作为Pricing的基础
forward rate agreement

Type of rate

Treasury rates

investors earns on T-bills and T-bonds
as risk free rate

LIBOR

London interbank offered rate
unsecured short-term borrowing rate between banks(AA banks)，无抵押的
published each business day, widely used

一般认为AA等级银行的违约率很低

随机选取AA等级银行，去掉最高价最低价，以防止被操纵

Overnight rates

隔夜利率，在美国市场一般指Federal funds rate

Repo rate

secured borrowing rate，有抵押的
agree to sell securities to other company now and buy them back later at a slight high price，即以短期证券为抵押进行借贷。
隔夜借贷风险较低

OIS(overnight indexed swap)

金融危机之前，一般用LIBOR作为RFR，之后使用OIS
this means fixed rate for a period is exchange for the geometric average of the overnight rates during the period
interest rate swap, fixed-floating
continually refreshed one-day
报价时使用1 month LIBOR，期间不会变；而OIS每日重新商定，违约风险更低

Before credit crisis, LIBOR is RFR, after that, OIS is RFR.

For treasury rate is influenced by tax and regulatory factors, usually too low to be RFR

Measuring Interest Rate

无特殊说明，一般用连续复利

Periodic compounding

A for FV, P for PV

If R is compounded once per annum, the terminal value is:
$$
A=P(1+R)^n
{\tag1}
$$
If R is compounded m times per annum, the FV is:
$$
A=P(1+\frac{R}{m})^{mn}
{\tag2}
$$

Continuous compounding

If the limit as m tends to infinity is known as continuous compounding:
$$
A=P(1+\frac{R}{m})^{mn}=Pe^{Rn}
{\tag3}
$$
The terminal value is:
$$
A=Pe^{Rn}
{\tag4}
$$
The present value is:
$$
P=Ae^{-Rn}
{\tag5}
$$

Continuous VS Periodic

连续复利的转换，掌握
$$
R_c=mln(1+\frac{R_m}{m})
{\tag6}
$$

$$
R_m=m(e^{R_C/m}-1)
{\tag7}
$$

Spot interest rate

zero-coupon rate
observe the price of the bonds
use the prices of bonds to calculate

Bond Pricing

Coupon bearing bond: T-bond, T-note and corporate bond
Bond yield: YTM
Par yield(not important)

Bootstrap Method

Example: PPT slide 11

Forward Interest Rate

future zero-coupon rate implied by today’s term structure interest rate
$$
R_F=\frac{R_2T_2-R_1T_1}{T_2-T_1}
{\tag8}
$$

Forward rate agreements

An OTC agreement that a certain interest rate will apply to either borrowing or lending money during a specified future period of time
Long: Suppose to borrow money
Short: Suppose to lend money
Underlying asset: LIBOR

FRA

$R_k$: the agreed rate in FRA (at 0)
$R_M$: the actual LIBOR observed at $T_1$, for the period $T_1-T_2$
$R_k$ and $R_M$ are all measured with compounding frequency reflecting the length of the period they apply to $T_1-T_2$
$T_1$结算，计算得失盈亏

pay of f for long position(pay rate)

at $T_2$: $L(R_M-R_k)(T_2-T_1)$

at $T_1$: PV of the amount above, which is $\frac{L(R_M-R_k)(T_2-T_1)}{1+R_M(T_2-T_1)}$

pay off for short position(receive)

at $T_2$: $L(R_k-R_M)(T_2-T_1)$

at $T_1$: PV of the amount above, which is $\frac{L(R_k-R_M)(T_2-T_1)}{1+R_M(T_2-T_1)}$

Long party will benefit from rate increasing - suppose to borrow
Short party will benefit from rate decreasing - suppose to lend

衍生工具 Chap 4