I can automatically save the notebook into HTML after running code. However, sometimes the results are generated too quick, so the output HTML does not have the output in the last cells.

I'm wondering, if it is possible to tell the file to save itself?

Something like

# In last cell
current_filename = 'my_file.ipynb'
save_current_notebook(current_filename)
output_HTML(current_filename)

Now I can get away with:

display(Javascript("IPython.notebook.save_notebook()"),            
        include=['application/javascript'])