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# Lecture 6 2012 ir models and embedded options

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### Lecture 6 2012 ir models and embedded options

1. 1. 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
2. 2. Stochastic Process <ul><li>Describing the behavior of interest rate movement </li></ul><ul><li>Must incorporate statistical properties </li></ul><ul><ul><li>Drift （漂流） </li></ul></ul><ul><ul><li>Volatility （浮动） </li></ul></ul><ul><ul><li>Mean reversion （回归平均值） </li></ul></ul><ul><li>The equation that describes the behavior is called stochastic differential equations (SDEs). </li></ul><ul><li>Can the SDEs describe interest rate of all maturities? </li></ul><ul><li>One-Factor Model: short-term rate drives all maturities </li></ul>
3. 3. Continuous-Time Stochastic Process <ul><li>dr = bdt + σdz </li></ul><ul><li>dr = change in short term rate </li></ul><ul><ul><li>The change in the short-term rate in two intervals are independent </li></ul></ul><ul><li>b = drift, mean, or expected rate change </li></ul><ul><li>σ = standard deviation (or volatility) </li></ul><ul><li>dz = a random process </li></ul><ul><ul><li>Standard normal </li></ul></ul><ul><ul><li>When multiplied by σ, it says the effect on short-term rate is scaled by σ </li></ul></ul>
4. 4. Modification of the specification <ul><li>Ito Process </li></ul><ul><li>dr = b(r, t) dt + σ(r,t)dz </li></ul><ul><ul><li>Rate and time dependent. </li></ul></ul><ul><li>Mean Reversion </li></ul><ul><ul><li>b(r, t) = -α(r – long-term stable mean value) </li></ul></ul><ul><li>Specifying the volatility </li></ul><ul><ul><li>σ(r, t)dz = σr ϒ dz </li></ul></ul><ul><ul><li>Vasicek Model: ϒ = 0  σ(r, t) = σ </li></ul></ul><ul><ul><li>Dothan Model: ϒ = 1,  σ(r, t) = σr </li></ul></ul><ul><ul><li>Cox-Ingersoll-Ross Model: ϒ = 0.5,  σ(r, t) = σ sqrt(r) </li></ul></ul>
5. 5. Arbitrage-Free vs Equilibrium Models <ul><li>Arbitrage-Free </li></ul><ul><ul><li>Model must match the price of the reference instrument </li></ul></ul><ul><ul><li>Ho-Lee </li></ul></ul><ul><ul><li>Hull-White </li></ul></ul><ul><ul><li>Kalotay-Williams-Fabozzi (we use this one in this course) </li></ul></ul><ul><ul><li>Black-Karasinki </li></ul></ul><ul><ul><li>Black-Derman-Toy </li></ul></ul><ul><ul><li>Heath-Jarrow-Morton </li></ul></ul><ul><li>Equilibrium model </li></ul><ul><ul><li>Economic factors are assumed to affect interest rate process </li></ul></ul><ul><ul><li>Vasicek model </li></ul></ul><ul><ul><li>CIR </li></ul></ul>
6. 6. <ul><li>我们的讨论假设以下利率期限结构： </li></ul>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
7. 7. <ul><li>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 </li></ul><ul><li>Suppose the interest rate will involve as described in the table below with a probability of 0.5 reaching each node. </li></ul>利率二叉树模式 Binomial Model of Interest Rate t = 0 t=1 (6 months) 4.5% 3.99% 4.00%
8. 8. <ul><li>给予利率的演化途径： </li></ul><ul><ul><li>1 年期利息债券的价格如何演化？ </li></ul></ul><ul><ul><li>今天的价格应该是多少？ </li></ul></ul>利率二叉树模式 Binomial Model of Interest Rate 注意： 角注 U = 上 U p, D = 下 D own UU = two ups; UD = Up and then down or down and then up; DD = two downs. t = 0 t=1 (6 months) t=2 (12 months) P UU2 =100 P U1 = P 0 = P UD2 =100 P D1 = P DD2 =100
9. 9. 利率二叉树模式 Binomial Model of Interest Rate 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?
