If we have constant influx of fiat and constant influx of bitcoins, they would form an equilibrium price. If bitcoin influx halves and nothing else changes, the equilibrium price will double.
So reward halving will cause price doubling.
Unfortunately, there are number of assumptions in this reasoning. We assume that influx of fiat won't change. We assume that price is currently fluctuates around the equilibrium. Both of these assumptions are wrong. Influx of fiat can change tenfold, any direction (up, down or sidewise

). Rally/crash can move price very far from the equilibrium. And even influx of bitcoins can change - miners can stop selling or long-time holders will decide to sell. Scale of this can be order of magnitude more, than the halving. Halving can change prices twofold, while a rally - twenty-fold.
Having said all this, we can still assume, that halving will double the prices. When? IMO, earlier than in 15 months time. But it won't look like sharp doubling. Because of this: Say, you are a miner. You are selling all coins you produce straightaway, to cover running expenses and interest. But if you know that the coin price will double in a year, it would make sense to borrow money and pay from them, while saving the coins. If you borrow even at 20% p/a it already makes sense to stop selling coins now and get 80% profit from it. The same is true about speculators. They should start accumulating already. So it is possible that the doubling is already priced in.
IMO, it is not yet priced in, because 15 months is an eternity in the bitcoin world. Borrowing for the eternity is too risky.

So, the price doubling will happen somewhere early next year and will be very gradual, maybe not even noticeable on the background of rallies and crashes.