利率的类别--潘登同学的期权、期货及其他衍生品学习笔记
利率种类
- 国债利率（Treasury Rate）
- 政府以本国货币发行的国库券或国债利率（The interest rate on treasury bills or bonds issued by a government in its own currency）
- 投资者将资金投资于国库券或国债时所获得的收益率（The rate of return investors earn when they invest their money in Treasury bills or Treasury bonds）
- 经常被作为无风险利率（Often referred to as the risk free rate）
- 隔夜利率（Overnight Rate）
- 应用于银行之间的无担保借贷（Applied to unsecured lending between banks）
- 目的：银行需按照中央银行要求调整准备金（Banks need to adjust reserves in accordance with central bank requirements）一天结束时，有些金融机构在准备金账户中有资金盈余，有些有缺口（At the end of the day, some financial institutions have a surplus of funds in their reserve accounts and some have a deficit）
- 在美国被称为联邦基金利率（Known as the Fed Funds Rate in the US）
- 生效的联邦基金利率是中介（经纪商）交易利率的加权平均值（The effective federal funds rate is a weighted average of intermediary (broker) transaction rates）
- 央行可以会通过干预自己的交易以提高或降低隔夜利率（Central banks may intervene in their own transactions to raise or lower overnight rates）
- 回购利率（Repo Rate）
- 有抵押借贷利率（Secured Lending Rate）故比隔夜利率低（lower than the overnight rate）
- 回购协议（Repurchase Agreement）
- 拥有证券的金融机构同意以 X 的价格出售这些证券并在未来(通常是第二天)以稍高的价格 Y 购买它们的协议（An agreement in which financial institutions that own securities agree to sell those securities at price X and to buy them at a slightly higher price Y in the future (usually the next day)）
- 回购利率是根据 X 和 Y 之间的差值计算的（The repo rate is calculated from the difference between X and Y）
无风险利率（Risk Free Rate）
指的是1 个月期国库券利率（1-month T-Bill rate）。通常国债利率 (Treasury Rate) 被认为是人为刻意调低的（Usually the treasury rate is considered to be artificially lowered）
- 银行不需要为国债保留资本（Banks are not required to reserve capital for treasury bonds）
- 国债在美国享有优惠的税收待遇（Treasury bonds enjoy favorable tax treatment in the US）
- 无风险利率代理（Proxy of Risk free rate）：LIBOR（SOFR（美国担保隔夜融资利率）、SONIA(英镑隔夜指数平均),ESTER(欧元短期利率),SARON(瑞士平均隔夜利率)，TONUS(东京平均隔夜利率)
连续复利与离散复利转换
| 符号 | 定义 | 英文 |
|---|---|---|
| $R_c$ | 连续复利利率 | the continuously compounded interest rate |
| $R_m$ | （同样的利率）离散复利 | the same interest rate but compounded m times per year |
| $m$ | 离散复利复利次数 | The number of compound interest of discrete compound interest |
| $A$ | 名义本金 | Nominal principal |
| $t$ | 期限（年） | term |
$$ Ae^{R_ct} = A(1+\frac{R_m}{m})^{mt} \ \Rightarrow R_c = mln(1+\frac{R_m}{m}) \quad or \quad R_m = m(e^{R_ct}-1) $$
- 10% 半年复利（一年计2次复利）相当于多少连续复利？(How much continuous compounding is equivalent to 10% semi-annual compounding (Compound interest twice a year)?) $$ e^{R_c} = (1+\frac{10\%}{2})^{2} \ R_c = 9.758\% $$
- 连续复利的 8% 相当于多少季度复利（一年计4次复利）？(How much quarterly compounding is equivalent to 8% of continuous compounding (Compound interest four times a year)?) $$ e^{8\%} = (1+\frac{R_m}{4})^{4} \ R_m = 8.08\% $$
