Document:

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                                                                Exhibit 10.13(b)

                                                                   DIRECTOR FORM

                             DELEK US HOLDINGS, INC.
                          2006 LONG-TERM INCENTIVE PLAN
                             STOCK OPTION AGREEMENT

      AGREEMENT, made as of this _ day of _________, 200_ (the "Grant Date"), by
and between Delek US Holdings, Inc., a Delaware corporation (the "Company"), and
__________________ (the "Optionee").

                              W I T N E S S E T H:

      WHEREAS, pursuant to the Delek US Holdings, Inc. 2006 Long-Term Incentive
Plan (the "Plan"), the Company desires to grant to the Optionee, and the
Optionee desires to accept, an option to purchase shares of the Company's common
stock, $0.01 par value (the "Common Stock"), upon the terms and conditions set
forth in this Agreement and the Plan. Capitalized terms used but not defined
herein shall have the meanings ascribed to such terms in the Plan.

      NOW, THEREFORE, the parties hereto agree as follows:

      1. Grant. The Company hereby grants to the Optionee an option (the
"Option") to purchase up to ______ shares of Common Stock, at a purchase price
per share of ________ U.S. dollars ($______). This Option is not intended to
qualify as an "incentive stock option" within the meaning of Section 422 of the
Internal Revenue Code of 1986, as amended.

      2. Term of Option. The term of this Option shall be for a period of ten
(10) years from the date hereof, subject to earlier termination as provided
herein and the Plan.

      3. Vesting of Option. Except as otherwise provided herein or the Plan,
this Option shall become vested and exercisable with respect to ________ of the
shares of Common Stock subject thereto on each of the following dates
____________________, provided that the Optionee remains in continuous service
with the Company or its affiliates through each applicable vesting date.

      4. Termination of Service.

            (a) Death or Disability. If the Optionee's service with the Company
or its affiliates is terminated due to the Optionee's death or Disability (as
defined below), then: (i) that portion of this Option, if any, that is vested
and exercisable on the date of termination shall remain exercisable by the
Optionee (or, in the event of death, the Optionee's beneficiary) during the one
(1) year period following the date of termination but in no event after
expiration of the stated term hereof and, to the extent not exercised during
such period, shall thereupon terminate, provided that, in the event of a
termination due to Disability, if the Optionee dies during such one (1) year
period, then the Optionee's beneficiary may exercise this Option, to the extent
vested and exercisable by the Optionee immediately prior to his or her death,
for a period of one (1) year following the date of death but in no event after
expiration of the stated term hereof, and
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(ii) that portion of this Option, if any, that is not exercisable on the
date of termination shall thereupon terminate. For purposes of this Agreement,
"Disability" shall mean the inability of an Optionee to perform the customary
duties of the Optionee's service with the Company or its affiliates by reason of
a physical or mental incapacity or illness which is expected to result in death
or to be of indefinite duration.

            (b) Other Termination. If the Optionee' s service with the Company
or its affiliates is terminated for any reason other than those set forth in
Section 4(a) above, then: (i) that portion of this Option, if any, that is
vested and exercisable on the date of termination shall remain exercisable by
the Optionee during the one hundred eighty (180) day period following the date
of termination but in no event after expiration of the stated term hereof and,
to the extent not exercised during such period, shall thereupon terminate, and
(ii) that portion of this Option, if any, that is not vested and exercisable on
the date of termination shall thereupon terminate.

      5. Exercise and Payment. Once vested and exercisable, the Optionee may
exercise this Option in whole or in part by delivering to the Secretary of the
Company (a) a written notice of such exercise specifying the number of shares of
Common Stock that the Optionee has elected to purchase and (b) payment in full
of the exercise price and the withholding taxes due in connection with the
exercise, unless other arrangements satisfactory to the Company are made for the
satisfaction of such payment(s). The exercise price and applicable tax
withholding obligation may be paid (i) in cash or by check, (ii) at the
discretion of the committee, (1) by the delivery of previously-owned shares of
Common Stock, (2) by means of a cashless exercise procedure, or (3) in any other
form of legal consideration that may be acceptable to the Committee, or (iii) by
a combination of the foregoing.

      6. Rights as Stockholder. No shares of Common Stock shall be issued
hereunder until full payment for such shares has been made and the applicable
tax withholding obligation satisfied. The Optionee shall have no rights as a
stockholder with respect to any shares covered by this Option until the date a
stock certificate for such shares is issued to the Optionee.

      7. Nontransferability. This Option may not be pledged, hypothecated or
otherwise encumbered or subject to any lien, obligation or liability of the
Optionee to any party (other than the Company or an affiliate thereof), or
assigned or transferred (collectively, "Transferred") by the Optionee other than
by will or the laws of descent and distribution or to a beneficiary upon the
death of the Optionee, and this Option may be exercised during the lifetime of
the Optionee only by the Optionee or his or her guardian or legal
representative. Any attempt by the Optionee or any other person claiming
against, through or under the Optionee to cause this Option or any part of it to
be Transferred in any manner and for any purpose shall be null and void and
without effect upon the Company, the Optionee or any other person.

