Nothing gets Alan Davidson, the outspoken president of a small New York brokerage firm, more riled than the name Mary Schapiro. "She went around the country telling people she was a hawk on regulations," he huffs.

Schapiro, 44, president of NASD Regulation Inc., the self-regulatory unit of the National Association of Securities Dealers Inc., has spent the past few years putting a very public face on the in-house cops of the NASD, the parent company of the Nasdaq Stock Market.

She brought in a team of former Securities and Exchange Commission staffers, forcing changes in the clubbier atmosphere of the old NASD, where members policed one another and the punishments often amounted to wrist slaps, giving Nasdaq an image as the Wild, Wild West of stock markets.

Now, as the home for many Internet companies, Nasdaq is the hottest market around, with the Nasdaq composite index up 63 percent this year. Several securities lawyers credit Schapiro's efforts with helping to turn around Nasdaq's image. That, in turn, has helped clear the way for a pivotal NASD board meeting today, where the board will discuss a plan to spin off Nasdaq and form a privately held, for-profit corporation.

Even so, the new regulatory tone has unnerved some securities firms.

"We have to ask: Where is the 'self' in self-regulation?" a securities executive said. "These ex-SEC guys have never worked in the industry. They seem to view every brokerage firm as potential crooks."

Schapiro is well aware of the criticism, even appearing eager to talk at length about it. "I know that for a number of our firms, the nature of self-regulation has changed a lot. It had to change," she said. "There were real issues that had to be addressed in a serious and forthright way by this institution."

Frank Zarb, NASD chairman, said he hears complaints, too. But "three years ago, the SEC said you are lousy regulators," he said. "You are going to have teeth and you are going to go out and bite. And if you don't, we're going to take action."

If the board does eventually approve the plan to create a for-profit company, it will thrust Schapiro and her regulatory unit, known as NASDR, into an even more highly visible position. Under the plan to be discussed today, the NASD plans to sell shares to members and to major companies that have stocks trading on Nasdaq.

It will be the job of the NASDR to referee the expected conflicts between the profit-making goals of a new Nasdaq and the stock market's duties to keep out reckless brokers, poorly funded ones, fast-talking scam artists and market manipulators.

In fact, this role is so pivotal to the health of the nation's stock markets, the Securities and Exchange Commission has signaled that Nasdaq and the New York Stock Exchange will have to guarantee the funding and independence of their self-regulatory organizations, known as SROs, before becoming for-profit corporations.

The SEC also has said it may eventually merge the different SROs to form a national organization that would handle everything except daily trading activities. Many brokers and securities lawyers say NASDR is being studied as a potential model for such a national SRO.

"I would like to see the role of NASDR become more central to all market centers. If Nasdaq privatizes, and the Big Board privatizes, today's NASDR ought to be the infrastructure of a single dominant self-regulator," said Neal Sullivan, a lawyer with the firm Bingham Dana.

The NASD recruited Schapiro in 1995 in the midst of a scandal and a showdown with the SEC. The SEC concluded that Nasdaq traders were fixing prices and manipulating the market and the NASD in-house cops did nothing. Schapiro, who had served as an SEC commissioner and head of the U.S. Commodity Futures Trading Commission, had developed a reputation as straightforward, savvy and tough regulator.

Unlike many other financial industry regulators, Schapiro has not been lured by the fast paced excitement of Wall Street or the online trading world. She is the kind of person who gets excited about new turbocharged software designed to slice and dice and find the bad guys among the NASD's 5,600 members. Flipping through graphs, grinning widely, Schapiro eyes light up. "We're so excited about this software," she said repeatedly.

When she was hired, her mission was to convince the SEC and public investors that Nasdaq could police itself.

"They've gotten tougher, and any time an organization gets tougher, people are going to complain it's too tough. . . . Sometimes the fines they insist upon may not be proportional to the harm inflicted but rather are meant to send a message," said Richard Phillips, a securities lawyer at Kirkpatrick & Lockhart.

Schapiro said she is "an enormous fan of self-regulation." She said her staff consults frequently with industry participants to design rules and requirements. "Things work because we haven't left it all to the government. We have incorporated people in industry who can bring real world experience and true understanding of markets and products and trading to solving real problems," she said.

But she acknowledges that the disciplinary process have gotten tougher.

And Schapiro thinks things need to change further. She is a fan of the SEC's national self-regulatory organization idea. She thinks as competition between stock markets heats up, the old self-regulatory structure may not work.

For example, Schapiro said, firms are under legal obligation to get "best execution" for their customers, which generally means best price, but also includes other factors such as speed.

A broker may send his trades in Microsoft Corp. stock to a Nasdaq competitor, instead of to Nasdaq. "If a Nasdaq examiner were to come in and question why I'm sending all my orders to one of the competitors . . . instead of to Nasdaq, I would wonder if I was being criticized because I really didn't get best execution or because I sent it to a rival," Schapiro said.

IN PROFILE: Mary L. Schapiro

* Position: President of NASD Regulation Inc., an independent subsidiary of the National Association of Securities Dealers reponsible for regulating brokerage firms, their registered representatives and the Nasdaq Stock Market.

* Age: 44

* Birthplace: New York

* Education: BA, Franklin and Marshall College, 1977; law degree, with honors, George Washington University, 1980.

* Career highlights: Trial lawyer; counsel to the Commodity Futures Trading Commission chairman; Securities and Exchange Commission member and acting chair; chairman, CFTC.

* Personal: Married to Charles Cadwell; two daughters.

* Other: Trustee, Franklin and Marshall College

CAPTION: Mary Schapiro appeared at a news conference in Manhattan with U.S. District Attorney Robert Morgenthau in January 1997 to announce that 53 people had been charged in a stockbroker-licensing test scam.

CAPTION: Mary Schapiro, the NASD's top regulator, helped clean up Nasdaq's image.