CEV, very briefly

CEV has a formulation of sigma(t). It's a function of the current underlier price. A power function, to model the volatility smile...

SABR is related to CEV...

Hul White -- learning notes

Hull white is simpler than LMM. Both Arbitrage free.

LMM has too many factors. Hull White can be one-factor or 2-factor.

OIS instruments in YC bootstrapping - Jeff

Jeff's lecture notes has numerous practical tips, such as
*

fwd libor rate != instantaneous rate

Fwd libor rate , like a 20Y forward start 3-month Libor has an accrual period of 3 months. The inst fwd rate has an underlying loan with accrual period shrunk to 1 picosecond...

zero curve doesn't mean discount curve

Zero curve usually means the zero bond yield curve. I think it's usually upward sloping.

 

I think in STIRT/Sprite zero curve means the discount factor curve, so it's decreasing with maturity and always below 1.0.

 

A YC can be interchangeably represented as zero curve, discount curve, fwd curve (i.e. inst fwd rate) etc.

Gaussian HJM, briefly

... is a subset of HJM models.

An HJM model is Gaussian HJM if vol term is deterministic. Note "vol" term means the coefficient of the dW term. Every Brownian motion must always refer to an implicit measure. In this case, the RN measure.

How about the drift term i.e. the "dt" coefficient? It too has to be deterministic to give us a Gaussian HJM.

Well, Under the RN measure, the drift process is determined completely by the vol process. Both evolve with time, but are considered slow-moving [1] relative to the extremely fast-moving Brownian Motion of "dW". Extremely because there's no time-derivative of a BM

[1] I would say "quasi constant"

Language is not yet precise so not ready to publish on recrec...

modified duration vs duration, briefly

http://en.wikipedia.org/wiki/Bond_duration

MacD / (1+ r/k)

So in the limit, when the compound frequency k goes to infinity (i.e. continuous compound), this becomes same as MacD.

arb-free IR model

... must model the (evolution of) entire YC, rather than some points on it, like (the evolution of) one Libor rate. This is a main theme of the lectures on Black's model, forward measure, HJM etc.

 

For more details, See the post on HJM

Yuri on yield curve models

HJM hard to calibrate. Usually using European swaptions.

swaps illustration diagrams -- how to read

This write-up covers IRS, x-ccy swap...

These block diagrams are popular and partially useful, but beginners often don't realize:

* initial context -- typically a corporation has a periodic liability, or an investor has a periodic income.
** We had better ignore all the other arrows first.

* the motivation -- typically to convert the initial single arrow to other arrows. The swap contract adds 2 arrows, one of them cancelling out the pre-existing arrow.
** we had better focus on the 3 arrows and ignore other parts of the diagram.

yield curve , according to Jeff

Jeff's lecture notes (in 0x\pdf) has detailed explanations on

1) EUR OIS YC bootstrapping using specific OIS instruments
2) Libor YC under OIS discounting -- so OIS curve + libor curve needed.
3) Libor curve for a non-default tenor, such as 6M or 2M

lots of "root-finding"... but not too hard.

a YC (or a term structure) can be represented as a series of
* spot disc factors
* fwd disc factors
* spot interest rates
* fwd interest rates

Rebanato - good author on fixed income models

recommended by Sian Hwee.

Ronnie said Black model is popular (largely due to simplicity, and historical reason), and many option products are quoted in terms of vols implied from the Black model. 

TA seems to agree that the advanced models (beyond the Black model) are still needed but indeed harder than the earlier lectures before the Black model.

buying (i.e. long) a given interest rate

Tony (FX lecturer) pointed out "buying" any variable means executing at the current "level" and hope the "level" moves up. (Note a mathematician would point out an interest rate is not directly tradeable, but never mind.)

Therefore, buying an interest rate means borrowing at a (rock bottom) rate.

Wrong intuition --- "locking in the interest income stream".

Eg: Say gov bond interest is super low, we would borrow now, and hope for a rise.

Eg: Say swap rate is super low, we would lock it in -- pay fixed and lock in the floating income stream, and hope for the swap rate and floating stream both to rise.

ED future = a fwd interest rate

What about TED?

par swap rate drop means ...@@

For a given tenor (say 1Y)

 

I think treasury yield rise (or drop) has a simpler interpretation....

 

I think Libor ED deposit rate drop (or rise) has another simple interpretation .... and has a credit element.

 

Libor par swap rate drop has a non-trivial interpretation....

 

OIS swap rate is even more complicated...

yield curves - y-axis defaults to fwd rates

In theory inst fwd rates, actually discrete fwd rates...

 

fwd rate is a better choice than zero bond rates or par swap rates...

 

See Jeff's lecture notes

Libor + 50 bps -- never in IRS

Whenever we see something like "3M Libor + 50 bps", I feel it shouldn't be part of an IRS contract. IRS should use Libor + 0 on the floating side.

 

Such an interest rate could be the floating loan interest rate offered to a corporation by a bank.

bid/off quote in terms of interest rates

if a mkt maker GS advertises to Bbbborrow at 7.7% (annualized), GS is Bbbbidding at 7.7%

Interest rate is like a price to pay. AAAAsk is ABove BBBBid. Market maker would Ask higher interest, like 7.78%.

100-leveraged long Treasury position

The LTCM case and Mark's slide P7.43 illustrate very high leverage used in treasury trading. Say i buy $100m of a T bond AA. Immediately i repo it and get $99m, so I only use up $1m of my capital. 

However, if I were to keep the repo for a few months, i would pay quite a bit of repo fee.

4th data source to a yield curve - year-end "turn"

See http://www.jonathankinlay.com/Articles/Yield%20Curve%20Construction%20Models.pdf
for more details.

The year-end turn of the yield curve is defined as the sudden jump in yields during the change of the year. This usually happens at the end of the calendar year, reflecting increased market activity related to
year-end portfolio adjustments and hedging activity....When there is a year turn(s), two discount curves are
constructed: one for turn discount factors and one for the discount factors calculated from the input instruments after adjustments and the discount factor at any time is the multiplication of two.