Facebook Inc. isn't publicly traded yet, but that hasn't stopped investors like Leon Cohen from trying to buy the social network's stock now.

The 83-year-old resident of Aventura, Fla., would like to purchase 10,000 or more of Facebook's private, before-initial-public-offering shares. For the past two months, Mr. Cohen said he has been lingering by the phone "like a lady in waiting" to get his hands on Facebook stock through a broker.

"We expect Facebook to do very well in the aftermarket," he said, referring to the expected public trading of the stock, "so I want to get in and buy the stock before it comes out." Mr. Cohen, who retired 15 years ago after selling a lighting-fixtures company in Brooklyn, N.Y., last year bought shares of Zynga Inc., Groupon Inc. and LinkedIn Corp. at their IPO prices through a different broker. After weeks of fruitless waiting for pre-IPO Facebook shares, he also has begun seeking them through private-market auctions.

Strong interest from investors wanting to get in early has sparked a 24% surge in Facebook's share price on the "secondary market" for private-company shares since just before the social network filed on Feb. 1 for an IPO. As of February 22, Facebook's per-share price on the secondary market was $42, up from $34 on Jan. 20, according to SharesPost Inc., which regularly conducts auctions of private-company shares. That has boosted the company's indicated value to about $105 billion.

Facebook's trajectory in private-company trading underscores the immense appetite for the social network's stock. The price rise since its IPO filing is outpacing that of other Web companies after they filed for their IPOs last year, according to some brokers and people familiar with the matter.