Option Agreement

Exhibit 10.3

Option Agreement

This OPTION AGREEMENT (this “Agreement”) is made by and between Endurance Specialty Holdings Ltd., an exempted company organized under the laws of Bermuda (the “Company”) and John R. Charman (the “Executive”), as of May 28, 2013.

WHEREAS, the Executive is to become the Chairman and Chief Executive Officer of the Company, and the Company desires to increase the incentive of the Executive to exert his utmost efforts to improve the business and increase the assets of the Company; and

WHEREAS, as a material inducement to the Executive’s agreeing to accept employment with the Company, the Company wishes to grant the Executive this option award set forth in this Agreement in accordance with the employment inducement award exception set forth in Section 303A.08 of the New York Stock Exchange Listed Company Manual.

NOW, THEREFORE, for good and valuable consideration, the receipt and sufficiency of which are hereby acknowledged, the parties hereto agree as follows:

1. Definitions. Capitalized terms used and not defined in this Agreement shall have the respective meanings ascribed to such terms in the Employment Agreement, dated May 28, 2013, between the Company and the Executive, as amended from time to time (the “Employment Agreement’). In addition to the foregoing, the following definitions shall be applicable within this Agreement.

a. “Fair Market Value” of an Ordinary Share means, (i) if the Ordinary Shares are at the time listed on a national securities exchange (as defined under the U.S. Securities Exchange Act of 1934, as amended (a “Stock Exchange”), the closing selling price per Ordinary Share on the date in question on such Stock Exchange, as such price is officially quoted in the composite tape of transactions on such Stock Exchange or, if there is no closing selling price for the Ordinary Shares on the date in question then the closing selling price on the last preceding date for which such quotation exists or (ii) if the Ordinary Shares are not at the time listed on a Stock Exchange, the amount determined by the Board to be the fair market value of the Ordinary Shares based upon a good faith attempt to value the Ordinary Shares accurately.

b. “Shareholder Agreement” means the Shareholder Agreement, dated May 28, 2013, among the Company, Dragon Global Holdings Limited and the Executive, as amended from time to time.

2. Grant of Option. The Company hereby grants to the Executive the right and option to purchase (the “Option”) up to 800,000 of the Company’s ordinary shares, par value $1.00 per share (the “Ordinary Shares”) at an exercise price of US$48.20 per Share (the “Exercise Price”), pursuant to and subject to the terms and provisions set forth in this Agreement. The Option shall be a non-qualified stock option.

3. Exercise of Option.

a. In order to exercise the Option, the Executive (or the Executive’s estate or legal representative in the case of a permitted exercise of the Option following the Executive’s death or Disability) shall notify the Company in writing addressed and delivered in accordance with Section 14 as to the number of whole Ordinary Shares which Executive desires to purchase under the Option, which notice shall be in the form and containing such representations, warranties and agreements as are required to comply with applicable law and the Shareholder Agreement. If someone other than the Executive exercises the Option, then such person must submit documentation reasonably acceptable to the Company verifying that such person has the legal right to exercise the Option. The Option, to the extent vested, may be exercised on multiple occasions in tranches of not less than 1,000 shares.

b. The Exercise Price of the Option shall be payable in full at the time of exercise. The Exercise Price shall be payable by one or more of the following methods, to the extent permitted by applicable statutes and regulations:

i. in cash or by certified or bank check at the time the Option is exercised;

ii. by delivery to the Company of other Ordinary Shares, duly endorsed for transfer to the Company, with a Fair Market Value on the date of delivery equal to the Exercise Price (or portion thereof) due for the number of Ordinary Shares being acquired;

iii. through a “cashless exercise program” established with a broker;

iv. by reduction in the number of Ordinary Shares otherwise deliverable upon exercise of such Option by the largest whole number of Ordinary Shares having a total Fair Market Value on the date of exercise that does not exceed the aggregate Exercise Price at the time of exercise;

v. by any combination of the foregoing methods; or

vi. in any other form of legal consideration that may be acceptable to the Board.

c. As soon as practicable after the receipt of notice to exercise the Option and the corresponding Exercise Price, the Company shall, in accordance with the Executive’s written instructions, issue the Ordinary Shares to the Executive (or his nominee) as fully-paid Ordinary Shares and register the Executive (or his nominee) in its register of shareholders as the holder of the Ordinary Shares and shall tender to the Executive, a certificate issued in the Executive’s name evidencing the Ordinary Shares purchased by the Executive hereunder or shall instruct the Company’s transfer agent to electronically transfer such Ordinary Shares to the Executive’s account.

