How is zaroff able to finance his life style?

From the text:

"After the debacle in Russia I left the country, for it was imprudent for an officer of the Czar to stay there. Many noble Russians lost everything. I, luckily, had invested heavily in American securities, so I shall never have to open a tearoom in Monte Carlo or drive a taxi in Paris."

Zaroff descended from nobility and wisely invested his fortune in American securities. His investments provide him with the money necessary to finance his extravagant lifestyle on the island.