<PAGE>
                                                                    Exhibit 10.2

                                 ZIX CORPORATION
                             STOCK OPTION AGREEMENT

      THIS STOCK OPTION AGREEMENT ("Agreement") is made and entered into as of
the date set forth on the signature page attached hereto (the "Signature Page")
with respect to the stock options granted by Zix Corporation, a Texas
corporation (the "Company"), to the Optionee ("Optionee") listed on the
signature page hereto.

      WHEREAS, 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 non-qualified
stock option ("Option") to purchase up to the total number of shares of the
Common Stock set forth on the Signature Page at the price per share (the "Option
Price") as set forth on the Signature Page on the terms and conditions and
subject to the restrictions as set forth in this Agreement and the provisions of
the applicable Zix Corporation stock option plan (which is incorporated herein
by reference) (the "Plan"), which is referenced on the Signature Page. All
defined terms contained herein shall have the meanings ascribed to them in the
Plan, except as otherwise provided herein.

1.    DEFINITIONS.

      a.    Disability. "Disability" shall mean any medically determinable
physical or mental impairment that, in the opinion of the Committee, based upon
medical reports and other evidence satisfactory to the Committee, can reasonably
be expected to prevent the Optionee from performing substantially all of his or
her customary duties of employment (with or without reasonable accommodation)
for a continuous period of not less than 12 months.

      b.    Resignation. "Resignation" shall mean the voluntary termination by
the Optionee of his or her employment relationship with the employing Subsidiary
and, if applicable, Company under circumstances other than voluntary Retirement.

      c.    Retirement. "Retirement" shall mean the termination of Optionee's
employment in accordance with the requirements of a written retirement plan,
policy or rule of the Company that has been duly adopted by the Company or
employing Subsidiary, as applicable.

2.    TERM OF OPTION. The term of this Option shall expire on the expiration
date set forth in the Signature Page (the "stated term"), except as such term
may be otherwise shortened by the other provisions of the Plan or this
Agreement.

                                       1
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3.    EXERCISE OF OPTION.

      a.    Exercise. This Option shall become exercisable in increments as set
forth in the Signature Page. Except as provided in the Plan, the Option shall
not be exercisable unless Optionee shall, at the time of exercise, be an
employee of the Company or a Subsidiary, 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 expiration or termination of this Option. However, this
Option may not be exercised as to less than 100 shares at any one time (or the
remaining shares then purchasable under this Option, if less than 100 shares).

      b.    Adjustment. In the event there is any adjustment to the Common Stock
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 Option
Price, or both.

      c.    Method of Exercise. This Option may be exercised only by written
notice (the "Exercise Notice") by the Optionee 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 this 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 this Option is being exercised and, if the shares for which this 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.

      This Option may be exercised either by tendering cash in the amount of the
Option Price or, with the Company's consent, by tendering shares of Common Stock
(which may include shares previously acquired upon exercise of options granted
under the Plan). The Exercise Notice shall be accompanied by payment of the
aggregate Option Price of the shares purchased by cash, a certified cashier's
check or, at the Company's option, by delivery of shares of Common Stock having
a Fair Market Value on the date immediately preceding the exercise date equal to
the Option Price.

      If the shares to be purchased are covered by an effective registration
statement under the Securities Act of 1933, as amended, any option granted under
the Plan may be exercised by a broker-dealer acting on behalf of an 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.

                                       2
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      The certificates for shares of Common Stock as to which this 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. An
option exercise shall be valid only if the Optionee makes payment or other
arrangements relating to the withholding tax obligations discussed in Paragraph
8. In the event the person exercising this 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 this Option.

4.    TERMINATION OF OPTION.

      In the event an Optionee ceases to be an employee of either the Company or
a Subsidiary of the Company due to death, Retirement, Resignation, Disability or
termination by the Company for any reason other than "cause" (such five events
each being a "Qualified Termination"), this Option may be exercised by the
Optionee or his or her estate, personal representative or beneficiary to the
fullest extent that the Optionee was entitled to exercise the same on the day
immediately prior to such termination (i) at any time within the one-year period
commencing on the day next following such termination if such termination is due
to the death of the Optionee; (ii) at any time within the thirty-day period
commencing on the day next following the effective date of such termination if
such termination is due to the Resignation of the Optionee; or (iii) at any time
within the six-month period commencing on the day next following such
termination in the case of any other Qualified Termination (or in any such case
in (i), (ii) or (iii) above, if shorter, only for the remaining stated term of
this Option). In the event that the Optionee's employment is terminated for any
reason other than a Qualified Termination, this Option shall automatically
expire simultaneously with such termination. For purposes of this Paragraph,
"cause" shall have the meaning given such term in the Severance Agreement, dated
July 1, 2003, between Optionee and the Company. For purposes of this Agreement,
Optionee shall be considered to be an employee of the Company until the
expiration of the "consulting period," as such term is defined in Section 8 of
the Employment Agreement, dated June 1, 2004, between the Company and Optionee.

