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 contract expiry date (three N@FT) ALL follow LogNormal distributions.
* Black model is unable to model the term structure. I think it doesn’t eliminate arbitrage. 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…
* HJM uses (inst) fwd rate, which is continuously compounded. Some alternative term structure models use the “short rate” i.e. the extreme version of spot overnight rate. Yet other models  use the conventional “fwd rate” (i.e. compounded 3M loan rate, X months forward.)
 the Libor Mkt Model
* HJM is mostly under RN measure. The physical measure is used a bit in the initial SDE…
* Under RN measure, the fwd rate follows a BM (not a GBM) with instantaneous drift rate and instantaneous variance both time-dependent but slow-moving. Since it’s not GBM, the N@T is Normal, not LG
** However, to use the market-standard Black’s formula, the discrete fwd rate has to be LN
* HJM is the 2nd generation term-structure model and one of the earliest arbitrage free model. In contrast, the Black formula is not even an interest rate model.
[[Hull]] is first a theoretical / academic introductory book. He really likes theoretical stuff and makes a living on the theories.
As a sensible academic, he recognizes the (theory-practice) “gaps” and brings them to students’ attention. but I presume many students have no spare bandwidth for it. Exams and grades are mostly on the theories.
update — use “bank account” …
Beginners like me often intuitively use cash positions when replicating some derivative position such as a fwd, option or swap.
I think that’s permissible in trivial examples, but in the literature, such a cash position is replaced by a bond position or a MMA. I think the reason is, invariably the derivative position has a maturity, so when we lend or borrow cash or deploy our savings for this replication strategy, there’s a fixed period with interest . It’s more like a holding a bond than a cash position.
Background – in mathematical finance, DF is among the most basic yet practical concepts. Forward contracts (including equity fwd used in option pricing, FX fwd, FRA…) all rely directly on DF. DF is part of most arbitrage discussions including interview questions.
When we talk about a Discount Factor value there are always a few things implicit in the context
* a valuation date, which precedes
* a cash flow date,
* a currency
* a financial system (banking, riskfree bond…) providing liquidity, which provides
* a single, consistent DF value, rather than multiple competing values.
*  There's no uncertainty in this DF value, as there is about most financial contracts
– almost always the DF value is below 1.0
– it's common to chain up 2 DF periods
An easily observable security price that matches a DF value is the market price of a riskless zero-coupon bond. Usually written as Z_0. Now we can explain  above. Once I buy the bond at this price today (valuation date), the payout is guaranteed, not subject to some market movement.
In a math context, any DF value can be represented by a Z_0 or Z(0,T) value. This is the time-0 price of some physical security. Therefore, the physical security “Z” is a concrete representation of the abstract _concept_ of discount factor.