A basic assumption in BS and most models can be loosely stated as
“Daily stock returns are normally distributed, approximately.” I used
to question this assumption. I used to feel that if the 90% confidence
interval of an absolute price change in IBM is $10 then it will be
that way 20 years from now. Now I think differently.
When IBM price was $1, daily return was typically a few percent, i.e.
a few cents of rise or fall.
When IBM price was $100, daily return was still a few percent, i.e. a
few dollars of rise or fall.
So the return tends to stay within a narrow range like (-2%, 2%),
regardless of the magnitude of price.
More precisely, the BS assumption is about log return i.e. log(price
relative). This makes sense. If %return is normal, then what is a
The touted feature of a “tradable” doesn’t impress me. Now I feel this feature is useful IMHO only for option pricing theory. All traded assets are supposed to follow a GBM (under RN measure) with the same growth rate as the MMA, but I’m unsure about most of the “traded assets” such as —
– IR futures contracts
– weather contracts
– range notes
– a deep OTM option contract? I can’t imagine any “growth” in this asset
– a premium bond close to maturity? Price must drop to par, right? How can it grow?
– a swap I traded at the wrong time, so its value is decreasing to deeply negative territories? How can this asset grow?
My MSFM classmates confirmed that any dividend-paying stock is disqualified as “traded asset”. There must be no cash coming in or out of the security! It’s such a contrived, artificial and theoretical concept! Other non-qualifiers:
eg: spot rate
eg: price of a dividend-paying stock – violates the self-financing criteria.
eg: interest rates
eg: swap rate
eg: future contract’s price?
eg: coupon-paying bond’s price
My professors emphasized repeatedly
* first generation IR model is the one-factor models, not Black model.
* Black model initially covered commodity futures
* However, IR traders adopted Black’s __formula__ to price the 3 most common IR options
** bond options (bond price @ expiry is LN
** caps (libor rate @ expiry is LN
** swaptions ( swap rate @ expiry is LN
** However, it’s illogical to assume the bond price, libor ate, and swap rates on the upcoming expiry date (3 N@FT) all follow LN distributions.
* Black model is unable to model the term structure. I would say that a proper IR model (like HJM) must describe the evolution of the entire yield curve with N points on the curve. N can be 20 or infinite…