Relief from stock-market anxiety is always just a telephone call away -- provided you can get through to your broker. Unless he is underemployed, or unless your account is so big that he promptly hangs up on everyone else, this isn't always an easy chore.

Now, however, you can order trades by letting your fingers do the walking across the keyboard rather than calling on the phone. Consult prices, make your decisions, and execute -- all on a personal computer.

The process has its advantages. "We like using our computer to make our stock transactions because there's no one calling us on the phone and bothering us," remarks Evelyn Merwin, a retired certified public accountant in Canon City, Colo. "We can price some stocks in the evening if we want to and initiate the transaction when the stock market opens the next morning."

Merwin and her husband, who own an Osborne I computer equipped with a modem, subscribe to The Desk Top Broker, a service marketed since last August by the San Francisco discount-brokerage house C. D. Anderson & Co. To use the service, Merwin hooks her computer up to Anderson's via a phone connection, answers a few prompts that appear on the screen, and indicates her instructions. The order is displayed back on the screen for confirmation, and once confirmed it is transmitted directly to the broker's wire room for execution. C. Derek Anderson, president of C. D. Anderson, points out that "with The Desk Top Broker, you never get a busy signal."

Market orders are supposed to go through in 5 to 10 minutes. Once they do, investors' portfolios are automatically updated, and written confirmations are mailed out. Investors trading on the system can indicate market or limit orders, good-till-canceled or day orders, and stop-loss, stop-buy, or stop-sell orders. They can also initiate margin or short sales. The cost of all this includes an annual $195 subscription fee; per-minute rates of from 10 cents to 40 cents, depending on the time of day; and Anderson's discount-level brokerage commission.

Merwin's only complaint with the system focuses on the U.S. Postal Service rather than C. D. Anderson. "It recently took seven days for us to receive a transaction slip, and the [Securities and Exchange Commission] requires that the trade be paid for within five business days," she says. Other subscribers have encountered internal problems. Sam Liberto, an administrator at Columbia University in New York City, complains, "I can't get real-time quotes for many options. Instead of the real-time price, my screen often reads 'not available.' I also have trouble finding out if an order has gone through. With The Desk Top Broker I sometimes have to wait two hours for a response."

Presumably, bugs like these can be worked out of the system. Longer-term issues, though, are not all resolved. For security, investors use a double password of their own choosing, which they can change whenever they want. But if they hit the wrong key or if an order somehow doesn't go through, no one is quite sure who will be responsible for the error, the user or the broker. "There's the question of responsibility that has yet to be legally tested," says John Wolfe, assistant editor of TeleServices Report, a Washington, D.C.-based newsletter that analyzes home delivery of financial and shopping services.

The potential for problems, in any event, hasn't stopped other brokerage houses from investigating personal-computer systems of their own. Fidelity Brokerage Services Inc., the country's second-largest discount broker, began offering stock and option trading in March, and plans to offer mutual fund buying and selling late in 1984. Both Anderson and Fidelity work through a privately held computer-services company, Trade*Plus Inc., in, Palo Alto, Calif., which originated the stock-trading software and which hopes to sell similar products to other discount brokers.

Full-service brokerages, for the most part, have steered clear of computerized execution of trades for obvious reasons: Letting people make their own trades would sidestep the relationship between broker and customer that the houses count on to generate commissions.

Currently, however, one full-service broker does offer home brokerage, albeit not through Trade*Plus. North American Investment Co., an East Hartford, Conn.-based firm sells a service dubbed NAICO-NET, which home-computer owners can tie into through such on-line information systems as Delphi, CompuServe, and The Source. Investors can fill out an application with the firm right at the computer, without first ordering any software or making a telephone call.

Unlike the discount broker services, NAICO-NET provides investors with advice and research on securities followed by the firm's analysts, and it does not limit its customers to stock and option trading. In addition, there is no subscription fee -- users pay only for the on-line service they use -- and NAICO charges a commission slightly lower than that charged by other full-service brokers.

Sam Liberto, however, likes C. D. Anderson's system despite its problems -- precisely because he doesn't have to listen to a broker's advice. "I like using my computer for a lot of my stock and option transactions," he says, "because I don't have to explain to my broker what I'm doing. This way I don't make my investment decisions based on what he says."