SUPPLEMENTAL LETTER

Exhibit 4.20

SUPPLEMENTAL LETTER

To: DryShips Inc.

Trust Company Complex

Ajeltake Road

Ajeltake Island

Majuro

The Xxxxxxxx Xxxxxxx XX 00000

From: HSH Nordbank XX

Xxxxxxx-Xxxxxxxxx-Xxxxx 00

X-00000 Xxxxxxx

Xxxxxxx

Dear Sirs

9 February 2012

1 Background.

(A) By a loan agreement dated 31 March 2006 (as supplemented, amended and restated from time to time, the “Senior Loan Agreement”) and made between (i) DryShips Inc. as borrower (the “Borrower”), (ii) the banks and financial institutions listed in Part A of Schedule 1 thereto as lenders (the “Senior Leaders”), (iii) the banks and financial institutions listed in Part B of Schedule 1 thereto as swap banks (together, the “Swap Banks” and each a “Swap Bank”) (iv) ourselves as agent, lead arranger, lead bookrunner and security trustee, (v) ourselves and Bank of Scotland plc (“BOS”) as joint underwriters and (vi) BOS as joint bookrunner, it was agreed that the Senior Lenders would make available to the Borrower a term loan and short-term credit facilities of (originally) up to US$518,750,000 (the “Senior Loan”) in aggregate.

(B) By two ISDA master agreements (each on the 1992 ISDA Master Agreement (Multicurrency-crossborder) form) and each dated 31 March 2006 (the “Master Agreement”) made between the Borrower and a Swap Bank, the Borrower has entered into or will enter into certain Designated Transactions (as such term is defined in the said Loan Agreement) pursuant to separate Confirmations (as such term is defined in the said Loan Agreement).

(C) By a letter (the “First Letter”) dated 19 May 2011 we:

(i) notified the Borrower there was a shortfall (the “Shortfall”) at that time in the security cover required to be maintained pursuant to clause 15.1 of the Senior Agreement; and

(ii) called on the Borrower, pursuant to clause 15.1 of the Senior Loan Agreement within 14 days of the date of the First Letter, either:

(a) to provide, or ensure that a third party provides, additional security which, in the opinion of the Majority Lenders (as defined in the Senior Loan Agreement), has a net realisable value at least equal to the shortfall and which, if it consists of or includes a Security Interest (as defined in the Senior Loan Agreement), covers such asset or assets and is documented in such terms as we may, with authorisation from the Majority Lenders (as defined in the Senior Loan Agreement), approve or require; or

(b) prepay in accordance with clause 8 of the Senior Loan Agreement such part (at least) of the Senior Loan as will eliminate the shortfall.

(D) Following the Borrower’s non-compliance with the terms of the First Letter and clause 15.1 of the Senior Loan Agreement, by a letter dated 7 June 2011 we reserved the rights of the Creditor Parties under the Senior Loan Agreement and all other Finance Documents (as defined in the Senior Loan Agreement).

(E) The Borrower has requested and the Lenders agreed to receive certain additional security to rectify the Shortfall.

(F) This Agreement sets out the terms and conditions on which the Lenders agree to:

(i) receive certain additional security to rectify the Shortfall; and

(ii) the consequential amendments to the Loan Agreement and the other Finance Documents in connection with those matters.

2 Agreement and amendments to the Senior Loan Agreement. Subject to the satisfaction of the conditions of this Letter and with effect from the date of this Letter, the Senior Loan Agreement shall be amended as follows:

(a) by adding the following new definitions in clause 1.2 thereof:

““Additional Security Period” has the meaning given in Clause 15.10;

“Approved Stock Exchange” means NASDAQ or any other stock exchange acceptable to the Agent (acting on the instructions of the Majority Lenders);

“Ocean Rig” means Ocean Rig UDW Inc. a corporation incorporated in The Xxxxxxxx Islands whose registered office is at Trust Company Complex, Ajeltake Road, Ajeltake Island, Majuro, the Xxxxxxxx Islands;

“Required Shares” has the meaning given in Clause 15.10;

“Shares Pledge” means, in respect of the Required Shares, a pledge and security agreement in respect of such shares executed by the Borrower in favour of the Security Trustee pursuant to Clause 15.10 in such form as the Lenders may approve or require; and

“Shortfall” means, during the Additional Security Period, any shortfall in the security cover required to be maintained pursuant to Clause 15.1;”;

(b) by adding to the definition of “Finance Documents” in clause 1.1 thereof a new sub-paragraph (w) as follows:

“(w) the Shares Pledge;”

and by re-designating the existing sub-paragraph (w) in the definition of “Finance Documents” in clause 1.1 thereof as a new sub-paragraph (x);

(c) by adding therein the following new clauses 11.27 and 15.10:

“11.27 Listing of Ocean Rig. The Borrower shall procure that at least 23.9 per cent, of the share capital of Ocean Rig is listed and continues to be listed in an Approved Stock Exchange.”;

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15.10 Additional security during the Additional Security Period. The Borrower undertakes with the Lenders:

(a) to execute and deliver, and to cause Ocean Rig to execute and deliver, on the first date of the Additional Security Period the Shares Pledge and the Control Agreement referred to therein in respect of the Required Shares; and

(b) deliver to the Agent such other documents equivalent to those referred to in paragraphs 3, 4 and 5 of Schedule 5, Part A as the Agent may require in connection with the execution of the Shares Pledge; and

(c) to ensure that any additional security given pursuant to this Clause 15.10 shall remain in full force and effect until the last date of the Additional Security Period.