-2-

d. The Option may not be exercised (i) until the Option has vested in accordance with the schedule set forth in Section 4 or earlier pursuant to the terms of Section 6, (ii) unless all requirements under applicable law have been met or (iii) after termination of the Option in accordance with Section 10.

4. Vesting Schedule. Absent any acceleration of vesting of the Option pursuant to Section 6 or termination of the Option pursuant to Section 10, the right to exercise the Option shall vest, on the dates and in the amounts as set forth below:

5. Rights of a Shareholder. The Executive shall not be entitled to the privileges of share ownership in respect of Ordinary Shares which are subject to issuance upon exercise of the Option, including but not limited to the right to vote or receive dividends, until the issuance of such Ordinary Shares following exercise of the Option.

6. Accelerated Vesting of Option.

a. Death or Disability.

i. In the event the Executive’s employment with the Company is severed as a result of the Executive’s death or Disability, 50 percent of that portion of the Option, if any, that is not vested in accordance with Section 4 as of the date of the Executive’s Date of Separation from Service shall be vested on such Date of Separation from Service.

ii. In the event the Executive shall die or become subject to a Disability within 30 days of the Executive’s Date of Separation from Service as a result of a termination by the Company for Cause or the Executive severing his employment with the Company without Good Reason, 50 percent of that portion of the Option, if any, that is not vested in accordance with Section 4 as of the date of the Executive’s death or Disability shall be vested on the date of the Executive’s death or Disability, as the case may be.

b. Termination of the Executive without Cause or by the Executive with Good Reason other than During a Change in Control Period. In the event of a termination of the Executive’s employment with the Company other than during a Change in Control Period by the Company without Cause or by the Executive with Good

-3-

Reason, 50 percent of that portion of the Option, if any, that is not vested in accordance with Section 4 as of the date of the Executive’s Date of Separation from Service without Cause or for Good Reason shall be vested as of the Date of Separation from Service.

c. Termination of the Executive without Cause or by the Executive with Good Reason During a Change in Control Period. In the event of a Change in Control and the termination of the Executive’s employment with the Company during a Change in Control Period by the Company without Cause or by the Executive with Good Reason, the entire unvested portion of the Option, if any, shall vest as of the Executive’s Date of Separation from Service.

7. Repayment or Forfeiture. Any benefits the Executive may receive hereunder may be subject to repayment or forfeiture as may be required in order to comply with the requirements of the U.S. Securities and Exchange Commission or any applicable law, including the requirements of the Dodd-Frank Wall Street Reform and Consumer Protection Act, or any securities exchange on which the Ordinary Shares are traded, as may be in effect from time to time.

8. Legends. All certificates representing any of the Ordinary Shares issued upon exercise of the Option, to the extent required by the Shareholder Agreement, shall have endorsed thereon the following legend:

“The transferability of this certificate and the ordinary shares represented hereby are subject to the restrictions, terms and conditions (including restrictions against transfer) contained in a Shareholder Agreement entered into between the registered owner of such ordinary shares and the Company.”

