[SalesForce] Remove an option from a Visual force SelectList SelectOption

I have had the same problem. The solution was to use tha native html select list and an visualforce hidden field. After each change on the selectlist the currently selected value is saved to the hidden field:

<!-- Here is the native select list. 
     It is empty, we fill it later with own options.
     Notice, that by onchange the selected value will be send to the hidden field. -->
<select id="someList" onchange="jQuery('[id$=selectedItemHidden]').val(jQuery(this).val());">
</select>

<!-- Hidden field with a reference to the apex variable -->
<apex:inputHidden value="{!selectedAccount}" id="selectedItemHidden"/>

<!-- Here we fill the select list with some options -->
<script>
for(var i = 0; i < optionsArray.length; i++)
{
    var opt = new Option(optionsArray[i],optionsArray[i]);
    jQuery("select#someList").append(opt);
}
<script>

You can pre fill the html select list with options from the database:

Controller:

public class MyClass{
    // Here we define the list with accounts
    public List<Account> accounts { get; set; }

    // In constructor we will read the accounts
    public MyClass(){
        accounts = [ Select Id, Name From Account Limit 10 ];
    }
}

Page:

<!-- Here we just use repeat function to iterate the list -->
<select id="someList">
    <apex:repeat value="{!accounts}" var="a">
        <option value="{!a.id}">{!a.name}</option>
    </apex:repeat>
</select>

It seems like you have not initialized applicationOptions list in your constructor where you are adding values to select options, try initializing as

applicationOptions = new List<SelectOption>();