Lecture 6 2012 ir models and embedded options

固定收益课题 6 ：

利率模型 / Interest rate modeling附有嵌入式期权债券 / Bonds with Embedded Options

陈国辉博士／南洋理工大学商学院

Stochastic Process• Describing the behavior of interest rate movement• Must incorporate statistical properties

• Drift （漂流）• Volatility （浮动）• Mean reversion （回归平均值）

• The equation that describes the behavior is called stochastic differential equations (SDEs).

• Can the SDEs describe interest rate of all maturities?• One-Factor Model: short-term rate drives all maturities

Continuous-Time Stochastic Process

dr = bdt + σdz

•dr = change in short term rate• The change in the short-term rate in two intervals are

independent

•b = drift, mean, or expected rate change•σ = standard deviation (or volatility)•dz = a random process

• Standard normal• When multiplied by σ, it says the effect on short-term rate is

scaled by σ

Modification of the specification• Ito Process

dr = b(r, t) dt + σ(r,t)dz• Rate and time dependent.

• Mean Reversion

b(r, t) = -α(r – long-term stable mean value)• Specifying the volatility

σ(r, t)dz = σrϒdz• Vasicek Model: ϒ = 0 σ(r, t) = σ• Dothan Model: ϒ = 1, σ(r, t) = σr• Cox-Ingersoll-Ross Model: ϒ = 0.5, σ(r, t) = σ sqrt(r)

Arbitrage-Free vs Equilibrium Models• Arbitrage-Free

• Model must match the price of the reference instrument• Ho-Lee• Hull-White• Kalotay-Williams-Fabozzi (we use this one in this course)• Black-Karasinki• Black-Derman-Toy• Heath-Jarrow-Morton

• Equilibrium model• Economic factors are assumed to affect interest rate process• Vasicek model• CIR

我们的讨论假设以下利率期限结构：

利率二叉树模式Binomial Model of Interest Rate

期限Maturity

现货利率Zero Rate

利息债券的价格Price of Zero

6 months 3.99% 98.04402

12 months 4.16% 95.96628

18 months 4.33% 93.77641

•假设利率将如以下的二叉树演化，向上和向下的概率皆为 0.5•Suppose the interest rate will involve as described in the table below with a probability of 0.5 reaching each node.

t = 0 t=1 (6 months)

4.5%

3.99%

4.00%


给予利率的演化途径：• 1 年期利息债券的价格如何演化？• 今天的价格应该是多少？

t = 0 t=1 (6 months)t=2 (12 months)

PUU2 =100

PU1 =

P0 = PUD2 =100

PD1 =

PDD2 =100

注意：

角注 U = 上 Up, D = 下 Down

UU = two ups; UD = Up and then down or down and then up; DD = two downs.


98.03922 0.04/2)(1

100 P

79951.970.045/2)(1

100 P

D1

U1

95.9662896.0994 0.0399/2)(1

97.91937 P0

依我们模式推导出来的价格为什么不同于市场价格？Our model price is different from the market price. What’s wrong?

97.91937 3922)(0.5)(98.0 9951)(0.5)(97.7Value Expected


•我们假设的错误的概率•称此概率为捏造概率 (pseudo-probabilities) 或中性风险概率 (risk-neutral probabilities)•要找到正确的概率，我们必须先假设债券市场价格是正确的。那就是模式的目标价格应该是市场价格 :

0.339 p-1 and 0.661 p

:pfor solving

96628.95

20.0399

1

p)-98.03922(1 97.79951p



•不可套利的价格 Arbitrage Free Pricing•如果我们小心的选择概率，我们就能准确为债券定价。If we carefully chose the probabilities, we can price the zero-coupon bond correctly.•中性风险意味着投资者的风险偏向不存在定价的模式内This probabilities are called risk neutral probabilities because it can be used to price a bond under the risk neutral context, ie., investor’s preference for risk does not come into the equation. •一般我们也可以假定概率＝ 0.5 ，让利率变动We may also fix the probabilities (commonly, it is fixed at 0.5) and vary the interest rate path. We will do that in our application example.•注意：这并不意味着真实的利率和概率Remember: we do not mean that they are the actual interest rates or probabilities.•重要的是模式能不能准确的给资产定价What is most essential is that the path we specified must satisfy the condition that the model price equals the market price.