      8. Compliance with Law; Transfer Orders; Legends. The Company will not be
obligated to issue or deliver shares of Common Stock pursuant to this Option
unless the issuance and delivery of such shares complies with applicable law,
including, without limitation, the Securities Act of 1933, as amended, the
Securities Exchange Act of 1934, as amended, and the requirements of any stock
exchange or market upon which the Common Stock may then be listed, and shall be
further subject to the approval of counsel for the Company with respect to such
compliance. All certificates for shares of Common Stock delivered under this
Option shall

                                       2

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be subject to such stock-transfer orders and other restrictions as the Company
may deem advisable under the rules, regulations, and other requirements of the
Securities and Exchange Commission, any stock exchange or market upon which the
Common Stock may then be listed, and any applicable federal or state securities
law. The Company may cause a legend or legends to be placed on any such
certificates to make appropriate reference to such restrictions.

      9. No Rights Conferred. Nothing contained in the Plan or this Agreement
shall confer upon the Optionee any right with respect to the continuation of his
or her service with the Company or its affiliates or interfere in any way with
the right of the Company and its affiliates at any time to terminate such
service or to increase or decrease, or otherwise adjust, the other terms and
conditions of the Optionee's service.

      10. Provisions of the Plan. The provisions of the Plan, the terms of which
are incorporated in this Agreement, shall govern if and to the extent that there
are inconsistencies between those provisions and the provisions hereof. The
Optionee acknowledges receipt of a copy of the Plan prior to the execution of
this Agreement.

      11. Miscellaneous. This Agreement shall be binding upon and shall inure to
the benefit of the parties hereto and their respective successors and permitted
assigns. This Agreement shall be governed by and construed in accordance with
the laws of the State of Delaware, without regard to its principles of conflicts
of law. This Agreement constitutes the entire agreement between the parties with
respect to the subject matter hereof and, except as otherwise provided in the
Plan, may not be modified other than by written instrument executed by the
parties.

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      IN WITNESS WHEREOF, this Agreement has been executed as of the date first
above written.

                                       DELEK US HOLDINGS, INC.

                                       By: _______________________
                                           Name:
                                           Title:

                                       By: _______________________
                                           Name:
                                           Title:

                                       OPTIONEE:

                                       ___________________________

                                       4<PAGE>
                                                                EXHIBIT 10.13(c)

                                                                    OFFICER FORM

                           DELEK US HOLDINGS, INC.
                          2006 LONG-TERM INCENTIVE PLAN
                             STOCK OPTION AGREEMENT

      AGREEMENT, made as of this _ day of _________, 200_ (the "Grant Date"), by
and between Delek US Holdings, Inc., a Delaware corporation (the "Company"), and
__________________ (the "Optionee").

                             W I T N E S S E T H:

      WHEREAS, pursuant to the Delek US Holdings, Inc. 2006 Long-Term Incentive
Plan (the "Plan"), the Company desires to grant to the Optionee, and the
Optionee desires to accept, an option to purchase shares of the Company's common
stock, $0.01 par value (the "Common Stock"), upon the terms and conditions set
forth in this Agreement and the Plan. Capitalized terms used but not defined
herein shall have the meanings ascribed to such terms in the Plan.

      NOW, THEREFORE, the parties hereto agree as follows:

      1. Grant. The Company hereby grants to the Optionee an option (the
"Option") to purchase up to ______ shares of Common Stock, at a purchase price
per share of ________ U.S. dollars ($______). This Option is not intended to
qualify as an "incentive stock option" within the meaning of Section 422 of the
Internal Revenue Code of 1986, as amended.

      2. Term of Option. The term of this Option shall be for a period of ten
(10) years from the date hereof, subject to earlier termination as provided
herein and the Plan.

      3. Vesting of Option. Except as otherwise provided herein or the Plan,
this Option shall become vested and exercisable with respect to ________ of the
shares of Common Stock subject thereto on each of the following dates
____________________, provided that the Optionee remains in continuous
employment or other service with the Company or its affiliates through each
applicable vesting date.

      4. Termination of Employment or Other Service.

            (a) Death or Disability. If the Optionee's employment or other
service with the Company or its affiliates is terminated due to the Optionee's
death or Disability (as defined below), then: (i) that portion of this Option,
if any, that is vested and exercisable on the date of termination shall remain
exercisable by the Optionee (or, in the event of death, the Optionee's
beneficiary) during the one (1) year period following the date of termination
but in no event after expiration of the stated term hereof and, to the extent
not exercised during such period, shall thereupon terminate, provided that, in
the event of a termination due to Disability, if the Optionee dies during such
one (1) year period, then the Optionee's beneficiary may exercise this Option,
to the extent vested and exercisable by the Optionee immediately prior to his or
her death, for a period of one (1) year following the date of death but in no
event after expiration of
<PAGE>
the stated term hereof, and (ii) that portion of this Option, if any, that is
not exercisable on the date of termination shall thereupon terminate. For
purposes of this Agreement, "Disability" shall mean the inability of an Optionee
to perform the customary duties of the Optionee's employment or other service
with the Company or its affiliates by reason of a physical or mental incapacity
or illness which is expected to result in death or to be of indefinite duration.