10. 10. <ul><li>我们假设的错误的概率 </li></ul><ul><li>称此概率为捏造概率 ( pseudo-probabilities) 或中性风险概率 ( risk-neutral probabilities) </li></ul><ul><li>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 : </li></ul>利率二叉树模式 Binomial Model of Interest Rate
11. 11. 利率二叉树模式 Binomial Model of Interest Rate <ul><li>不可套利的价格 Arbitrage Free Pricing </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul>
12. 12. 为附有嵌入式期权债券定价 Pricing Bonds with Embedded Options <ul><li>什么是可赎回债券？ </li></ul><ul><li>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. </li></ul><ul><li>什么是可回售债券 ? </li></ul><ul><li>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. </li></ul><ul><li>可赎回债券的价格 ＝ 纯债券价格 ( Straight bond) – 赎回权（认购权）（ call option ） </li></ul><ul><li>可回售债券的价格 ＝ 纯债券价格 ( Straight bond) ＋ 回售权（认沽权）（ put option ） </li></ul>
13. 13. 赎回期 <ul><li>一般可赎回债券只能在发行后的几年后才能开始赎回 </li></ul><ul><li>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 </li></ul>
14. 14. 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
15. 15. 可赎回债券定价 Valuation Model for Callable Bond <ul><li>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 </li></ul><ul><li>我们将用利率二叉树模式为可赎回债券定价 </li></ul><ul><li>我们先考虑如何建立利率二叉树。假设 : </li></ul><ul><ul><li>票息 = 4% ，每年付息一次 </li></ul></ul><ul><ul><li>不可赎回 Noncallable </li></ul></ul><ul><ul><li>今天的现货利率 = 3.5% </li></ul></ul><ul><ul><li>平价债券 P ar bond </li></ul></ul>
16. 16. 可赎回债券定价 Valuation Model for Callable Bond
17. 17. 可赎回债券定价 Valuation Model for Callable Bond <ul><li>P ：代表在每一交点 (node) 的未来现金流的现在值 </li></ul><ul><li>利率的波动率 (sigma or volatility, vol) ＝ 10 ％ p.a. </li></ul><ul><li>根据上图的利率树，我们应用倒行反复过程 (backward iteration process): </li></ul>
18. 18. <ul><li>步骤 1: </li></ul><ul><ul><li>假设 r D1 ＝ 4.5% </li></ul></ul><ul><li>步骤 2: </li></ul><ul><ul><li>r U1 = r D1 exp(2xsigma) = 5.496% </li></ul></ul><ul><li>步骤 3: </li></ul><ul><ul><li>计算 P U1 = 1/2 {104/(1+0.05496) + 104/(1+0.05496)} = 98.582 </li></ul></ul><ul><ul><li>计算 P D1 = 1/2 {104/(1+0.045) + 104/(1+0.045)} = 99.522 </li></ul></ul><ul><li>步骤 4: </li></ul><ul><ul><li>计算 P: = 1/2 {(98.582+4)/(1+0.035) + (99.522+4)/(1+0.035)} = 99.567 </li></ul></ul><ul><li>步骤 5: </li></ul><ul><ul><li>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 r D1 . </li></ul></ul><ul><li>假设经过不断的反复尝试我们发现当 r D1 = 4.074 % ， P ＝ 100 </li></ul>可赎回债券定价 Valuation Model for Callable Bond
19. 19. 可赎回债券定价 Valuation Model for Callable Bond
20. 20. 可赎回债券定价 Valuation Model for Callable Bond <ul><li>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 ： </li></ul>
21. 21. <ul><li>假设一 3 年的可赎回债可在任何一交点以 100 赎回 </li></ul><ul><li>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 </li></ul><ul><li>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 。 </li></ul><ul><li>Notice the values in bold and italic. </li></ul>可赎回债券定价 Valuation Model for Callable Bond
22. 22. 可赎回债券定价 Valuation Model for Callable Bond
23. 23. <ul><li>最终我们的模式给的价格是 101.432. </li></ul><ul><li>那么期权值多少呢？ </li></ul><ul><ul><li>Value of a call = value of non-callable - value of callable </li></ul></ul><ul><ul><li>V alue of the call = 102.075 - 101.432 = 0.643 </li></ul></ul>可赎回债券定价 Valuation Model for Callable Bond
24. 24. 传统收益率价差 Traditional Yield Spread <ul><li>收益率价差 Yield Spread = 非国债 YTM 或 YTC （ Non-Treasury YTM or YTC ） - 国债 YTM （ Treasury YTM ） </li></ul><ul><li>举例： </li></ul><ul><ul><li>Price YTM </li></ul></ul><ul><ul><li>国债 96.6133 9.15% </li></ul></ul><ul><ul><li>企业债 87.0798 10.24% </li></ul></ul><ul><ul><li>收益率价差 109 basis point </li></ul></ul><ul><li>缺点： </li></ul><ul><ul><li>没考虑利率期限结构 </li></ul></ul><ul><ul><li>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 </li></ul></ul>
25. 25. 有效价差 ( 或叫静态价差） Effective Spread (or Static Spread) <ul><li>假设一 5 年 8.8% 债券，价格＝ 99 </li></ul><ul><li>应用国债现货利率 </li></ul><ul><ul><li>下页展示理论价格太高了 </li></ul></ul><ul><ul><li>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. </li></ul></ul>