Background — “Drift” sounds simple and innocent, but no no no.

* it requires a probability measure

* it requires a numeraire

* it implies there’s one or (usually) more random process with some characteristics.

It’s important to estimate the drift. Seems essential to derivative pricing.

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BA = a bank account paying a constant interest rate, compounded daily. No uncertainty no pr-distro about any future value on any future date. $1 today (time-0) becomes exp(rT) at time T with pr=1 , under any probability measure.

MMA = money market account. More realistic than the BA. Today (time-0), we only know tomorrow’s value, not further.

Z = the zero coupon bond. Today (time-0) we already know the future value at time-T is $1 with Pr=1 under any probability measure. Of course, we also know the value today as this bond is traded. Any other asset has such deterministic future value? BA yes but it’s unrealistic.

S = IBM stock

Now look at some tradable asset X. It could be a stock S or an option C or a futures contract … We must must, must assume X is tradable without arbitrage.

—- Under BA measure and cash as numeraire.

X_{0}/B_{0} = E (X__{T}/B__{T}) = E (X__{T})/B__{T} =>

E (X__{T})/X_{0 }= B__{T}/B_{0}

Interpretation – X__{T} is random and non-deterministic, but its expected value (BA measure) follows the _*same*_ drift as BA itself.

—- Under BA measure and using BA as numeraire or “currency”,

X_{0}/B_{0} = E (X__{T}/B__{T})

Interpretation – evaluated with BA as currency, the value of X will stay constant with 0 drift.

—- Under T-measure and cash numeraire

X_{0}/Z_{0} = E (X__{T}/Z__{T}) = E (X__{T})/$1 =>

E (X__{T})/X_{0 }= 1/Z_{0}

Interpretation — X__{T} is random and non-deterministic, but its expected value (Z measure) follows the _*same*_ drift as Z itself.

—- Under T-measure and using Z as numeraire or “currency”,

X_{0}/Z_{0} = E (X__{T}/Z__{T})

Interpretation – evaluated with the bond as currency, the value of X will stay constant with 0 drift.

—- Under IBM-measure and cash numeraire

X_{0}/S_{0} = E (X__{T}/S__{T})

Interpretation – can I say X follows the same drift as IBM? No. The equation below doesn’t hold because S_T can’t come out of E()!

!wrong —> E (X__{T})/X_{0 }= S__{T}/S_{0} ….. wrong!

—- Under IBM-measure and IBM numeraire… same equation as above.

Interpretation – evaluated with IBM stock as currency, the value of X will stay constant with 0 drift.

Now what if X is non-tradable i.e. not the price process of a tradable asset? Consider random variable X = 1/S. X won’t have the drift properties above. However, a contract paying X_T is tradeable! So this contract’s price does follow the drift properties above. See http://bigblog.tanbin.com/2013/12/tradeablenon-tradeable-underlier-in.html