CapStar Financial Holdings, Inc. Non-Qualified Stock Option Agreement by and between the Company and Timothy K. Schools, dated May 22, 2019

Exhibit 10.3

CAPSTAR FINANCIAL HOLDINGS, INC.

NON-QUALIFIED STOCK OPTION AGREEMENT

THIS AGREEMENT is entered into by and between CapStar Financial Holdings, Inc., a Tennessee corporation (the “Company”), and Timothy K. Schools (the “Participant”) on this the 22^nd day of May, 2019.

W I T N E S S E T H:

1.Grant of Option. The Company grants to Participant the Option to purchase from the Company Fifty thousand (50,000) fully paid and non-assessable shares of the common stock, $1.00 par value (“Stock”) of the Company at a price of Fourteen and 835/1000 dollars ($14.835) per share, subject to the vesting provisions in Section 2, with such price being not less than the Fair Market Value of the Stock on May 22, 2019, the date that this Option was awarded  (the “Date of Grant”). This Option is subject to all of the terms, conditions, and provisions hereof and the CapStar Financial Holdings, Inc. Stock Incentive Plan (the “Plan”).

2.Vesting. The Option shall become vested incrementally with respect to the shares of Stock described in Section 1 as follows:

(a) 16,667 shares of Stock on or after May 22, 2020;

(b) an additional 16,667 shares of Stock on or after May 22, 2021; and

(c) an additional 16,666 shares of Stock on or after May 22, 2022.

Notwithstanding any provision of this Agreement to the contrary, the Option is only exercisable to the extent that it has become vested.

3.Transferability. This Option is not transferable or assignable, except by will or by the laws of descent and distribution and shall be exercisable during Participant’s lifetime, only by him. Any attempt to alienate, assign, pledge, hypothecate, or otherwise dispose of the Options, except as provided for herein or in the Plan, or attempted levy of any attachment, execution, or similar process upon the rights or interest hereby conferred shall be void ab initio and the Committee may take any action it deems appropriate to prevent such attempted disposition.

4.Exercise of Option. The Option may be exercised at any time, in whole or in part, to the extent that it has become vested under Section 2. The right to exercise this Option shall expire ten (10) years after the Date of Grant (the “Expiration Date”).

(a)Termination of Provision of Services. If the Participant ceases to provide services to the Company and its Affiliates for any reason other than death or disability (as defined in section 22(e)(3) of the Internal Revenue Code (“the Code”)), the unvested portion of the Option shall thereupon terminate and the Participant may exercise the vested portion of the Option for a period of three months thereafter or, if sooner, until the Expiration Date. Thereafter, the Option shall terminate and cease to be exercisable.

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(b)Disability. If the Participant ceases to provide services of the Company or one of its Affiliates by reason of a Disability, the unvested portion of the Option shall thereupon terminate and the Participant may exercise the vested portion of the Option for a period of twelve months thereafter or, if sooner, until the Expiration Date. Thereafter, the Option shall terminate and cease to be exercisable.

(c)Death. Upon the death of the Participant, the unvested portion of the Option shall thereupon terminate, except that the Option may be exercised by the Participant’s legal representatives, heirs, legatees or distributes may exercise the vested portion of the Option for a period of twelve months thereafter or, if sooner, until the Expiration Date. Thereafter, the Option shall terminate and cease to be exercisable.

5.Method of exercise. Any exercise of the Option shall be accompanied by a written notice to Company specifying the number of shares of Stock as to which the Option is being exercised that is accompanied by payment of the exercise price and arrangements for minimum required tax withholdings. Payment of the exercise price shall be made in cash or in other consideration that is acceptable to the Committee.

6.Change in Control. Notwithstanding the terms of the Plan, a Change in Control will not be deemed to occur unless and until the Board takes action to confirm that an event or transaction that is described as a Change in Control under the Plan has resulted in an actual change in control of the Company, as determined by the Board in its sole discretion. If the Board deems a Change in Control Event to have occurred, the Participant’s right to exercise this Option will be determined by the Committee in accordance with terms of the Plan.

7.Securities Act of 1933. Unless at the time of exercise of this Option there is an effective registration statement filed with the Securities and Exchange Commission under the 1933 Act, with respect to the sale of the shares of stock issuable upon exercise of this Option, the Participant’s right to exercise this Option shall be subject to the delivery to the Company upon such exercise of a letter, in form satisfactory to the Company’s counsel: (a) representing that the Participant intends to acquire the shares of stock issuable upon such exercise for investment for his own account and without a view to the resale or distribution thereof; and (b) agreeing that such shares shall not be sold or transferred by him in the absence of an effective registration statement filed with the Securities and Exchange Commission under the 1933 Act with respect to such transfer or an opinion of counsel satisfactory to the Company that such sale or transfer is not required to be registered under the 1933 Act or any applicable state securities law.

8.Subject to Provisions of Plan. The Options provided for herein are granted pursuant to the Plan and are subject to all the terms and conditions and provisions of the Plan. The terms that are defined in the Plan shall have the same meanings when used herein, except where the context clearly requires otherwise. A copy of the Plan is attached hereto and made a part hereof as if fully set out herein.

9.Withholding. As a condition to any exercise of the Option, Participant shall promptly remit in full to the Company the minimum amount of federal and (if any) state income and employment tax withholding that Company is required to remit to the Internal Revenue Service or applicable state department of revenue in accordance with the then-current provisions

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of the Code and applicable state law. The Company shall withhold from the Stock to be delivered a number of shares that is sufficient to cover the minimum required tax withholdings due on exercise, based on the Fair Market Value of Stock upon exercise, unless alternate arrangements for tax withholdings has been made by the Participant.

10.General. This Agreement shall be construed and interpreted according to the laws of the State of Tennessee. The foregoing contains the entire and only agreement between the parties respecting the subject matter hereof, and any representation, promise, or condition in connection therewith not incorporated herein shall not be binding upon either party. The headings of the various sections of this Agreement are for convenience of reference only, and shall not modify, define, limit or expand the express provisions of this Agreement. This Agreement shall be binding upon and inure to the benefit of any successor or successors of the Company.

11.Acknowledgment. Participant acknowledges receipt of a copy of the Plan, a copy of which is attached hereto, and represents that Participant is familiar with the terms and provisions thereof. Participant agrees to accept as binding, conclusive, and final all decisions and interpretations of the Committee on any questions arising under the Plan.

[Execution Page Follows]

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EXECUTION PAGE

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

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