Stock markets will be looking forward nervously to October 1999, the last October of this millennium. The omens are not necessarily good. October 1929 saw the start of the great crash which led to the depression of the early 1930s. The 1987 crash happened over two days - October 16th and 19th - and in both October 1997 and October 1998 trouble brewed in world equity markets.

On October 27th 1997 the US stock market had its biggest one-day fall. In 1998, October 8th saw the low point of the Dow for the year - it was some 20 per cent down on the peak reached three months earlier.

Can October 1999 be any different? According to Prof Tim Congdon, a member of the Bank of England's Monetary Policy Committee (MPC), the US stock market is heading for a big fall. That could well happen just next month, although the precise timing is "inherently unknowable".

According to a paper from Prof Congdon's Lombard Street Research, tension exists between company valuations, which are far in excess of any norm, and unsustainable macroeconomic trends.

The paper reviews a number of benchmarks and finds that on the most optimistic assumptions, the stock market is heading for a 30 per cent fall. The most pessimistic scenario would see the market halve in value.

A similar exercise involving British and Irish companies would probably also reveal an overvaluation but of a much smaller amount.

Irrespective of this, however, European stock markets could not remain immune to any significant decline in the US and serious falls could also be expected here, knocking billions off the value of listed companies.

Prof Congdon warns that the domestic US economy is overheating while the growth in the external deficit, where the US is still acting as importer to the rest of the world, cannot continue. "October 1999 may or may not break the mould but sooner or later - a collapse in the American stock market will be part of a return to a more balanced macroeconomic situation."

He also questions the view that a collapse in share prices would trigger a recession, believing that it may simply correct the growth in domestic demand which has been a hallmark of many economies over recent years. This is a view obviously shared by the majority on the MPC given its decision last week to increase British interest rates, just before inflation fell to a 37-year low.

According to this analysis there are a number of different ways of measuring valuations and hence the "correct" level for a stock market. Two of the most simple benchmarks are price earnings ratios and the dividend yield. Both of these measures suggest that to restore "normal" valuations, US equities need to halve.

Slightly more complicated benchmarks introduce bond yields or long-term interest rates into the equation. This "yield ratio" also predicts that US share prices ought to halve. Two other benchmarks of bonds-to-earnings imply that earnings need to fall by at least a quarter to restore a normal valuation.

However, investors may not need to sell everything. There is also a very strong argument that the old models of equity valuation are failing to capture changes in the world, particularly increased productivity and low inflation. This is the new-paradigm argument and it is becoming increasingly popular, although no one can explain exactly how it works.

But at its core is the belief that inflation as an issue is dead, while US industry has become much more flexible and very adaptable.

One of its most famous converts is the Federal Reserve chairman Mr Alan Greenspan. Only three years ago Mr Greenspan and his Fed team were making similar noises to Prof Congdon is today. Since then he has decided that this was probably wrong. Certainly the US market has proved very sustainable and chances are that it will remain sustainable going forward.

Nevertheless, whether or not equity markets are overvalued, there is a good possibility that next month could once again be volatile. There are already identifiable risks. Global interest rates are under upward pressure with Britain already having raised rates, and there is a chance that the Fed may do the same at the beginning of October.

At the same time there are concerns about the millennium bug and there is a distinct risk of a flight to quality. Already some of the largest US banks have said they will not pay any return on money on deposit over the New Year, fearing a tidal wave of liquidity.

There are also rumoured serious losses at some of the world's largest hedge funds which are said to have made serious stock-picking errors in recent months, leading to large falls in valuations.

On the other hand long-term players will look at the fundamentals. So far it still appears that the economic background is favourable, inflation is under control and corporate earnings are still looking good.

Whether investors choose to sit it out or panic at the first sign of serious selling remains to be seen.