While the SIM shown can be physically cut to nano size, the problem is that you will need to cut part of the actual contacts off to make it fit. And if you bugger that up, good bye SIM.
Apart from that, nano SIMs are 0.67mm thick while older SIMs (micro etc.) are 0.76mm thick. So even if you cut your old SIM to nano size, it still might not fit in your phone.
The safest, and probably sanest, option is to visit your cellular provider's local store and ask them for assistance. They should be able to copy the contents of your current SIM to a new nano one but if they do this, ensure they do not keep the old SIM! That way you just need to pay for the cost of the new nano SIM, and maybe the store's time.