为附有嵌入式期权债券定价Pricing Bonds with Embedded Options

•什么是可赎回债券？•发行人有权力在特定的时间里以特定的价格赎回债券A callable bond is a bond which gives the rights to the issuer to call back the bond on a certain date or period at a predetermined price.•什么是可回售债券 ?•债券持有人有权力在特定的时间里以特定的价格把债券回售给发行人A puttable bond is a bond which gives the rights to the investors to put back the bond on a certain date or period at a predetermined price to the issuer.

可赎回债券的价格＝纯债券价格 (Straight bond) – 赎回权（认购权）（ call option ）可回售债券的价格＝纯债券价格 (Straight bond) ＋回售权（认沽权）（ put option ）

赎回期• 一般可赎回债券只能在发行后的几年后才能开始赎回• 比如一 20 年的债券如果只能在 10 年后才能赎回，那么一

般以此形式表示： 20NC10

可赎回债券的价值利率关系Price-Yield Relationship of a Callable Bond

Noncallable Bond

Price

Yield

Callable Bond

可赎回债券定价Valuation Model for Callable Bond

• 只有在利率有波动的情况下，可赎回债券的期权才会有价值

• 我们将用利率二叉树模式为可赎回债券定价• 我们先考虑如何建立利率二叉树。假设 :

• 票息 = 4% ，每年付息一次• 不可赎回 Noncallable• 今天的现货利率 = 3.5%• 平价债券 Par bond


t=0 t=1 t=2

C=104

rUU2=rDD2e4(sigma)

=?

PU1=?

C=4

P0=? rU1 =rD1e2(sigma)

=? C=104

C=0

r0 = 3.5% PD1=? rDU2 =rDD2e2(sigma)

=?

C=4rD1 =?

C=104

rDD2 =?


•P ：代表在每一交点 (node) 的未来现金流的现在值•利率的波动率 (sigma or volatility, vol) ＝ 10 ％ p.a.•根据上图的利率树，我们应用倒行反复过程 (backward iteration process):

nodecurrent at r

Couponnode low previousat P

nodecurrent at r

Coupon nodehigh previousat P

2

1 P

• 步骤 1:• 假设 rD1 ＝ 4.5%

• 步骤 2:• rU1 = rD1 exp(2xsigma) = 5.496%

• 步骤 3:• 计算 PU1 = 1/2 {104/(1+0.05496) + 104/(1+0.05496)}

= 98.582• 计算 PD1 = 1/2 {104/(1+0.045) + 104/(1+0.045)}

= 99.522• 步骤 4:

• 计算 P:= 1/2 {(98.582+4)/(1+0.035) + (99.522+4)/(1+0.035)}= 99.567

• 步骤 5:• 比较 P 值和市值 (=100), 很明显我们的假设是错误的。我们必

须重新来过假设另一 rD1.• 假设经过不断的反复尝试我们发现当 rD1 = 4.074% ， P ＝ 100


t=0 t=1 t=2

C=104

rUU2=rDD2e4(sigma)

=?

PU1=99.071

C=4

P0=100 rU1 =rD1e2(sigma)

=4.976% C=104

C=0

r0 = 3.5% PD1=99.929 rDU2 =rDD2e2(sigma)

=?

C=4rD1 =4.074%

C=104

rDD2 =?


t=0 t=1 t=2

PUU2=98.588C = 5.25

rUU2=rDD2e4(sigma)

=6.757%

PU1=99.461C=5.25

P0=102.075 rU1 =rD1e2(sigma)

=4.976% PUD2=99.732

C=0 C = 5.25

r0 = 3.5% PD1=101.333 rDU2 =rDD2e2(sigma)

=5.532%

C=5.25rD1 =4.074%

PDD2=100.689C = 5.25rDD2 =4.530%

•我们继续开展建立二叉利率树。假设我们用一 3 年 5.25% 的债券，价格＝ 102.075 ：


• 假设一 3 年的可赎回债可在任何一交点以 100 赎回• 定价的步骤向为美式期权定价，当其基础资产带红利时我

们必须检查是否应该行使权力• 对可赎回债定价时，如果在一交点的值 P 大过赎回价时，

赎回价取代 P 。• Notice the values in bold and italic.


t=0 t=1 t=2

PUU2=98.588

C = 5.25

rUU2=rDD2e4(sigma)

=6.757%

PU1=99.461

C=5.25

P 0=101.432 rU1 =rD1e2(sigma)

