To remove the option with value: "option1":

$("#selectBox option[value='option1']").remove();

You will have to set values in your options, though:

<option value="option1">option1</option>
<option value="option2">option2</option>
<option value="option3">option3</option>

Adding value properties also has the benefit that you can change the displayed value, while sending something the code understands back to the server:

<option value="option1">Mercedes</option>
<option value="option2">Volvo</option>
<option value="option3">Volkswagen</option>

The server will still receive option1, for example.

For removing the last element simply do:

jQuery("select option:last").remove();

Of course you can change the selector to whatever you want, it must only point to one or multiple option elements.

To remove a specific option with index then you must use .eq() or :eq and .remove()

$('select option:eq(2)').remove();

Using simple JavaScript with your given example:

<script type="text/javascript">
    var oList = document.getElementsByTagName("select")[0] ;
    var oOption = oList.options[2]; // to select the third childelement
    oList.removeChild(oOption); // to remove the childelement
</script>

This removes the third entry of the first select in the DOM. It would be a good idea to give your select an id like this:

<select id="sel"> ... </select>

This way you can select the element like this:

var oList = document.getElementById("sel") ;

Or with using jQuery use

$("#sel option:eq(2)").remove();

to remove the third item again.

You should get this option by a selector...for example using an id, and then use the romove() function. For example:

<select>
  <option id="a">a</option>
  <option>b</option>
</select>

JQuery:

$("#a").remove();