      After the Optionee's death, this Option shall be exercisable only by the
executor or administrator of the Optionee's estate, or if the Optionee's estate
is not in administration, by the person or persons to whom the Optionee's rights
shall have passed by the Optionee's will or under the laws of descent and
distribution of the state where the Optionee was domiciled at the date of death.

5.    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 this 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).

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6.    STATE AND FEDERAL SECURITIES REGULATION. No shares shall be issued by the
Company upon the exercise of this Option unless and until any then-applicable
requirements of state and federal laws 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
this 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 this
Option. This 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 this 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 this Option or
the issue or purchase of shares under this Option, this 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 this Option or the issue or
purchase of shares under this Option.

7.    MODIFICATION OF OPTIONS. At any time and from time-to-time the Committee
may execute an instrument providing for modification, extension, or renewal of
any outstanding option, provided that no such modification, extension or renewal
shall impair this Option in any respect without the written consent of the
holder of this Option.

8.    WITHHOLDING OF TAXES. The Company may make such provisions and take such
steps as it may deem necessary or appropriate for the withholding of any taxes
which the Company or any Subsidiary is required by any law or regulation of any
governmental authority, whether federal, state or local, domestic or foreign, to
withhold in connection with any option, including, but not limited to, the
withholding of the issuance of all or any portion of the shares of Common Stock
subject to this Option until the Optionee reimburses the Company or the
applicable Subsidiary for the amount the Company or the applicable Subsidiary is
required to withhold with respect to such taxes, canceling any portion of the
issuance in an amount sufficient to reimburse the Company or the applicable
Subsidiary for the amount it is required to so withhold, or taking any other
action reasonably required to satisfy the withholding obligation of the Company
or the applicable Subsidiary.

9.    CONTINUED EMPLOYMENT 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 in employment with the Company or any of its Subsidiaries
or affect the right of the Company or a Subsidiary to terminate the employment
of the Optionee with or without cause.

                                       4
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10.   NON-COMPETITION COVENANTS.

      a.    The provisions of this subparagraph a. shall apply both during
normal working hours and at all other times including, but not limited to,
nights, weekends and vacation time, while Optionee is employed by the Company or
any Subsidiary. Optionee shall not directly or indirectly (i) engage in any
employment, business, or activity that is competitive with the business of the
Company or any Subsidiary, (ii) assist any other person or organization in
competing with, or in preparing to engage in competition with, the business of
the Company or any Subsidiary. Direct competition shall include, but not be
limited to, the design, development, production, promotion or sale of products,
software, or services competitive with those of the Company or any Subsidiary.
In addition, Optionee shall not directly or indirectly (i) engage in any
employment, business, or activity that is competitive with either (A) the
proposed business of the Subsidiary that employs Optionee ("Employing
Subsidiary") or (B) any proposed business of any of the Company's other
Subsidiaries (the "Non-Employing Subsidiaries") of which Optionee has actual
knowledge, or (ii) assist any other person or organization in competing with, or
in preparing to engage in competition with, either (A) the proposed business of
the Employing Subsidiary or (B) any proposed business of any Non-Employing
Subsidiary of which Optionee has actual knowledge.

      b.    The provisions of this subparagraph b. shall apply during Optionee's
employment with the Company or any Subsidiary and for a period of six months
after Optionee ceases to be employed by the Company or any Subsidiary. Optionee
shall not directly or indirectly solicit to conduct any Competitive Business
with, or conduct any Competitive Business with, any (i) then-current customer of
the Employing Subsidiary or (ii) any person that has been a customer of the
Employing Subsidiary within the six months prior to the time of Optionee's
separation from employment. The phrase "Competitive Business" means the line(s)
of business(es) conducted by the Employing Subsidiary.

      c.    The provisions of this subparagraph c. shall apply during Optionee's
employment with the Company or any Subsidiary and for a period of 12 months
after Optionee's separation from employment. Optionee shall not directly or
indirectly solicit to hire, or cause to be hired, any employee of the Company or
any Subsidiary as an employee or agent of, or consultant to, any business
enterprise that Optionee is associated with.