=4.976% PUD2=99.732

C=0 C = 5.25

r0 = 3.5% P D1=100 rDU2 =rDD2e2(sigma)

=5.532%

C=5.25rD1 =4.074%

P DD2=100

C = 5.25rDD2 =4.530%


•最终我们的模式给的价格是 101.432.•那么期权值多少呢？

• Value of a call = value of non-callable - value of callable• Value of the call = 102.075 - 101.432 = 0.643


传统收益率价差Traditional Yield Spread

• 收益率价差 Yield Spread = 非国债 YTM 或 YTC （ Non-Treasury YTM or YTC ） - 国债 YTM （ Treasury YTM ）

• 举例：Price YTM

国债 96.6133 9.15%

企业债 87.0798 10.24%

收益率价差 109 basis point

• 缺点：• 没考虑利率期限结构• 如果债券存有期权，那么现金流将会有变化

in the case of callable, expected interest rate may alter the cash flows of a bond

有效价差 ( 或叫静态价差） Effective Spread (or Static Spread)

• 假设一 5 年 8.8% 债券，价格＝ 99• 应用国债现货利率

• 下页展示理论价格太高了• 如果我们在没一个利率点上价 1.69935% 的有效价差，我们就

能取得 99.

Period Cash Flow Spot Rate PV of CF

1 4.4 7.00000% 4.25122 4.4 7.04999% 4.10553 4.4 7.09998% 3.96284 4.4 7.12498% 3.82515 4.4 7.13998% 3.69226 4.4 7.16665% 3.56227 4.4 7.19997% 3.43518 4.4 7.26240% 3.30779 4.4 7.33315% 3.1820

10 104.4 7.38977% 72.6319Theoretical Price 105.9556


Period Cash Flow Spot Rate Effective Spr PV of CF

1 4.4 7.00000% 1.69935% 4.21662 4.4 7.04999% 1.69935% 4.03893 4.4 7.09998% 1.69935% 3.86684 4.4 7.12498% 1.69935% 3.70215 4.4 7.13998% 1.69935% 3.54446 4.4 7.16665% 1.69935% 3.39187 4.4 7.19997% 1.69935% 3.24428 4.4 7.26240% 1.69935% 3.09859 4.4 7.33315% 1.69935% 2.9566

10 104.4 7.38977% 1.69935% 66.9401Theoretical Price 99.0000


Time

Rate

Spot Curve

Spot Rate + Effective Spread


• 有效价差比传统价格优越吗？• 是的，因为它并入了利率期限结构• 但是如果期限短的话，两个利差的差别不大• 还有，如果利率曲线很陡，差别加大

• 注意：有效价差是一无波动率利率价差Static Spread is zero-volatility spread.


期权调整价差Option-Adjusted Spread (OAS)

• 回顾先前的例子，假设 3 年 5.25% 可赎回债的市场价格是 100 而不是 101.432 。这意味着债券被低估了 1.432.

• 要将这价格的差别转变为收益率，我们在每一交叉点加一价差，然后重新定价。

• 此价差叫期权调整价差 (option-adjusted spread ， OAS). • OAS 用来调和市场价• 所以如果你买了被低估的债券， OAS 代表多余的回报

(incremental return) ( 在基础利率下但是已调整过期权的存在）（ over the benchmark after adjusted for the call option ）

有效久期和凸性Effective Duration and Convexity

由于有期权的存在，修正久期和凸性不适于用来衡量可赎回债的价格风险（因为当利率变化时，现金流夜开始变化）•We should use effective duration and convexity:

20

0-

y))((P

P2P - P Convexity Effective

y))(2(PP - P

0

- Duration Effective

有效久期和凸性Effective Duration and Convexity• 步骤 1:

• 计算 OAS• 步骤 2:

• 将利率曲线往上移动几个基点Shift the yield curve up by some small basis point.

• 步骤 3:• 根据新曲线建立二叉树• Construct the binomial tree based on the new yield curve.

• 步骤 4:• 在每个交叉点上加上 OAS• Add OAS to each node of interest rate.

• 步骤 5:• 计算出 P+

• Find the bond price, P+.• 步骤 6:

• 重新步骤 2 － 5但在步骤 2 将曲线往下移动几个基点• Repeat step 2 to step 5 except in step 2, shift the yield curve down by same amount of

basis point and find P-.

Lecture 6 2012 ir models and embedded options

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Transcript of Lecture 6 2012 ir models and embedded options