在期权定价中，利率几乎总是用连续复利表示(In option pricing, interest rates are almost always expressed in terms of continuously compounded)
零息利率、债券收益率与票面收益率
- 零息利率(Zero-coupon Interest Rate)
- T 时长零息利率 (The T zero-coupon interest rate/zero rate)
- 从现在开始并持续时间T 的投资所赚取的利率(The interest rate earned on an investment starting now and lasting T)
- 所有利息和本金都在 T 实现(All interest and principal are realized at T)
- 没有中间付款(No intermediate payment)
- 用恰当的零息利率对每笔现金流进行贴现，从而计算债券的现金价格（Calculate the bond's cash price by discounting each cash flow with the appropriate zero-coupon rate）
- 收益率曲线是零息利率构成的
- T 时长零息利率 (The T zero-coupon interest rate/zero rate)
- 债券收益率（Bond Yield）
- 使债券现金流量的现值 (Present Value) 等于债券市场价格的贴现率(Discount Rate)（Make the present value of the bond's cash flow equal to the discount rate of the bond's market price）
- 票面收益率（Par Yield）
- 针对某一特定期限的票面收益率，指的是使债券价格等于其面值的票面利率（The Par Yield for a specific period refers to the Coupon Rate that makes the bond price equal to its face value）
使用Bootstrap方法计算零息利率曲线
由交易者在出售与本金分开的国债息票时共同确定 （Determined jointly by traders when selling Treasury coupons separate from principal）
Bootstrap Method假设
- 假设1：零息利率曲线 (Zero Curve) 在使用重复抽样法确定的点之间是线性的（Assumption 1: The Zero Curve is linear between points determined using repeated sampling）
- 假设2：零息利率曲线在第一个点之前是水平的，在最后一个点之后是水平的（Assumption 2: The zero curve is horizontal before the first point and after the last point）
| Principal | Time to Maturity | Annual Coupon(paid semi-annually) | Price | calculate |
|---|---|---|---|---|
| 100 | 0.25 | 0 | 97.5 | $97.5=\frac{100}{1+\frac{R}{4}} \quad\Rightarrow R=10.26\%$ |
| 100 | 0.50 | 0 | 94.9 | $94.9=\frac{100}{1+\frac{R}{2}} \quad\Rightarrow R=10.75\%$ |
| 100 | 1.00 | 0 | 90.0 | $90.0=\frac{100}{(1+\frac{R}{2})^2} \quad\Rightarrow R=10.82\%$ |
| 100 | 1.50 | 8 | 96.0 | $\frac{4}{1+\frac{10.75}{2}} + \frac{4}{(1+\frac{10.82x`}{2})^2}+\frac{104}{(1+\frac{R}{2})^3}=96.0\quad\Rightarrow R=10.97\%$ |
| 100 | 2 | 12 | 101.6 | $\frac{6}{1+\frac{10.75}{2}} + \frac{6}{(1+\frac{10.82x`}{2})^2}+\frac{6}{(1+\frac{10.97\%}{2})^3}+\frac{106}{(1+\frac{R}{2})^4}=101.6\quad\Rightarrow R=11.1\%$ |
根据期限结构计算远期利率
- 假设在时间$T_1$和$T_2$的零息利率分别是$R_1$和$R_2$ (两个利率连续复利)(Suppose the zero interest rates at time $T_1$ and $T_2$ are $R_1$ and $R_2$ respectively (both are continuous compounding))
- 那么$T_1$和$T_2$之间的远期利率为(Then the forward rate between $T_1$ and $T_2$ is) $$ R_F = \frac{R_2T_2-R_1T_1}{T_2-T_1} = R_2 + (R_2-R_1)\frac{T_1}{T_2-T_1} $$
| 年份 | n年投资的年化零息利率 | 第n年的远期利率 |
|---|---|---|
| 1 | 3.0 | - |
| 2 | 4.0 | $\frac{4.02-3.01}{2-1}=5.0$ |
| 3 | 4.6 | $\frac{4.63-4.02}{3-2}=5.8$ |
| 4 | 5.0 | $\frac{5.04-4.63}{4-3}=6.2$ |
