There is a certain irony in me answering this question in the negative but here goes.

Is there anything to stop a historian from making prophecies or speculating about the future? No. If something occurs that a historian has claimed would occur can they realistically attribute that to their historical wisdom? Probably. It still wouldn't necessarily be true. It is much easier to ensure your prophecy was correct after the fact, and even better to make the prophecy later and then attribute it once it has come to past - although more carefully kept records are making such a possibility increasingly difficult.

The greatest myth is that history is a series of cyclical events. It's not. The axiom: 'those who do not learn history are doomed to repeat it' is not meant to signify that those who study history can predict the future but that those who do not pay attention to precedent will be more inclined to make basic and avoidable errors.

Historians make their judgements and comments about the past based, hopefully, on most if not all of the evidence avaiable to them (including relying on the sagacity of their peers and predecessors to fill out the gaps) I don't think any of them would ever consider themselves worldly enough to predict the future based on their very refined knowledge of the past.

Of Asimov's non-robot/foundation stories there is one in particular which is relevant here. This is 'Alexander the God', in Gold [PDF] an anthology published posthumously in 1995.

Asimov's character is molding himself as a financial Alexander the Great enabled by his trusty steed Bucephalus (his computer). This is not technically 'psychohistory' but instead the ability to process and contextualise the modern market by drawing a comparison from all of the world's history as saved in the computer's 'brain'. It's number crunching simulation based on not only an apparently perfect knowledge of the present but the past as well.

There are a couple of problems here. The first is that almost any period before, perhaps, the twentieth-century would come up with large swathes of 'insufficient data'. There are massive leaps of logic and intuiton and, well, accepted speculation to weave together the gaps in the sources. If any computer were capable of processing every possible outcome in the present and comparing it to the past then I would suggest that it might as well not bother. You will already possess sufficient influence and power to dominate the world without the necessity of 'predicting' the market anyway. You could make it happen.

No historian possesses, or will ever possess, the ability to hold all of the conflicting disciplines, history, and modern data to predict the future because on both sides there will be insufficient data. Fot my part I'll predict whatever you want me to. It doesn't mean it's going to happen (although I'll happily accept the accolades and carefully ensure that my next prediction will be fulfilled after my death). People can identify macro-trends and blather on about historical inevitability to their hearts content. It is far easier to confirm a prophecy once it has been completed than it is to have confidence in a prophecy before it has been fulfilled.

History is not a predictive but a contextualising tool. We can understand that humans operate in certain ways when exposed to certain pressures or influences (in fact in 'disciplinary' terms anthropology and sociology are both far better at this than 'history' - although the boundaries between disciplines are rapidly dissolving). History should ground our delusions of grandeur and remind us constantly that 'Empires wax and wane; states cleave asunder and coalesce'. History is ultimately what forms the structures (mental and physical) that we inhabit and the culture which we adore or despise. Understanding history helps us understand ourselves at a particular juncture. It does not tell us where we, or our descendants, will go.

You could. Predictions based on historical data is very common in economics, where they use historical data to uncover trends.

Of course, there is a problem with this line of reasoning. The philosopher (and certainly not a historian) Taleb Nassim has used the metaphor of the turkey keeping track of days being fed (see his The Black Swan Theory). The data collected by the turkey shows that he has been fed 999 consecutive days. The data would suggest that he'll be fed on the 1000th day. What the model does not know is that the 1000th day is the day before Thanksgiving. That's what /u/TheGreenReaper7 means by insufficient data.

Another way to show the insufficiency of historical data for predicting the past is by employing historical data, methods and philosophy to predict the future. While a fun dinner-table game, 'what-if?' or virtual history is impossible. Or, rather, it's possible but absolutely meaningless:

"What if Gavrilo Princip had spend five minutes longer ordering a sandwich?"

Answer: He would have missed Franz Ferdinand and/or bought a different sandwich.

Would it have prevented World War 1? Unlikely, but we simply can't be certain. Contexts would have changed, and in history the context is everything.