9. Tax Withholding.

a. Subject to Section 9(b), it shall be a condition to the obligation of the Company to issue Ordinary Shares to the Executive upon the exercise of the Option that the Executive pay to the Company, upon demand (but no later than 10 days following such demand), such amount as may be requested by the Company for the purpose of satisfying any liability to withhold income or other taxes. In the event any such amount so requested is not paid, the Company may refuse to issue Ordinary Shares to the Executive upon the exercise of the Option. Unless the Board shall in its discretion determine otherwise, payment for taxes required to be withheld may be made in cash, or in whole or in part, (i) by withholding Ordinary Shares deliverable upon exercise of the Option having a fair market value less than or equal to the required tax withholding liability and/or (ii) tendering to the Company Ordinary Shares held by the Executive having a fair market value less than or equal to the required tax withholding liability, in each case as determined by the Company in its sole discretion.

b. Section 9(a) shall not apply in respect of any taxes in respect of the Option which arise in Bermuda or in any jurisdiction outside Bermuda, where the tax liability outside Bermuda arises as a result of a requirement of the Company that the Executive provides services to the Company and its subsidiaries and affiliates outside Bermuda. The Company shall be responsible for and will duly and promptly report for and pay any payroll taxes in respect of the Option which arise in Bermuda.

-4-

10. Termination of the Option. This Agreement and the Option shall terminate (except in relation to any portion of the Option validly exercised before the relevant time) upon the earliest to occur of the following events:

a. 5:00 p.m. (Atlantic Time) on the ten (10) year anniversary of the date of this Agreement;

b. 5:00 p.m. (Atlantic Time) on the two (2) year anniversary of the Executive’s Date of Separation from Service in the event that the Executive’s employment with the Company is severed by the Company without Cause, by the Executive for Good Reason or on account of the Executive’s death or Disability;

c. 5:00 p.m. (Atlantic Time) six (6)calendar months after the Executive’s Date of Separation from Service in the event the Executive’s employment with the Company is severed otherwise than as specified in Section 10(b);

d. The termination of this Agreement by an instrument in writing signed by the parties hereto; or

e. As may be required under applicable law.

Upon termination of this Agreement, all rights and obligations of the Executive and the Company hereunder shall cease.

11. Agreement to Perform Necessary Acts. Each party to this Agreement agrees to perform any further acts and execute and deliver any documents that may be reasonably necessary to carry out the provisions of this Agreement.

12. Registration of Shares. The Company shall use commercially reasonable efforts to file and maintain an effective Registration Statement on Form S-8 under the United States Securities Act of 1933, as amended, with respect to the Ordinary Shares issuable upon exercise of the Option. The Company will maintain at all relevant times sufficient authorized but unissued shares in order to comply with its obligations to deliver Ordinary Shares upon exercise of any portion of the Option under this Agreement.

13. No Limitation on Rights of the Company. This Agreement shall not in any way affect the right or power of the Company to make adjustments, reclassifications or changes in its capital or business structure or to merge, consolidate, dissolve, liquidate, sell or transfer all or any part of its business or assets.

-5-

14. Miscellaneous.

a. Notices. Any notice or other communication required or permitted hereunder shall be in writing and shall be delivered personally or sent by certified, registered, or express mail, postage prepaid. Any such notice shall be deemed given when delivered to each party at the following addresses (or to such other address as may be designated in a notice given in accordance with this Section):

(i) if to the Company:

Endurance Specialty Holdings Ltd.

Wellesley House

90 Pitts Bay Road

Pembroke HM 08

Bermuda

Attn: General Counsel

(ii) if to the Executive, to the most recent primary residence address listed for the Executive in the employment records of the Company.

b. Failure to Enforce Not a Waiver. The failure of the Company or the Executive to enforce at any time any provision of this Agreement shall in no way be construed to be a waiver of such provision or of any other provision hereof.

c. Governing Law. This Agreement shall be governed by and construed according to the laws of Bermuda without giving effect to the choice of law principles thereof. Subject to Section 14(d), the parties submit to the non-exclusive jurisdiction of the courts of Bermuda in respect to matters arising hereunder.

d. Arbitration. All disputes, controversies or claims arising out of, relating to or in connection with this Agreement or the Option, or the breach, termination or validity thereof, shall be finally settled by arbitration. The arbitration shall be conducted in accordance with the arbitration procedures set forth in Section 7.7 of the Employment Agreement.

