Can a normal SIM be modified to work as a nano-SIM? I just puchased an iPhone5 and want to use my current SIM from my existing phone. Is there a way to convert my existing SIM to nano-SIM?
No. You could cut it to the same dimensions, but the risk to damage the chip is high. Additionally the nano sim is slightly thinner, so it still wouldn't be a good fit.

I suggest you exchange your sim card with your operator. Note that there exist adapters to fit a nano sim in a larger form factor though.