Friday, January 31, 2014

change measure but using cash numeraire (drift

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.
   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

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