# 期权、期货及其他衍生产品

# 第1章引言

# 1.1 交易所市场

交易所的清算中心负责已达成共识的交易的具体手续，是交易员的中间，对交易风险负责

# 电子交易市场

由公开喊价（open outcry system）向电子交易（electric trading）的转变
电子交易促成高频交易和算法交易的发展

系统风险指当一家金融机构宣布破产时，它所产生的连锁反应会导致其他金融机构破产，从而威胁整个金融系统的稳定性。

# 1.2 场外市场OTC

中央交易对手（CCP）扮演清算中心的角色
开始受到监管约束

# 市场规模

场外市场规模远远大于交易所市场

# 1.3 远期合约（forward contract）

# 1.3.1 远期合约的收益

多头： $S_T-K$
空头： $K-S_T$

# 1.3.2 远期价格和即期价格

套期保值的可能性
基差风险

# 1.4 期货合约

期货合约在交易所进行
期货合约标准化
保证金机制与每日结算过程

# 1.5 期权合约

期权在交易所和OTC均有交易

# 1.6 交易员的种类

对冲者：减小风险，对冲后的实际结果并不一定能保证比不对冲好
投机者：下注。杠杆效应
套利者无风险盈利，“空手套白狼”

# 第2章期货市场的运作机制

# 2.1 平仓

对一个合约平仓就是进入一个与初始交易相反的头寸
期货价格和即期价格联系在一起是因为期货合约有最终实物交割的可能性

# 2.2 期货合约的价格

# 2.2.1 （标的）资产

可以是实物资产可以是金融资产
可以存在替代品，价格有变化

# 2.2.2 合约规模

每一份合约中交割资产的数量

# 2.2.3 价格和头寸的限额

涨停：交易所决定大多数合约价格每天的变动限额，与前一天收盘价相比，某一天期货价格上涨的金额等于每日价格限额

# 2.3 期货价格收敛到即期价格

套利使这一过程发生
随交割日期临近，期货价格收敛至即期价格

# 2.4 保证金账户

参考Zastawniak, T., & Capinski, M. (2003). Mathematics for Finance: An Introduction to Financial Engineering. Springer. 书中关于保证金运作机制的介绍
信用风险：不能追加保证金的交易被平仓

# 2.5 折扣（haircut）

证券常常被用作抵押品，但是在决定将证券作为抵押品的价值时，通常会将其市值降低一定数量，所降低的数量就是haircut

# 2.6 市场报价

# 2.6.1 开盘价、最高价、最低价、结算价

开盘价：当天交易开始后立即成交的期货合约价格
结算价：用于计算每天合约的盈亏以及所需要的保证金数量

# 2.6.2 期货价格的规律

正常市场：交割价格是期限的递增函数
反向市场：合约价格是期限的递减函数

# 2.7 远期合约和期货合约的比较

远期合约	期货合约
交易双方间的私下合约	在交易所内交易
未被标准化	标准化
通常指名一个交割日	有一系列交割日
在合约到期时结算	每日结算
通常会发生实物或现金交割	合约到期前通常会被平仓
有信用风险	几乎没有信用风险

# 第3章利用期货的对冲策略（Hedge-and-Forget Strategy）

# 3.1 Short Hedge and Long Hedge

# 3.1.1 Short Hedge

一般而言，拥有某种资产并期望在将来卖出资产时，选择空头对冲

# 3.1.2 Long Hedge

将来买入资产并想在现在把价格锁定

# 3.2 基差风险（Basis Risk）

引起基差风险的几种情况：

需要对冲价格风险的资产与期货合约的标的资产可能并不完全一样
对冲者可能确定无法买入或卖出资产的准确时间
对冲者可能需要在期货到期月之前将期货平仓

# 3.2.1 基差（Basis）

基差=被对冲资产的即期价格-用于对冲的期货合约价格

$Basis = S_t-F_{t,T}$

# 3.2.2 交叉对冲

给对冲者带来风险的资产不同于对冲的合约标的资产，会带来更大的基差风险
定义 $S_2^*$ 为期货合约标的资产在 $t_2$ 时期的价格； $S_2$ 是被对冲资产在 $t_2$ 的价格。通过对冲，公司确保购买或出售资产的价格为

S_2+F_1-F_2

上式可变形为 $F_1+(S_2^*-F_2)+(S_2-S_2^*)$ , 基差被分成了2个部分

# 3.3 交叉对冲

对冲比率(hedging ratio)是指持有期货合约的头寸数量与资产风险敞口数量的比率

当期货标的资产与被对冲资产一样时，对冲比率应取1.0

交叉对冲时，对冲者采用的对冲比率应当使被对冲后头寸价格变化的方差达到最小

# 3.3.1 计算最小方差对冲比率

最小方差对冲比率取决于即期价格的变化与期货价格变化之间的关系
证明： $h^*$ $h^{\times}$ 是 $\Delta S$ $Δ S$ 对 $\Delta F$ $Δ F$ 进行线性回归所产生的最优拟合直线
- 进入 $N_a$ 资产多头， $N_f$ 期货合约空头
- hedge ratio: $h=\frac{N_f}{N_a}$

