Edit in JSFiddle

$(function(){
  setInterval(function(){
    var divUtc = $('#divUTC');
    var divLocal = $('#divLocal');  
    //put UTC time into divUTC  
    divUtc.text(moment.utc().format('YYYY-MM-DD HH:mm:ss'));      
    
    //get text from divUTC and conver to local timezone  
    var localTime  = moment.utc(divUtc.text()).toDate();
    localTime = moment(localTime).format('YYYY-MM-DD HH:mm:ss');
    divLocal.text(localTime);        
  },1000);
});

UTC<br/>
<div id="divUTC"></div><br/>
Your Local Time with respect to above UTC time<br/>
<div id="divLocal">
</div>