long-term daily stock returns ~ N(m, sigma)

Label: intuitiveFinance

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

-150% return?

rolling fwd measure#Yuri

(label: fixedIncome, finMath)


In my exam Prof Yuri asked about T-fwd measure and the choice of T.

I said T should match the date of cashflow. If a deal has multiple cashflow dates, then we would need a rolling fwd measure.  See [[Hull]

However, for a standard swaption, I said we should use the expiry date of the option. The swap rate revealed on that date would be the underlier and assumed to follow a LogNormal distro under the chosen T-fwd measure.

John Hull: estimate default probability from bond prices

label: credit

The arithmetic on P524-525 could be expanded into a 5-pager if we were to explain to people with high-school math background…


There are 2 parts to the math. Part A computes the “expected” (probabilistic) loss from default to be $8.75 for a notional/face value of $100. Part B computes the same (via another route) to be $288.48Q. Equating the 2 parts gives Q =3.03%.


Q3: How is the 7% yield used? Where in which part?


Q4: why assume defaults happen right before coupon date?

%%A: borrower would not declare “in 2 days I will fail to pay the coupon” because it may receive help in the 11th hour.


–The continuous discounting in Table 23.3 is confusing

Q: Hull explained how the 3.5Y row in Table 23.3 is computed. Why discount to  the T=3.5Y and not discounting to T=0Y ?


The “risk-free value” (Column 4) has a confusing meaning. Hull mentioned earlier a “similar risk-free bond” (a TBond). At 3.5Y mark, we know this risk-free bond is scheduled to pay all cash flows at future times T=3.5Y, 4Y, 4.5Y, 5Y. We use risk-free rate 5% to discount all cash flows to T=3.5Y. We get $104.34 as the “value of the TBond cash flows discounted to T=3.5Y”


Column 5 builds on it giving the “loss due to a 3.5Y default, but discounted to T=3.5Y”. This value is further discounted from 3.5Y to T=0Y – Column 6.

Part B computes a PV relative to the TBond’s value. Actually Part A is also relative to the TBond’s value.


In the model of Part B, there are 5 coin flips occurring at T=0.5Y   1.5  2.5  3.5  4.5 with Pr(default_0.5) = Pr(default_1.5) = … = Pr(default_4.5) = Q. Concretely, imagine that Pr(flip = Tail) is 25%. Now Law of total prob states


100% = Pr(05) + Pr(15) + Pr(25) + Pr(35) + Pr(45) + Pr(no default). If we factor in the amount of loss at each flip we get


Pr(05) * $65.08 + Pr(15) * $61.20 + Pr(25) * $57.52 + Pr(35) * $54.01 + Pr(45) * $50.67 + Pr(no default, no loss) + $0 == $288.48Q

so-called tradable asset – disillusioned

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

Black’s formula isn’t interest rate model, briefly

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…

mean reversion in Hull-White model

The (well-known) mean reversion is in drift, i.e. the inst drift, under physical measure.

(I think historical data shows mean reversion of IR, which is somehow related to the “mean reversion of drift”….)

When changing to RN measure, the drift is discarded, so not relevant to pricing.
However, on a “family snapshot”, the implied vol of fwd Libor rate is lower the further out accrual startDate goes. This is observed on the market [1], and this vol won’t be affected when changing measure. Hull-White model does model  this feature:
<!–[if gte msEquation 12]>σte-a T-t <![endif]–>
[1] I think this means the observed ED future price vol is lower for a 10Y expiry than a 1M expiry.

HJM, again

HJM’s theory started with a formulation containing 2 “free” processes — the drift (alpha) and vol (sigma) of inst fwd rate 

<!–[if gte msEquation 12]>df­T=α(t) dt+σ(t) dW<![endif]–>    

, both functions of time and could be stochastic.
Note the vol is defined differently from the Black-Scholes vol.
Note this is under physical measure (not Q measure). 
Note the fwd rate is instantaneous, not the simply compounded.
We then try to replicate one zero bond (shorter maturity) using another (longer maturity), and found that the drift process alpha(t) is constrained and restricted by the vol process sigma(t), under P measure. In other words, the 2 processes are not “up to you”. The absence of arbitrage enforces certain restrictions on the drift – see Jeff’s lecture notes.
Under Q measure, the new drift process [1] is completely determined by the vol process. This is a major feature of HJM framework. Hull-white focuses on this vol process and models it as an exponential function of time-to-maturity:
<!–[if gte msEquation 12]>σte-a T-t <![endif]–> 
That “T” above is confusing. It is a constant in the “df” stochastic integral formula and refers to the forward start date of the (overnight, or even shorter) underlying forward loan, with accrual period 0.
[1] completely unrelated to the physical drift alpha(t)
Why bother to change to Q measure? I feel we cannot do any option pricing under P measure.  P measure is subjective. Each investor could have her own P measure.
Pricing under Q is theoretically sound but mathematically clumsy due to stochastic interest rate, so we change numeraire again to the T-maturity zero bond.
Before HJM, (I believe) the earlier TS models can’t support replication between bonds of 2 maturities — bond prices are inconsistent and arbitrage-able