e. Amendments. This Agreement may be amended or modified at any time by an instrument in writing signed by the parties hereto.

f. Investment Intent. Recipient represents that the Option is being acquired by him for investment and that he has no present intention to transfer, sell or otherwise dispose of the Option, except in compliance with applicable securities laws, and the parties agree that the Option is being acquired in accordance with and subject to the terms, provisions and conditions of this Agreement.

g. Adjustment of Ordinary Shares Issuable Upon Exercise of Option. The Company hereby confirms that (i) in the event the outstanding Ordinary Shares of the Company shall be changed into an increased number of shares, through a share dividend or bonus issue or a split-up of shares, or into a decreased number of shares, through a combination of shares, then immediately after the record date for such change, the number of Ordinary Shares issuable upon exercise of the Option in

-6-

accordance with this Agreement shall be proportionately increased, in case of such stock dividend or bonus issue or split-up of shares, or proportionately decreased, in case of such combination of shares; and (ii) in the event that, as result of a reorganization, sale, merger, amalgamation, consolidation or similar occurrence, there shall be any other change in the Ordinary Shares of the Company, or of any shares or other securities into which such Ordinary Shares shall have been changed, or for which they shall have been exchanged, then equitable adjustments to the Ordinary Shares issuable upon exercise of the Option in accordance with this Agreement (including, but not limited to, changes in the number or kind of Ordinary Shares then subject to the Option) shall be made.

h. Agreement Not a Contract of Employment. This Agreement is not a contract of employment, and the terms of employment of the Executive or the employment relationship of the Executive with the Company or any subsidiary of the Company shall not be affected in any way by this Agreement except as specifically provided herein. The execution of this Agreement shall not be construed as conferring any legal rights upon the Executive for a continuation of employment or employment relationship with the Company or any subsidiary of the Company, nor (without limitation to the rights of the Executive under this Agreement) shall it interfere with the right of the Company or any subsidiary of the Company to discharge the Executive and to treat the Executive without regard to the effect which that treatment might have upon the Executive as a holder of the Option.

i. Entire Agreement. This Agreement contains the entire understanding and agreement of the parties hereto concerning the subject matter hereof, and supersede all earlier negotiations and understandings, written or oral, between the parties hereto with respect thereto.

j. Captions. The captions and headings of the sections and subsections of this Agreement are included for convenience only and are not to be considered in construing or interpreting this Agreement.

k. Counterparts. This Agreement may be executed in counterparts and by facsimile or electronic reproduction, each of which when signed by the Company or the Executive will be deemed an original and all of which together will be deemed the same agreement.

l. Assignment. Neither party may assign its or his rights or duties under this Agreement. If any such assignment or delegation requires consent of any state securities authorities, the parties hereto agree to cooperate in requesting such consent. The Option may not be sold, pledged, assigned, hypothecated, transferred, or disposed of in any manner by the Executive other than by will or by the laws of descent or distribution and may be exercised, during the lifetime of the Executive, only by the Executive; provided, however, that the Executive may direct under Section 3(c) that Ordinary Shares deliverable upon the exercise of the Option shall be issued to his nominee. This Agreement shall inure to the benefit of the successors and permitted assigns of the parties and, subject to the restrictions set forth herein, be binding upon the Executive, his heirs, executors, administrators, successors and permitted assigns.

-7-

m. Severability. This Agreement will be severable, and the invalidity or unenforceability of any term or provision hereof will not affect the validity or enforceability of this Agreement or of any other term or provision hereof. Furthermore, in lieu of any invalid or unenforceable term or provision, the parties hereto intend that there be added as a part of this Agreement a valid and enforceable provision as similar in terms to such invalid or unenforceable provision as may be possible.

[Execution Page Follows]

-8-

IN WITNESS WHEREOF, the parties hereto have executed this Agreement as of the day and year first above written.

The undersigned hereby accepts and agrees to all the terms and provisions of the foregoing Agreement.

-9-