If you send the signal out through your computer's serial port (assuming it has one) on the RTS, or the DTR, output pin, it might directly generate a ±10V signal. You can measure the voltage at the RTS and DTR output pins on the serial port connector to find out. When idle, and the serial port is enabled, the voltage will be +V or -V. (±5 V, ±10 V, ±12 V, and ±15 V levels are commonly seen.) If the voltage is what you want, you can use LabView to easily toggle those outputs between the +V and -V states at will.
If you find that the voltage is ±5 V, you could toggle the RTS and DTR into opposite states simultaneously and take the voltage across those two pins to get ±10 V. If you find the voltage is higher than ±10 V, you can use resistors to divide the voltage down to a lower level, or use a pair of 10V zener diodes and series resistors to clamp the voltage to ±10 V.