Stock and bond prices soared today and short-term interest rates fell. The Dow Jones Industrial Average turned in its best one-day performance in 10 months.

Analysts said the broad rallies in all three markets--stocks, bonds and short-term securities--had more to do with the internal functioning of those markets than with the administration's economic programs or the increase in leading economic indicators announced this morning by the Commerce Department.

The Dow Jones Average of 30 industrial stocks jumped 21.59 points to 864.25, the biggest single-day increase for that key stock market indicator since last March 12, when the Dow index rose 22.15.

Trading volume was heavy, with 66.8 million shares changing hands on the New York Stock Exchange. That was a sharp rise from the 50.1 million shares bought and sold Wednesday and the heaviest trading day since last March 13.

There was an "explosive rally" in short-term interest rates, according to Nicholas Marrone of the Bank of New York. Three-month Treasury bills fell 60 basis points (each basis point is 1/100th of a percentage point) and one-year bills declined 30 basis points.

Long-term bonds rose in price. The increases ranged from $5 per $1,000 of face value to $15, Marrone said. But analysts said that the rise in the prices of bonds--long-term debt securities sold by companies and governments--seemed to be carried along by activity in the financial futures markets, where investors buy and sell contracts based on future expectations of interest rates. The rise in futures prices spilled over into the cash market for bonds themselves.

Larry Wachtel, of the brokerage firm Bache Halsey Stuart Shields, said there is no easy explanation for the strong performance by stocks today. Nearly 1,150 stocks rose in price today on the New York Stock Exchange, while about 175 fell.

He said the stock market began a long decline on Dec. 4. Even prolonged good or bad markets are punctuated by reversals. "Eight weeks is a long time for any market to go in the same direction," Wachtel said.

He said today's sharp price rise might have been triggered in part by the market's lackadaisical response Wednesday to President Reagan's State of the Union address.

Investors had hoped that Reagan would take steps to raise taxes to reduce the huge federal budget deficits that have been projected to occur for the next several years.

Instead, Reagan vowed to continue his program of tax cuts and rely on a stronger economy to push up revenues. Many analysts and traders felt that the Reagan speech should have caused a severe decline in stock prices because investors would be scared by the potential for higher interest rates threatened by bigger budget deficits. However, stock prices traded in an extremely narrow range on Wednesday and rallied near the end of trading.

That late rally impelled protective stock purchasing today by many traders who sell short. Short sellers bet on a decline in stock prices and sell stock they do not yet own at a specified price for delivery several days or weeks in the future. Short-sellers hope to make money by purchasing the stock when it must be delivered at a lower price than the seller will be paid. If stock prices rise, the short-seller takes a beating.

When stock prices did not fall Wednesday, many short-sellers got scared and began to buy today to minimize their losses.

Wachtel said the increase in the index of leading economic indicators, a tentative and perhaps incorrect signal that the current recession is winding down, had little impact on the stock market.

If anything, he said, its impact would have been negative because an early end to the recession would augur higher levels of business borrowing and, as a result, higher interest rates.

The Dow increase of 21.59 points, a 2.56 percent gain, was echoed by the New York Stock Exchange's own index, which rose 1.80 points, or 2.7 percent, to 68.58. The American Stock Exchange's index closed up 12.92 points to 292.59.

Marrone, of the Bank of New York, said there seemed to be an influx of foreign purchases of U.S. bonds that drove up prices initially (when bond prices rise, their interest yields fall). As in the stock market, short-sellers of bonds then began to buy to cover themselves.