      d.    Each non-competition covenant of Optionee contained in the preceding
provisions of this Paragraph 10 (the "non-competition covenant") shall be
construed as an agreement independent of any other provision of this Agreement
and the existence of any claim or cause of action of Optionee against the
Company or any Subsidiary, whether predicated on this Agreement or otherwise,
shall not constitute a defense to the enforcement by the Company or any
Subsidiary of such non-competition covenant.

      e.    The Company and Optionee have in good faith used their best efforts
to make each non-competition covenant contained in the preceding provisions of
this Paragraph 10 reasonable in both scope and in duration. It is not
anticipated, nor is it intended, by either party

                                       5
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to this Agreement that any court or other tribunal having jurisdiction over the
matter will find it necessary to reform any non-competition covenant to make it
reasonable in both scope and in duration, or otherwise. If any non-competition
covenant is deemed by a tribunal having jurisdiction over the matter to be
unlawful or unenforceable, such provision will be deemed severable from this
Agreement and such provision will be limited or eliminated to the minimum extent
necessary so that the remaining provisions of this Agreement shall otherwise
remain in full force and effect and be enforceable. Furthermore, in lieu of such
unlawful or unenforceable provision, there shall be added automatically as part
of this Agreement a provision as similar in terms as may be possible and be
enforceable.

      f.    Optionee is agreeing to the provisions of this Paragraph 10 in
consideration of the grant of this Option. The provisions of this Paragraph 10
shall be valid and enforceable by the Company and its Subsidiaries, regardless
of whether or not any of this Option granted hereunder actually becomes
exercisable, or whether or not Optionee actually exercises any rights under this
Option. In the event of any conflict or inconsistency between any provision of
this Paragraph 10 and any similar or analogous provision of any other agreement
(either currently in effect or that may be entered into in the future) between
Optionee, on the one hand, and the Company or any Subsidiary, on the other hand,
whichever provision is most favorable to the Company or such Subsidiary shall
govern.

11.   OPTION ISSUED PURSUANT TO PLAN. This Option is issued pursuant to and
subject to the terms and conditions and the restrictions as set forth in the
Plan, and in the event of any inconsistency, the provisions of the Plan shall
govern, provided that no amendment shall be made to the Plan subsequent to the
date hereof that impairs the Optionee's rights under this Option without the
Optionee's written consent.

12.   NO LIABILITY OF OPTION. This 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. This 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 Optionee. Without limiting the generality of the foregoing,
this Option may not be assigned, transferred (except as aforesaid), pledged or
hypothecated in any way (whether by operation of law or otherwise), and shall
not be subject to execution, attachment, or similar process, without the prior
written consent of the Company. Any attempted assignment, transfer, pledge, or
hypothecation 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.

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15.   ENTIRE AGREEMENT. By signing the Signature Page, the Optionee agrees to
the terms of this Option. 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.   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, LB 36, Dallas,
Texas 75204-2960: Attention: Treasurer, or any other address that may be given
by either party to the other party by notice pursuant to this Paragraph. Any
notice other than any Exercise Notice, if sent by registered or certified mail,
shall be deemed to have been given when received.

                                   ZIX CORPORATION

                                   By: /s/ Bradley C. Almond
                                       ---------------------------------
                                       Bradley C. Almond
                                       Vice President, Chief Financial
                                       Officer and Treasurer

                                       7
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                                                                  [ZIXCORP LOGO]

                                 Signature Page
                                       To
                     Zix Corporation Stock Option Agreement

Effective Date of Grant:  JUNE 1, 2004

Name of Optionee:         NUTKIS, DAN

No. of Shares:            164,000

Exercise Price:           $8.50

Name of Plan:             ZIX CORPORATION 2004 STOCK OPTION PLAN

Expiration Date:           MAY 31, 2009

Vesting Schedule:

<TABLE>
<CAPTION>
                                     Date Upon Which Right
Number of Shares                      To Purchase Accrues
----------------                      -------------------
<S>                                  <C>
 13,666                              15-JUN-04
 13,666                              15-JUL-04
 13,666                              15-AUG-04
 13,666                              15-SEP-04
 13,667                              15-OCT-04
 13,667                              15-NOV-04
 13,667                              15-DEC-04
 13,667                              15-JAN-05
 13,667                              15-FEB-05
 13,667                              15-MAR-05
 13,667                              15-APR-05
 13,667                              15-MAY-05
</TABLE>

/s/ Dan Nutkis                          Date: 7/30/04
-----------------------------
Optionee Signature

Sign and return to:
ZixCorporation, 2711 N. Haskell Avenue, Suite 2300, Dallas, Texas 75204-2960
Attn: Human Resources Department