| 5 | 5.3 | $\frac{5.35-5.04}{5-4}=6.5$ |
- 瞬时远期汇率(Instantaneous Forward Rate)
- 从T 开始的很短时间内的远期利率(The forward rate for a very short time period starting at T)
- R 是期限为T 的零息利率(R is the zero coupon rate with a term T) $$ R_F = R + T\frac{\partial{R}}{\partial{T}} $$
远期利率合约(Forward Rate Agreement)
- 远期利率合约/Forward Rate Agreement (FRA)
- 一种场外交易 (OTC) 合约(An over-the-counter (OTC) contract)
- 锁定未来借入或者借出资金时的利率(Lock in the interest rate when borrowing or lending funds in the future)
- 适用于特定时期的实际利率将兑换为约定利率 $R_K$，两者均适用于约定本金 (Agreed Principal, 𝐿)(The effective interest rate applicable for a specific period will be converted to the agreed rate $R_K$, both applicable to the Agreed Principal)
远期利率协议的价值是按利率$R_F$支付利息与按利率$R_K$支付利息之间的差额的现值(The value of a forward rate agreement is the present value of the difference between paying interest at the rate $R_F$ and paying interest at the rate $R_K$) $$ V_{FRA} = L(R_K-R_F)(T_2-T_1)*e^{-R_2T_2} $$ $R_2$为$T_2$期限连续复利的无风险零息利率。
- 一份远期利率协议 (FRA) 规定公司将在1.5至2年期间，以1亿美元的本金支付SOFR，其半年期约定收入利率5.8%（A Forward Rate Agreement (FRA) provides for ，The company will pay SOFR with a principal of US\$100 million in a period of 1.5 to 2 years, and its semi-annual agreed rate is 5.8%）
- 1.5至2年期间的远期 SOFR 半年期利率为 5%(Forward SOFR semi-annual rate of 5% for a period of 1.5 to 2 years)
- 连续复利的 2 年无风险利率 (SOFR) 为每年 4%( Continuously compounded 2-year risk-free rate (SOFR) of 4% p.a.)
- 这份远期利率协议的价值是多少？(以百万美元计)(What is the value of this forward rate agreement? (in millions of dollars))
$$ V_{FRA} = L(R_K-R_F)(T_2-T_1)e^{-R_2T_2}=100(5.8\%-5\%)0.5e^{-4\%*2}= $$
美元久期、修正久期与麦考利久期
$$ \frac{dP}{dy} = -\frac{1}{1+y}\sum_{t=1}^T\frac{t\times CF_t}{(1+y)^t} = - DD $$ DD则表示美元久期（dollar duration），美元久期表示收益率增加一个单位,债券价格减少多少美元。
将美元久期除以P，可以得到修正久期$D^$(modified duration) $$ D^ = \frac{DD}{P} = -\frac{dP/P}{dy} = \frac{1}{1+y}\sum_{t=1}^T\frac{\frac{t\times CF_t}{(1+y)^t}}{P} $$ 修正久期则表示收益率每增加一个单位，债券价格减少百分之几。
将修正久期乘上$(1+y)$,可以得到麦考利久期 $$ D = \sum_{t=1}^T\frac{\frac{t\times CF_t}{(1+y)^t}}{P} $$
- 麦考利久期是一种加权平均值，它将债券的各期现金流量的久期加权平均，用于衡量债券的平均到期时间。麦考利久期越长，债券的平均到期时间就越长。
- 修正久期都是最准确的衡量债券价格对利率变化敏感度的指标，而麦考利久期则更多地关注债券的到期时间。
- 三者关系如下 $$ DD = D^8 * P \ D^* = \frac{D}{1+y} $$
连续复利下的久期
连续复利下麦考利久期等于修正久期 $$ D =D^= \sum_{t=1}^T\frac{{t\times CF_t\times e^{-rt}}}{P} \ DD = D^P\ \triangle D = -D^*P\triangle y $$
凸度(Convexity)
债券的凸度 (Convexity), C 定义为(Bond Convexity, C is defined as) $$ 离散复利：C = \frac{1}{P}\frac{\partial^2{P}}{\partial{y^2}}={\sum_{t=1}^T\frac{\frac{t(t+1)\times CF_t}{(1+y)^{t+2}}}{P}}\ 连续复利：C=\frac{1}{P}\frac{\partial^2{P}}{\partial{y^2}}={\sum_{t=1}^T\frac{t^2CF_t*e^{-yt}}{P}}\ $$
使用久期和凸度计算债券价格变化 $$ \frac{\triangle P}{P} = -D^*\triangle y + \frac{1}{2}C(\triangle y)^2 $$