The searchbox is not the container component, it has not 'fit' property. The only way to fit a searchbox to its parent div container is to recreate it with 'width' and 'height' properties or call 'resize' method to resize it after created.
<div style="width:200px;height:30px;border:1px solid #ccc">
<input id="sb" data-options="prompt:'Please Input Value'"></input>
</div>
<script>
$(function(){
var p = $('#sb').parent();
$('#sb').searchbox({
width: p.width(),
height: p.height()
});
});
</script>