flattening： longer-term rates fall by more than shorter-term rates long长期
steepening: longer-term rates increase by more than shorter- term rates long短期
bond yields and return caculation
YTM
properties of YTM
YTM与价格呈反向关系、at premium coupon rate > YTM
term structure of spot rate is upward-sloping, forward>spot>YTM
downward-sloping:YTM>spot>forward
relationship between YTM and spot rate
coupon effect
upward: YTM与coupon反向
downward： YTM与coupon正向
all else equal lower coupon rate→ higher coupon rate
coupon rate 与 duration 反向
decomposition(分解） of bond profit/ loss
carrty-roll-down
rate change
spread change
applying duration convexity and DV01
duration
macaulay duration
zero coupon>discount>premium
modified duration
=macaulay duration/(1+periodic market yield)
dollar duration
= modified duration × full price
effective duration
\
portfolio duration
hedge
DV01 and DV01 hedge
DV01=DD×1bp=MD×P×1bp
Hedge Ratio=初始头寸的DV01（per100）÷对冲头寸的DV01（per100）
duration based hedge
分母对应合约标的资产
convexity
与YTM反向
与coupon反向
与T正向（T²成比例）
callable bond and puttable bond
callable=straght bond- call option 有利于issuer
利率与其duration正向，一开始负凸性
puttable=straight bond+put option 有利于买方
利率预期duration反向，一直都是正凸性
barbell and bullet strategy
barbell 凸性较大 短+长
bullet 凸性较小 中
期权定价
modeling non-parrel term structure shifts and hedging
key rate shifts
the rate of a given maturity is affected solely by its closet key rate
forwardbucket shifts
一个bucket上移1个bp
二叉树
风险中性概率
美式期权注意提前行权
BSM模型
the assumptions of BSM model
注意均值、方差服从对数正态分布
the formula without dividends
N(d1）=delta of call 1-N（d1）=-N（d1）
N(d2）=the probability that a call option will be exercised in a risk-neutral world
the formula with dividends
对S0按照相应的利率折现即可
美式期权
call一样美式put大于等于欧式put 先算欧式 再与选项对比即可
the greeks
Δ
the rate of change of the option price with respect to the price of the underlying asset
long call short put 的Δ都>0
at the money 处 Δ=0.5
Γ
the rate of change of the portfolio's delta with respect to the price of the underlying asset
long 都＞0
at the money 处最大 与T反向
gamma of stocks/forward = 0(线性产品）
Vega
the rate of change of the value of the option with respect to the volatility of the underlying asset
long 都＞0
与T正向 与根号T成比例
θ
到期时间
long都＜0
与T反向
ρ
利率
市场风险
measures of financial risk
coherent risk measures(ES)
montonicity(单调性） value高，risk低
subadditivity
homogeneity β>0, ρ(βX）=βρ(X)
translational invariance: risk is associated with the assets in portfolio
VAR
assumption
reteruns conform to a normal distributuon iid
每日波动率保持恒定（正态分布）
increase at increasing rate when degree of confidence increases
increases at decreasing rate when the holding period is longer
计算法则
平方根法则
different confidence level
expected shortfall(ES)
满足 subaddlitivity while VaR does not
tell us what to expect in bad states-it gives an idea of how bad might be-while VaR tells us nothing other than to expect a loss higher than the VaR itself
calculating and applying VaR
历史模拟法
最新一天的数据替换掉最旧一天的数据
VaR即是总天数×（1-VaR百分比）
ES即 除去VaR那一天后 最坏的数据的平均值
full revaluation method
burdensome but great advantage of accuracy
delta-normal model
只能用于线性产品 不能用于（MBS, Fixed-income securities with embedded option
蒙特卡罗模拟
providing correlations between risk factors can be defined in some way
measuring and monitoring volatiliity
参数法: imposes a specific distributional assumption on conditional asset reteruns
Historical stancdard deviation approach
use of historical time series data in order to determine the shape of the conditional distribution
EWMA
只需要前一期的收益率
、
GARCH(1,1)
是γ=0的special case of EWMA
VL是 long run average variance rate(均值回归） γ=(1-α-β)
通常用w=γVL VL=W/(1-α-β） α+β，回归速度越慢，越持久
unstable if α+β>1
非参数法:uses historical data directly, without imposing aspecific set of distributional assumptions
Historical simulation
MDE
implied-volatility based approach
优点
forward-looking（及时反映市场预测）
reacts immediately to market conditions
缺点
model independent
BSM model assume volatility is constant over the life of the option
options on the same underlying asset may trade different implied volatilities
emirical result indicate implied volatility is onaverage greater than realized volatility(隐含波动率通常更大）
信用风险
external and internal credit ratings
conditional and unconditional defuat probabilities
hazard rates
h把 is the average hazard rate between time zero and time t
the unconditional default probability between time 0 and time t is:
rating process
through-the cycle
时间长、稳定、经济好时 低估 经济差时 高高估
point-in-time
时间短
country risk
sources of country risk
GDP、political risk、legal、economic structure
impact of sovereign default
factors affecting sovergin ratings
sovereign credit sperad
操作风险
measuring credit risk
modelling credit risk capital
expected loss(ES)
the bank's capital is a buffer(缓冲器）against unexpected loss
the mean and standard deviation of credit losses
α与ρ正向关系，与根号ρ正相关
the Gaussian copula model
内部评级模型 使用映射
one-factor correlation model
相关系数少、半正定
risk allocation
对每笔信贷资产的风险进行加总
operational risk
approaches to calculate operational risk capital
basic indicator approach
三年年收入×15%
the standardized approach
分为8个业务条线，不同的业务条线给予不同的" beta factor"(18%,15%,12%)
advacned measurement approach(AMA)
8个业务条线给予7种不同类型的风险（risk sensivity 最好）
2 banks presented with the same data were liable to come up with quite different capital requirements under AMA
standardized measurement approach (SMA)
最新的方法 风险敏感且稳健(robustness)
measures to manage operational risk
loss frequency 服从泊松分布 loss severity 服从对数正态分布
estimation procedures
内外部数据结合 外部数据有可能偏大 因为只有较大的数据才会被披露
potential biases
reducing operational risk
risk control and aself assessment
key risk indicator
insurance
education and culture
stress testing
stress test looks at a much longer period wheras stress testung usually lokks at shorter period
one disadvantage of VAR and ES is that they are backward-looking
stress testing is designed to be forward-looking does not provide a probability distribution for losses
stress testing looks at relatively small number of scenarios(all bad) whereas VAR/ES looks at a wide range of scenarios(some good some bad)
