Gray box testing has limited knowledge of the system as an attacker would. The base code would remain confidential. This would further be enhanced by a Non-disclosure agreement (NDA) which is designed to protect confidential information.
Incorrect Answers:
A: A memorandum of understanding (MOU) documents conditions and applied terms for outsourcing partner organizations that must share data and information resources. They do not typically cover vulnerabilities and penetration / vulnerability testing. Furthermore, the CEO is concerned that IT staff lacks the knowledge to identify complex vulnerabilities.
B: A business partnership security agreement (BPA) is a legally binding document that is designed to provide safeguards and compel certain actions among business partners in relation to specific security-related activities. Black box testing is integrity-based testing that uses random user inputs. Code confidentiality is maintained but testing is limited.
D: White box testing requires full access to the code base as it involves validating the program logic. This does not test against vulnerabilities. Furthermore, the CEO is concerned that IT staff lacks the knowledge to identify complex vulnerabilities.
References:
Gregg, Michael, and Billy Haines, CASP CompTIA Advanced Security Practitioner Study Guide, John Wiley & Sons, Indianapolis, 2012, pp. 148, 167-168, 238-239
https://en.wikipedia.org/wiki/Non-disclosure_agreement
https://en.wikipedia.org/wiki/Gray_box_testing

Explanation
If an application has a known issue (such as susceptibility to buffer overflow attacks) and a patch is released to resolve the specific issue, then the best solution is always to deploy the patch.
A buffer overflow occurs when a program or process tries to store more data in a buffer (temporary data storage area) than it was intended to hold. Since buffers are created to contain a finite amount of data, the extra information - which has to go somewhere - can overflow into adjacent buffers, corrupting or overwriting the valid data held in them. Although it may occur accidentally through programming error, buffer overflow is an increasingly common type of security attack on data integrity. In buffer overflow attacks, the extra data may contain codes designed to trigger specific actions, in effect sending new instructions to the attacked computer that could, for example, damage the user's files, change data, or disclose confidential information. Buffer overflow attacks are said to have arisen because the C programming language supplied the framework, and poor programming practices supplied the vulnerability.

Explanation/Reference:
Reference: https://nvlpubs.nist.gov/nistpubs/legacy/sp/nistspecialpublication800-47.pdf