In this Clause 15.10:

“Additional Security Period” means the period commencing on 9 February 2012 and ending on 31 March 2012;

“Required Shares” ” means 10,000,000 issued and outstanding shares of common stock in Ocean Rig (including all other or additional stock or other securities or property paid or distributed in respect of the Required Shares by way of stock-split, spin-off, split-up, reclassification, combination of shares or similar arrangements) currently in the legal and beneficial ownership of the Borrower to be pledged in favour of the Security Trustee pursuant to this Clause 15.10 in order to rectify the Shortfall; and

(e) by construing all references therein to “this Agreement” where the context admits as being references to “this Agreement” as the same is amended and supplemented by this Letter.

3 Amendments to Finance Documents. With effect on and from the date of this letter each of the Finance Documents (other than the Senior Loan Agreement) shall be, and shall be deemed by this Agreement to have been, amended as follows:

(a) the definition of, and references throughout each of the Finance Documents to, the Senior Loan Agreement and any of the other Finance Documents shall be construed as if the same referred to the Senior Loan Agreement and those Finance Documents as amended and supplemented by this Letter; and

(b) by construing references throughout each of the Finance Documents to “this Agreement”, “this Deed”, “hereunder and other like expressions as if the same referred to such Finance Documents as amended and supplemented by this Letter.

4 Senior Loan Agreement and Finance Documents. The Borrower hereby agrees with the Lenders that the provisions of the Senior Loan Agreement and the Finance Documents shall be and are hereby re-affirmed and remain in full force and effect.

5 Representations and Warranties. The Borrower hereby represents and warrants to the Lenders that:

(b) this Letter comprises the legal, valid and binding obligations of the Borrower enforceable in accordance with its terms.

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6 Conditions. Our agreement contained in paragraph 2 of this Letter shall be expressly subject to the condition that we shall have received in form and substance as may be approved or required by us on or before the signature hereof:

(a) copies of resolutions passed at a meeting of the board of directors of the Borrower evidencing approval of this Letter and authorising appropriate officers or attorneys to execute the same;

(b) the original of any power of attorney issued in favour of any person executing this Letter and the Shares Pledge on behalf of the Borrower; and

(c) a duly executed original of the Shares Pledge (and any underlying documents, including without limitation, a duly executed original of the Control Agreement referred to in the Shares Pledge); and

(d) copies of all governmental and other consents, licences, approvals and authorisations as may be necessary to authorise the performance by the Borrower of its obligations under this Letter and the execution, validity and enforceability of this Letter.

7 Notices. Clause 28 (Notices) of the Senior Loan Agreement shall extend and apply to this Letter as if the same were (mutatis mutandis) herein expressly set forth.

8 Governing Law. This Letter shall be governed by and construed in accordance with English law and Clause 30 (Law and Jurisdiction) of the Senior Loan Agreement shall extend and apply to this Letter as if the same were (mutatis mutandis) herein expressly set forth.

Please confirm your acceptance to the foregoing terms and conditions by signing the acceptance at the foot of this letter.

Yours faithfully

/s/ XXXXXXXXX XXXXXXXXXXXX

XXXXXXXXX XXXXXXXXXXXX

for and on behalf of

HSH NORDBANK AG

(as Agent for and on behalf of all Creditor Parties)

Accepted and agreed

/s/ XXXXXXXX XXXXXX

XXXXXXXX XXXXXX

for and on behalf of

DRYSHIPS INC.

Dated: 9 February 2012

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DRYSHIPS INC.

Dated: 9 February 2012

COUNTERSIGNED this 9^th day of February 2012 for and on behalf of the below companies each of which, by its execution hereof, confirms and acknowledges that it has read and understood the terms and conditions of this supplemental letter, that it agrees in all respects to the same and that the Finance Documents to which it is a party shall remain in full force and effect and shall continue to stand as security for the obligations of the Borrower under the Senior Loan Agreement.

/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
WEALTH MANAGEMENT INC.				SAMSARA SHIPPING COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
MALVINA SHIPPING COMPANY LIMITED				ARLETA NAVIGATION COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
BORSARI SHIPPING COMPANY LIMITED				ONIL SHIPPING COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
FABIANA NAVIGATION COMPANY LIMITED				CELINE SHIPPING COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
XXXXXX SHIPPING COMPANY LIMITED				XXXXXX SHIPPING COMPANY LIMITED

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/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
ARGO OWNING COMPANY LIMITED				KRONOS OWNING COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
TETHYS OWNING COMPANY LIMITED				SELENE OWNING COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
DIONE OWNING COMPANY LIMITED				URANUS OWNING COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
TEMPO MARINE CO.				STAR RECORD OWNING COMPANY LIMITED
/s/ XXXXXXX XXXXX				/s/ XXXXXXX XXXXX
XXXXXXX XXXXX				XXXXXXX XXXXX
for and on behalf of				for and on behalf of
REA OWNING COMPANY LIMITED				PHOEBE OWNING COMPANY LIMITED

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