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## Wednesday, October 2, 2013

### drift under a given measure (but without dividing by its numeraire)

See post on using cash numeraire.

I think we can assume for each numeraire, there’s just one  probability measure. That measure defines the probability distribution of any price process.  We can use that measure to evaluate expectations, to talk about Normal/Lognormal or dW, and to evaluate “exponential” drift (the “m” below), assuming

dX = m X dt

Under the standard risk-neutral measure, the exponential drift is the same ( =r ) for all TRADEABLE assets, even though physical drift rates are not uniform. Specifically, the bank account itself (paying exponential short rate r) has a drift = r. So does the discount bond. So does a stock. So does a fwd contract. So does a vanilla call or binary call. So does an asset-or-nothing call.

At this point, we don’t need to worry about martingale or numeraire, though all the important results come from numeraire/MG reasoning.

I feel it’s important to remember drift is a __prediction__ about the future. It’s inherently based on some assumed probability distribution i.e. a probability measure. That probability distribution is derived from many live prices about T-expiry contracts.

Therefore, under another predicative probability distribution/measure, the predicted drift would differ.

The stock-measure is trickier. Take IBM. There exists an IBM measure. Under this measure, i.e. operating under this new (predictive) probability distribution, we can derive the (predicted) exponential drift rate of any asset’s price movement. Specifically, we can work out the predicted drift of the IBM price process. That drift is r + sigma^2, where

r:= exponential drift rate of the bank account i.e. money-market account. Consider it a physical drift but actulaly this is non-random and the same drift speed under any measure
Sigma:= the volatility of IBM. Same value under any measure.

 there might exists multiple, but I don’t bother.