Here you have 2 options:

a) provide a parentActivityName to your SecondActivity tag in AndroidManifest.xml like this:

 <activity
    ...
    android:name=".SecondActivity"
    android:parentActivityName=".MainActivity" >

b) override onOptionsItemSelected in SecondActivity like this:

@Override
public boolean onOptionsItemSelected(MenuItem item) {
    switch (item.getItemId()) {
    // Respond to the action bar's Up/Home button
    case android.R.id.home:
        onBackPressed();
        return true;
    }
    return super.onOptionsItemSelected(item);
}

I would suggest reading this guide for more information.

Here is your code

 public class EditProfileActivity extends AppCompatActivity {

        @Override
        protected void onCreate(Bundle savedInstanceState) {

            super.onCreate(savedInstanceState);
            setContentView(R.layout.edit_profile);
            Toolbar toolbar = (Toolbar)findViewById(R.id.profile_action_toolbar);
            setSupportActionBar(toolbar);
            setTitle("Edit Profile");
            ActionBar actionBar= getSupportActionBar();
            actionBar.setDisplayHomeAsUpEnabled(true);
        }

        @Override
        public boolean onOptionsItemSelected(MenuItem item) {
            int id = item.getItemId();
            if (item.getItemId() == android.R.id.home) {
                   finish();
            }

            return super.onOptionsItemSelected(item);
        }
    }     

You have to override onOptionsItemSelected and check the item's id, if it is equals with home button's id, just call onBackPressed method.

@Override
        public boolean onOptionsItemSelected(MenuItem item) {
            if (item.getItemId() == android.R.id.home) {
                onBackPressed();
            }
            return super.onOptionsItemSelected(item);
        }

You have to define what should happen when you click on that button, this can be done in your second activity's onOptionsItemSelected method. Notice the android.R.id.home constant which refers to the activity's back button that you want to use.

@Override
public boolean onOptionsItemSelected(MenuItem item) {
    switch (item.getItemId()) {

    case android.R.id.home:

        finish(); //close the activty
        return true;
    }
    return super.onOptionsItemSelected(item);
}

First of all, always see Android Guidelines http://developer.android.com/intl/pt-br/design/patterns/navigation.html to prevent Google blocks Android apps.

Try to add this code in your Activity

@Override
public boolean onCreateOptionsMenu(Menu menu) {
    // Inflate the menu; this adds items to the action bar if it is present.
    return true;
}

@Override
public boolean onOptionsItemSelected(MenuItem menuItem) {
    switch (menuItem.getItemId()) {
        case android.R.id.home:
            onBackPressed();
            break;
    }

    return super.onOptionsItemSelected(menuItem);
}

@Override
public void onBackPressed() {
    super.onBackPressed();
}