Your grandmother dies and leaves you 10,000 shares of stock. You have no broker and little interest in the stock market. You want to sell the stock and be done with it.

You may be a candidate for a discount broker.

Or perhaps you do all your own stock market research, laboriously plotting trend lines, profits and a multitude of other indices. You trust no one but yourself to manage your portfolio -- big or small -- and suffer grudingly the occasional phone call from your broker bearing advice.

You, too, may be a likely customer for a discount broker.

Until 1975, for all practical purposes, there were no discount brokers. All brokers who were members of the New York Stock Exchange used the same commission schedule to determine what minimum to charge a customer for buying or selling stocks. Even if your broker wasn't a member of the NYSE or another exchange, if you bought a stock listed on one of the nation's exchanges, he still had to charge at least the minimum fee.

But then the Securities and Exchange Commission changed all that.

Beginning in May 1975, brokers could no longer base their charges on an industrywide, preset commission schedule. Instead, the SEC ruled that rates had to be set by each broker, not by a cartel.

Big buyers and sellers of stocks, such as pension funds or insurance companies, whose trades were measured in thousands or tens of thousands of shares, used their market power to exact commission reductions, often sizable, from their brokers.

But the full-line brokers -- who support a full range of services, such as research, and employ a coterie of retail brokers who follow clients' portfolios, track market developments and advise customers to help ensure that their investment decisions are wise -- either were unable or unwilling to cut prices to many retail customers.

To satisfy the retail customer who was more interested in lower commissions on trades than in the other services the brokerage firms provide (generally for free), the so-called discount brokerage industry was born.

Discount brokerage firms generally offer no research services, no advice and usually assign a customer no individual broker. Although as years go on some so-called discounters have begun to add services, at the core a discount broker offers little but execution. At the customer's order, the broker will buy or sell a security.

But without the overhead incurred by full-line brokers, the discount broker offers the service at a commission that is sharply lower than the average retail customer will pay at a regular broker. The first discount brokers based their fees on the 1975 NYSE rates, offering the customer a "percentage off," or a discount, from the old schedule. Many still do. Hence the name discount broker.

Whether the discount brokerage firm today bases its fees on the old New York Stock Exchange schedule or on some other basis, the fundamental premise is the same: Because of its sharply lower costs, the discounter can charge from 25 to 70 percent less than traditional brokers. The size of the discount usually depends on the size of the transaction. Discounters usually charge a minimum fee per trade, between $25 and $35.

"Most full-cost brokers charge about 120 percent of the rates in effect on April 1, 1975," according to Darrell Brookstein, president of First Georgetown Securities Inc., a locally based discount brokerage house.

"Basically, we separate the transaction the buying or selling of securities from everything that comes before it," said John Lanza, manager of the Washington offices of Quick & Reilly Inc., the nation's second-largest discount brokerage house. Lanza said Quick & Reilly will execute trades and give a customer stock quotes and dividend payment dates. But Lanza's firm will not provide research reports, tips or investment help.

"We won't give advice," he said. "Facts we can give you. We stay away from things that aren't facts."

"We attract someone who knows his goals and objectives and is familiar with dealing in the stock market," according to Stephen C. Voss, president of Voss & Co. of Springfield, the area's first discount operation. Voss' firm began to offer discounts on trading in over-the-counter stocks (those not listed on any exchange) in 1973 and started offering lower fees on exchange-listed stocks in 1975, when the old NYSE rate schedule went by the boards.

"We are not for everyone. There is a place for both full-line and discount brokers. We assist that spectrum of people who know what they want to do," said Maurice Minerbi, vice president and manager of Charles Schwab & Co. Inc.'s Washington office. Schwab is the nation's biggest discounter, with 36 branches. Quick & Reilly has 28 operations across the country.

Brookstein, of First Georgetown, said there are three types of customers that frequent discounters. In addition to the individual who makes his or her own investment decisions, there is the individual who is unfamiliar with the stock market but needs to make a one-shot use of a broker. That person, such as the grandchild who inherits stock, does not need to pay full cost on a trade, he said.

There also are investors who use a full-line broker for most of their trading, but do research themselves in one particular area and use a discount broker to trade in that field, Brookstein said. The savings on even one sizeable trade can be thousands of dollars, he said.

There is a fourth type of customer, the kind that discounters don't mind but that drive full-line brokers crazy. They are the individuals who make full use of a major brokerage house's services, then go to a discounter for most of their trades.

"We've found out about some of our customers who buy a hundred shares of stock with us and a thousand shares of the same stock through a discounter," an official of one of the nation's biggest brokerage houses said. "When that happens, I sit down and have a serious talk with that client."

All discount brokers buy and sell stocks and corporate bonds. Most also deal in stock options. Some will buy and sell government securities or municipal bonds; some will not. Some sell mutual funds. Schwab is preparing to offer insurance and an interest-bearing checking account service similar to Merrill Lynch's well-known cash management accounts. But their bread and butter is stock trading, most officials of discount brokerage firms say.

Discount brokers say that they are a rapidly growing force in the retail securities business. Full-line brokers say that discounters appeal to a small segment of the market and that their impact has been small.

