If all you want to do is know if the date from your Front Matter is earlier than the system time, then you can use the ISO 8601 date format and rely on lexicographic ordering. This is kind of a hoax, but it will work on the example you provided.
It is important that this trick works both as site.timewell as the date of your Front Matter ( page.past_dateand page.future_datein the example below) in ISO 8601 format.
---
layout: default
past_date: 2015-03-02
future_date: 2016-03-02
---
{% capture currentDate %}{{ site.time | date: '%F' }}{% endcapture %}
{% capture pastDate %}{{ page.past_date | date: '%F' }}{% endcapture %}
{% capture futureDate %}{{ page.future_date | date: '%F' }}{% endcapture %}
<br>currentDate: {{currentDate}}
<br>PastDate earlier than currentDate? {% if pastDate < currentDate %}Yes{% else %}No{% endif %}
<br>FutureDate earlier than currentDate? {% if futureDate < currentDate %}Yes{% else %}No{% endif %}
Gives me the following result:
currentDate: 2015-07-12
Is PastDate earlier than currentDate? Yes
FutureDate , currentDate?