Lower short-term interest rates injected some optimism into both the stock and bond markets late today.

The Dow Jones industrial average, which plummeted to its lowest level in more than a year on Monday, recovered in late trading today to close up 1.72 points to 901.83. The Dow average measures the performance of 30 of the biggest and most widely held industrial stocks.

Other stocks did not fare so well. The number of stocks that closed lower on the New York Stock Exchange exceeded those that closed higher by 1,216 to 326. The New York exchange's own index fell 0.34 point to 72.58.

Even Dow stocks suffered in the early going, after the government's early morning announcement that consumer prices climbed 1.2 percent in July, their biggest increase in 16 months and far faster than most economists expected. The Dow average was down more than 9 points in early trading. But in the last hour of trading, major institutional buyers such as pension funds and insurance companies began buying again, according to Jerry Hinkle, head of trading at the brokerage firm Sanford C. Bernstein & Co.

Bond prices fell sharply in early trading after reaching record or near-record lows on Monday. But prices recovered as the day went on. The average bond with a face value of $1,000 fell about $2.50 today compared with a drop of between $20 and $25 on Monday.

Both stock and bond markets were buoyed by a fall in the so-called federal funds rate, the interest that banks charge each other for overnight loans of excess reserves. The rate, which was about 17 3/4 percent Monday, drifted down to near 16 percent this afternoon. The federal funds rate is affected mainly by the operations of the Federal Reserve system, the nation's central bank. Fed policies are designed to wring inflation out of the economy by holding down the availability of money and are considered one of the chief culprits in the stock and bond market malaise.

Bond prices rise or fall in inverse relationship to interest rates. When rates rise, prices fall. When interest rates are high, not only do stock investors find investments such as Treasury bills more attractive, they also find it too expensive to borrow money to buy stocks.

Investors also are worried that the high interest rates will plunge the economy into a severe recession, which would hurt corporate profits and add to future budget deficits at a time when the Reagan administration's tax cuts appear to be swelling the imbalance between government revenues and expenditures.

"The nemesis continues to be high interest rates," said Bernstein's Hinkle. But he said that the bad news of higher budget deficits, continued high interest rates and potential recession already are incorporated into stock prices. "The worst part appears to be over," he said.

Trading volume was heavier today than on Monday. On the New York exchange nearly 54.8 million shares changed hands compared with 46.8 million on Monday. Composite volume was 62.21 million shares.

The American Stock Exchange market index closed down 1.51 points at 351.21. Volume was 6.4 million shares compared with nearly 5.5 million on Monday.

On the over-the-counter market, the NASDAQ composite index closed at 197.78 off 2.98 points