"They sometimes force us to do a little discounting," said Leslie J. Silverstone, who runs the Dean Witter Reynolds Inc. office at 18th and K streets NW. "But on balance I see relatively little impact. They discount brokers may have forced out some marginal full-service firms and some individual brokers who were not pulling their weight at big firms, but on balance I don't think I have lost any business to them."

Surprisingly little data exists to support either view. A Securities and Exchange Commission report estimated in 1979 that discount brokers accounted for 4 to 6 percent of the market. The SEC recently rejected a staff proposal to study the discount brokerage industry, concluding that such a study would be of little but academic interest.

David Lefeve, vice president of Merrill Lynch, said that he would guess discount brokers account for no more than 5 to 6 percent of the Washington area market and perhaps 8 to 9 percent nationwide. But Lefeve said that is a seat-of-the-pants estimate.

George Cardy, manager of Olde & Co.'s Rockville office, said he thinks that discounters had 8 percent of the business in 1979 (compared with the SEC's estimate of 4 to 6 percent), more than 10 percent last year and will garner 20 to 25 percent of the business within a few years.

The area discount brokers have about 27,000 customers accounts among them, according to a Washington Post survey. As at most brokerage firms, many of those accounts are inactive. Some customers have accounts with more than one discount firm, and some have accounts both with a discount broker and a full-cost broker.

But the 27,000 accounts at the five discount brokers in the Washington area -- Quick & Reilly, Olde, Voss, First Georgetown and Schwab -- compare with 20,000 accounts at Silverstone's Dean Witter Reynolds Inc. branch.

According to a New York Stock Exchange study, in mid-1980 there were 754,000 investors in the Washington area in mid-1980. Assuming most investors have active or inactive accounts, discount brokers serve fewer than 4 percent of the area's investors. But no one has information on how more active or inactive discount brokerage customers are than full-line clients.

If discount brokers are to keep their rates low, according to some analysts, they will have to increase the number of customers they serve. Their income remains heavily dependent upon stock market volume and their costs of operation are increasing.

When the market is busy, discount brokers write lots of orders. When the market is slack they write fewer. The same is true at the full-line brokers. But in the last 10 years, full-line brokers have added many services beyond the traditional brokerage products to make themselves less dependent on volatile securities income. Brokers sell insurance, annuities and investment planning in addition to trading stocks, bonds and government securities. Many of the newer products are even more profitable than trading commissions.

In addition to having to cope with the vicissitudes of the stock market, discount brokers are facing rising costs. Clearing firms, which make sure that the stock gets from one customer to another, are raising their prices rapidly. Few discounters -- Quick & Reilly is an exception -- clear their own trades. In addition, rentals in major cities are rising rapidly, although discounters -- which rely heavily on newspaper advertising for new clients rather than the lure of a business district storefront -- may find it easier than major brokers to move their phone lines to lower-rent suburban office buildings.

"What I'm looking for is a shakeout in the discount business," said Voss of Springfield's Voss & Co. Already some big discounters have swallowed smaller ones. Schwab just bought out a small local firm, Brokers Exchange, as well as Kingsley, Boye & Southwood, a national discounter.

In their push to acquire new business, discounters are beginning to add services they used to eschew, services that in the end may cost them more money than merely taking orders and giving stock quotes.

For example, discounters traditionally have done no research. Brookstein at First Georgetown pointed out that most of the materials in its small lobby bookshelf are out of date. But Cardy, of the Detroit-based Olde & Co., perhaps the fastest growing national discounter, said his office not only keeps an up-to-date research library, but the firm publishes a monthly research letter.

"I'm not supposed to hold a customer's hand," Cardy said. But he also said that if a customer was about to do a transaction he felt would be a bad one, he would recommend against it -- much like the traditonal full-line, full-commissioned broker.

Voss has added computerized information-retrieval systems that can give a customer any news about a company in the last 90 days and bring an up-to-date balance sheet instantaneously. Voss said his firm often gives opinions and informs clients when there is news of importance to a stock they hold. The company even underwrote a small stock offering last year but has no plans to do it again soon.

"We should see the same developments in the discount brokerage industry that we saw in other discount industries. The firms come in and make a big splash with cut-down services and low prices. They make some inroads into the full-service market, then over time begin to add operations and services that make them less distinguishable from full-service firms," said a Wall Street economist who knows the securities industry well.

"It happened with discount department stores," he said. "Many cut-rate stereo stores now have knowledgeable salespersons. It will happen with discount brokers."

Discounters add services, he said, when the segment of the market they initially tried to serve is saturated. Since margins are low and competition is high, some rock-bottom discounters go out of business and others add services to attract a new breed of client. Those are the clients who are interested in services not provided by the lowest-cost discount houses but not in all the services provided by the more expensive full-line operations.

With competition lessened at the lower end, prices rise. But not to the level of the full-line operation.

"There always will be a market for discount brokers," said an executive of a major full-line firm. "Maybe one in 10 investors should use them. My only worry is that the one in 10 who could use them successfully does not, while the investors who do their own research and trade with discount brokers are precisely the people who don't have the time to do an effective job of it."