You can get JsonOutputFormatter from the BindingContext.OutputFormatters inside of the code of your controller. It allows you dynamically change the SerializerSettings.
Try to include using Newtonsoft.Json; in the controller code and to do the following inside of your controller action:
var f = BindingContext.OutputFormatters.FirstOrDefault(
            formatter => formatter.GetType() ==
                         typeof (Microsoft.AspNet.Mvc.Formatters.JsonOutputFormatter))
        as Microsoft.AspNet.Mvc.Formatters.JsonOutputFormatter;
if (f != null) {
    //f.SerializerSettings.Formatting = Formatting.Indented;
    f.SerializerSettings.NullValueHandling = Newtonsoft.Json.NullValueHandling.Ignore;
}
I included Formatting = Formatting.Indented just for my tests only, because one see the results immediately. You don't need it of cause.
UPDATED: I created the demo project using MVC Web Application with no Authentication. Then I added in HomeController the following metod
public object TestMethod()
{
    var testResult = new {
                         name = "Test",
                         value = 123,
                         nullableProperty = (string) null
                     };
    return testResult;
}
and changed the Launch URL of the project to Home/TestMethod and started the demo. I could see
{"name":"Test","value":123,"nullableProperty":null}
You don't need to add any additional using statements to use the code which I posted initially (one need just have the standard using Microsoft.AspNet.Mvc; and using System.Linq;), but the code could be more readable if you would have using Microsoft.AspNet.Mvc.Formatters; and using Newtonsoft.Json;. I added the using statements for Microsoft.AspNet.Mvc.Formatters and Newtonsoft.Json and modified the code to the following
public object TestMethod()
{
    var testResult = new {
                         name = "Test",
                         value = 123,
                         nullableProperty = (string) null
                     };
    var f = BindingContext.OutputFormatters.FirstOrDefault(
                formatter => formatter.GetType() == typeof (JsonOutputFormatter)) as JsonOutputFormatter;
    if (f != null) {
        f.SerializerSettings.Formatting = Formatting.Indented;
        f.SerializerSettings.NullValueHandling = NullValueHandling.Ignore;
    }
    return testResult;
}
The output results looks now as following
{
  "name": "Test",
  "value": 123
}
The standard code use "Newtonsoft.Json" in version 6.0.6. We can add "Newtonsoft.Json": "8.0.2" in dependencies to use the latest version of Newtonsoft.Json. See the problem with resolving of indirect dependencies which I reported in the issue and which is still opened.
You can download the test project from here.