<?xml version="1.0" encoding="us-ascii"?><InstanceReport xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><Version>2.4.0.8</Version><ReportLongName>1060 - Disclosure - ASSET RETIREMENT OBLIGATION</ReportLongName><DisplayLabelColumn>true</DisplayLabelColumn><ShowElementNames>false</ShowElementNames><RoundingOption /><HasEmbeddedReports>false</HasEmbeddedReports><Columns><Column FlagID="0"><Id>1</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

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&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;b&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt; FONT-WEIGHT: bold;" size="2"&gt;NOTE F&amp;#8212;ASSET RETIREMENT OBLIGATION&lt;/font&gt;&lt;/b&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;The estimated fair value of the future costs associated with dismantlement, abandonment and restoration of oil and gas properties is recorded generally upon acquisition or completion of a well. The net estimated costs are discounted to present values using a risk adjusted rate over the estimated economic life of the oil and gas properties. Such costs are capitalized as part of the related asset. The asset is depleted on the units-of-production method. The associated liability is classified in other long-term liabilities, net of current portion, in the accompanying Consolidated Balance Sheets. The liability is periodically adjusted to reflect (1)&amp;#160;new liabilities incurred, (2)&amp;#160;liabilities settled during the period, (3)&amp;#160;accretion expense, and (4)&amp;#160;revisions to estimated future cash flow requirements. The accretion expense is recorded as a component of depreciation, depletion and amortization. The Company has cash held in escrow with a fair market value of $3.2&amp;#160;million that is legally restricted for potential plugging and abandonment liability in the Wilmington field which is recorded in other assets in the Consolidated Balance Sheets. A reconciliation of the Company&amp;#8217;s asset retirement obligations is as follows:&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;
&lt;table style="text-align:left;WIDTH: 73.34%; BORDER-COLLAPSE: collapse; MARGIN-LEFT: 1in;" border="0" cellspacing="0" cellpadding="0" width="73%"&gt;

&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="78%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="3%"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 16.36%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="16%" colspan="2"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&lt;b&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 8pt; FONT-WEIGHT: bold;" size="1"&gt;June&amp;#160;30,&lt;/font&gt;&lt;/b&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="1%"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="78%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="3%"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="BORDER-BOTTOM: windowtext 1pt solid; BORDER-LEFT: medium none; PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 16.36%; PADDING-RIGHT: 0in; BORDER-TOP: medium none; BORDER-RIGHT: medium none; PADDING-TOP: 0in;" valign="bottom" width="16%" colspan="2"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&lt;b&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 8pt; FONT-WEIGHT: bold;" size="1"&gt;2013&lt;/font&gt;&lt;/b&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="1%"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="78%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="3%"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="BORDER-BOTTOM: medium none; BORDER-LEFT: medium none; PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 16.36%; PADDING-RIGHT: 0in; BORDER-TOP: medium none; BORDER-RIGHT: medium none; PADDING-TOP: 0in;" valign="bottom" width="16%" colspan="2"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&lt;b&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 8pt; FONT-WEIGHT: bold;" size="1"&gt;(in&amp;#160;thousands)&lt;/font&gt;&lt;/b&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="1%"&gt;
&lt;p style="TEXT-ALIGN: center; MARGIN: 0in 0in 0pt;" align="center"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="top" width="78%"&gt;
&lt;p style="TEXT-INDENT: -10pt; MARGIN: 0in 0in 0pt 10pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Balance at beginning of period&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="3%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.3%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="1%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;$&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 15.06%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="15%"&gt;
&lt;p style="TEXT-ALIGN: right; MARGIN: 0in 0in 0pt;" align="right"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;25,236&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="1%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="top" width="78%"&gt;
&lt;p style="TEXT-INDENT: -10pt; MARGIN: 0in 0in 0pt 10pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Liabilities incurred in current period&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="3%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 16.36%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="16%" colspan="2"&gt;
&lt;p style="TEXT-ALIGN: right; MARGIN: 0in 0in 0pt;" align="right"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;395&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="1%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="top" width="78%"&gt;
&lt;p style="TEXT-INDENT: -10pt; MARGIN: 0in 0in 0pt 10pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Liabilities settled in current period&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="3%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 16.36%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="16%" colspan="2"&gt;
&lt;p style="TEXT-ALIGN: right; MARGIN: 0in 0in 0pt;" align="right"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;(1,194&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="1%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;)&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="top" width="78%"&gt;
&lt;p style="TEXT-INDENT: -10pt; MARGIN: 0in 0in 0pt 10pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Accretion expense&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="3%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="BORDER-BOTTOM: windowtext 1pt solid; BORDER-LEFT: medium none; PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 16.36%; PADDING-RIGHT: 0in; BORDER-TOP: medium none; BORDER-RIGHT: medium none; PADDING-TOP: 0in;" valign="bottom" width="16%" colspan="2"&gt;
&lt;p style="TEXT-ALIGN: right; MARGIN: 0in 0in 0pt;" align="right"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;1,093&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; PADDING-TOP: 0in;" valign="bottom" width="1%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr style="padding:0;"&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 78.88%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="top" width="78%"&gt;
&lt;p style="TEXT-INDENT: -10pt; MARGIN: 0in 0in 0pt 10pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;Balance at end of period&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 3.4%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="3%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;
&lt;td style="BORDER-BOTTOM: windowtext 2.25pt double; BORDER-LEFT: medium none; PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.3%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; BORDER-TOP: medium none; BORDER-RIGHT: medium none; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="1%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;$&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="BORDER-BOTTOM: windowtext 2.25pt double; BORDER-LEFT: medium none; PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 15.06%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; BORDER-TOP: windowtext 1pt solid; BORDER-RIGHT: medium none; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="15%"&gt;
&lt;p style="TEXT-ALIGN: right; MARGIN: 0in 0in 0pt;" align="right"&gt;&lt;font style="FONT-FAMILY: Times New Roman; FONT-SIZE: 10pt;" size="2"&gt;25,530&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;
&lt;td style="PADDING-BOTTOM: 0in; PADDING-LEFT: 0in; WIDTH: 1.36%; PADDING-RIGHT: 0in; BACKGROUND: #cceeff; PADDING-TOP: 0in;" bgcolor="#CCEEFF" valign="bottom" width="1%"&gt;
&lt;p style="MARGIN: 0in 0in 0pt;"&gt;&amp;#160;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
&lt;/div&gt;</NonNumbericText><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>nonnum:textBlockItemType</ElementDataType><SimpleDataType>na</SimpleDataType><ElementDefenition>The entire disclosure for an asset retirement obligation and the associated long-lived asset. An asset retirement obligation is a legal obligation associated with the disposal or retirement from service of a tangible long-lived asset that results from the acquisition, construction or development, or the normal operations of a long-lived asset, except for certain obligations of lessees.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 410

 -SubTopic 20

 -Section 50

 -Paragraph 1

 -URI http://asc.fasb.org/extlink&amp;oid=6392692&amp;loc=d3e7535-110849



Reference 2: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 410

 -SubTopic 20

 -Section 50

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 -URI http://asc.fasb.org/extlink&amp;oid=6392692&amp;loc=d3e7569-110849



Reference 3: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 143

 -Paragraph 22

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



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