Hi, I am interested to know about London option exchange.
Are they a legitimate trading organisation?
What do you mean by legitimate? If you are asking if London Option Exchange has a license, the answer is no.
There are plenty of binary option brokers that are licensed, regulated and fair. You should read more here: http://scamoption.com/best-binary-brokers/
You can read a full review of London Option Exchange at: http://fakebroker.com/london-option-exchange-review/