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.
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.
X0/B0 = E (X_T/B_T) = E (X_T)/B_T =>
E (X_T)/X0 = B_T/B0
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”,
X0/B0 = 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
X0/Z0 = E (X_T/Z_T) = E (X_T)/$1 =>
E (X_T)/X0 = 1/Z0
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”,
X0/Z0 = 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
X0/S0 = 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)/X0 = S_T/S0 ….. 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