You could use an op-amp comparator using +15 and -5 on it's power rails. Bias -input to +2.5vdc and wire + input to arduino output pin. This is assuming the op-amp can sink and source the current requirements for your load, which you haven't stated.
Crazy idea (please validate before destroying something): A 7805? You will get 5V that would be used as Common. Then, between 20V and common you get 15v, and between common and "the other pole from the supply, formerly known as ground" you'll get -5V (play with your multimeter and place the black probe on common).
So now how will you turn it off? The base needs to be at -5V to turn it off and you can only change that signal between zero and +5.
Hint you need another transistor.
Do you want a "real" push pull or just a signal that goes from +15 to -5?
Attached is an example of the latter. For a real push pull you need another two transistors. Feed the output of this circuit through a resistor to the push pull pair that you posted.
There is a IC that can give you -5v with only 2 capacitors. ICL7660/MAX1044. You will need a 7805 to convert that 15v to 5v for its input. It can only handle 100mA. pdfserv.maxim-ic.com/en/ds/ICL7660-MAX1044.pdf