C. Almond
                                          Vice President Finance &
                                          Administration

OPTIONEE:

/s/ Antonio R. Sanchez III
-----------------------------
Antonio R. Sanchez III

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