            (b) Termination for Cause. If the Optionee's employment or other
service is terminated by the Company or its affiliates for Cause (as defined
below), then this Option (whether or not then vested and exercisable) shall
immediately terminate and cease to be exercisable. For purposes of this
Agreement, "Cause" shall mean the Optionee's (i) fraud, gross negligence or
willful misconduct involving the Company or its affiliates, (ii) conviction of,
or plea of nolo contendere to, a felony or crime involving moral turpitude, or
(iii) material breach of any written agreement between the Optionee and the
Company or any of its affiliates.

            (c) Other Termination. If the Optionee' s employment or other
service with the Company or its affiliates is terminated for any reason other
than those set forth in Section 4(a) or 4(b) above, then: (i) that portion of
this Option, if any, that is vested and exercisable on the date of termination
shall remain exercisable by the Optionee during the thirty (30) day period
following the date of termination but in no event after expiration of the stated
term hereof and, to the extent not exercised during such period, shall thereupon
terminate, and (ii) that portion of this Option, if any, that is not vested and
exercisable on the date of termination shall thereupon terminate.

      5. Exercise and Payment. Once vested and exercisable, the Optionee may
exercise this Option in whole or in part by delivering to the Secretary of the
Company (a) a written notice of such exercise specifying the number of shares of
Common Stock that the Optionee has elected to purchase and (b) payment in full
of the exercise price and the withholding taxes due in connection with the
exercise, unless other arrangements satisfactory to the Company are made for the
satisfaction of such payment(s). The exercise price and applicable tax
withholding obligation may be paid (i) in cash or by check, (ii) at the
discretion of the committee, (1) by the delivery of previously-owned shares of
Common Stock, (2) by means of a cashless exercise procedure, or (3) in any other
form of legal consideration that may be acceptable to the Committee, or (iii) by
a combination of the foregoing.

      6. Rights as Stockholder. No shares of Common Stock shall be issued
hereunder until full payment for such shares has been made and the applicable
tax withholding obligation satisfied. The Optionee shall have no rights as a
stockholder with respect to any shares covered by this Option until the date a
stock certificate for such shares is issued to the Optionee.

      7. Nontransferability. This Option may not be pledged, hypothecated or
otherwise encumbered or subject to any lien, obligation or liability of the
Optionee to any party (other than the Company or an affiliate thereof), or
assigned or transferred (collectively, "Transferred") by the Optionee other than
by will or the laws of descent and distribution or to a beneficiary upon the
death of the Optionee, and this Option may be exercised during the lifetime of
the Optionee only by the Optionee or his or her guardian or legal
representative. Any attempt by the Optionee or any other person claiming
against, through or under the Optionee to cause this Option or any

                                       2
<PAGE>
part of it to be Transferred in any manner and for any purpose shall be null and
void and without effect upon the Company, the Optionee or any other person.

      8. Compliance with Law; Transfer Orders; Legends. The Company will not be
obligated to issue or deliver shares of Common Stock pursuant to this Option
unless the issuance and delivery of such shares complies with applicable law,
including, without limitation, the Securities Act of 1933, as amended, the
Securities Exchange Act of 1934, as amended, and the requirements of any stock
exchange or market upon which the Common Stock may then be listed, and shall be
further subject to the approval of counsel for the Company with respect to such
compliance. All certificates for shares of Common Stock delivered under this
Option shall be subject to such stock-transfer orders and other restrictions as
the Company may deem advisable under the rules, regulations, and other
requirements of the Securities and Exchange Commission, any stock exchange or
market upon which the Common Stock may then be listed, and any applicable
federal or state securities law. The Company may cause a legend or legends to be
placed on any such certificates to make appropriate reference to such
restrictions.

      9. No Employment or Other Rights. Nothing contained in the Plan or this
Agreement shall confer upon the Optionee any right with respect to the
continuation of his or her employment or other service with the Company or its
affiliates or interfere in any way with the right of the Company and its
affiliates at any time to terminate such employment or other service or to
increase or decrease, or otherwise adjust, the other terms and conditions of the
Optionee's employment or other service.

      10. Provisions of the Plan. The provisions of the Plan, the terms of which
are incorporated in this Agreement, shall govern if and to the extent that there
are inconsistencies between those provisions and the provisions hereof. The
Optionee acknowledges receipt of a copy of the Plan prior to the execution of
this Agreement.

      11. Miscellaneous. This Agreement shall be binding upon and shall inure to
the benefit of the parties hereto and their respective successors and permitted
assigns. This Agreement shall be governed by and construed in accordance with
the laws of the State of Delaware, without regard to its principles of conflicts
of law. This Agreement constitutes the entire agreement between the parties with
respect to the subject matter hereof and, except as otherwise provided in the
Plan, may not be modified other than by written instrument executed by the
parties.

                                       3
<PAGE>
      IN WITNESS WHEREOF, this Agreement has been executed as of the date first
above written.

                                       DELEK US HOLDINGS, INC.

                                       By:
                                          -----------------------
                                          Name:
                                          Title:

                                       By:
                                          -----------------------
                                          Name:
                                          Title:

                                       OPTIONEE:

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

                                       4

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