\begin{aligned} Y_{t} &=S_{t} N_{a}-\left(f_{t}-f_{0}\right) N_{f} \\ &=S_{0} N_{a}+\left(S_{t}-S_{0}\right) N_{a}-\left(f_{t}-f_{0}\right) N_{f} \\ &=S_{0} N_{a}+\Delta S_{t} N_{a}-\Delta f_{t} N_{f} \\ &=S_{0} N_{a}+N_{a}\left(\Delta S_{t}-h \Delta f_{t}\right) \end{aligned}

\begin{array}{l}\operatorname{Var}\left(Y_{t}\right)=N_{a}^{2} \operatorname{Var}\left(\Delta S_{t}-h \Delta f_{t}\right) \\ \quad=N_{a}^{2}\left(\sigma_{S}^{2}+h^{2} \sigma_{f}^{2}-2 h \rho \sigma_{S} \sigma_{f}\right)\end{array}

h^*=\rho \frac{\sigma_s}{\sigma_f}

Hedging Effectiveness

\left[\operatorname{Var}\left(N_{\mathrm{a}} \mathrm{S}_{\mathrm{t}}\right)-\operatorname{Var}\left(\mathrm{Y}_{\mathrm{t}}\right)\right] / \operatorname{Var}\left(\mathrm{N}_{\mathrm{a}} \mathrm{S}_{\mathrm{t}}\right)=\rho^{2}

Because proportion of the exposure that should optimally be hedged is:

h=\frac{\rho \sigma_{S}}{\sigma_{F}}=\frac{\sigma_{S, F}}{\sigma_{F}^{2}}

Hedge ratio estimated from:

\Delta S_{t}=a+\beta \Delta F(t, T)+\varepsilon

# 3.3.2 最优合约数量

N^*=\frac{h^* Q_A}{Q_F}

# 3.3.3 尾随对冲

针对期货，存在每天结算和一系列持续1天的对冲交易问题
- $\hat{\rho}$ : 期货价格每天百分比变化和即期价格每天百分比变化之间的相关系数
- $\hat{\sigma}_F$ : 期货价格每天百分比变化的标准差
- $\hat{\sigma}_S$ : 即期价格每天百分比变化的标准差
- S: 即期价格
- F: 期货价格
期限为1天的对冲比率：

\hat{\rho} \frac{S\hat{\sigma}_S}{F \sigma_{F}}

下一天对冲需要持有的合约数量为

N^{*}=\hat{\rho} \frac{S \hat{\sigma}_{S} Q_{A}}{F \hat{\sigma}_{F} Q_{F}}=\hat{h} \frac{V_A}{V_F}

其中, $\hat{h}=\hat{\rho}\frac{\hat{\sigma}_S}{\hat{\sigma}_F}$

# 3.4 股指期货

股指：跟踪一个虚拟股票组合的价值变化，每个股票在组合中的权重等于股票组合投资于这一股票的比例
Price weighted vs. Value weighted
- Price weighted
- Value weighted

# 3.4.1 股票组合的对冲

若组合跟踪股票指数：

N^*=\frac{V_A}{V_F}

当股票不跟踪股票指数

N^*=\beta \frac{V_A}{V_F}

# 3.4.2 改变组合的 $\beta$

｜\beta-\beta^*｜\frac{V_A}{V_F}

# 3.6 向前滚动对冲 (stack and roll)

场景：对冲的期限比所有能够利用的期货期限长，对冲者必须对到期的期货平仓，同时再进入具有较晚期限的合约

# 第4章利率

# 4.1 Types of Rates

The interest rate applicable in a situation depends on the credit risk.

# 4.1.1 Treasury Rates

Treasury rates are the rates an investor earns on Treasury bills and Treasury bonds. These are the instruments used by a government to borrow in its own currency.

"Risk-free Rate"

# 4.1.2 LIBOR (银行间拆借利率)

LIBOR is usually considered to be an estimate of the short-term unsecured borrowing rate for a AA-rated ﬁnancial institution.

# 4.1.3 The Fed Funds Rate (隔夜拆借利率)

In the United States, ﬁnancial institutions are required to maintain a certain amount of cash (known as reserves) with the Federal Reserve. The reserve requirement for a bank at any time depends on its outstanding assets and liabilities. At the end of a day, some ﬁnancial institutions typically have surplus funds in their accounts with the Federal Reserve while others have requirements for funds. This leads to borrowing and lending overnight. This overnight rate is monitored by the central bank, which may intervene with its own transactions in an attempt to raise or lower it.

# 4.1.4 Repo Rates (回购利率)

Unlike LIBOR and federal funds rates, repo rates are secured borrowing rates. In a repo (or repurchase agreement), a ﬁnancial institution that owns securities agrees to sell the securities for a certain price and buy them back at a later time for a slightly higher price.

If structured carefully, a repo involves very little credit risk.

The most common type of repo is an overnight repo which may be rolled over day to day. However, longer term arrangements, known as term repos, are sometimes used. Because they are secured rates, a repo rate is generally slightly below the corresponding fed funds rate.

# 4.1.5 The Proxy of "Risk-free Rate"

Derivatives are usually valued by setting up a riskless portfolio and arguing that the return on the portfolio should be the risk-free interest rate. The risk-free interest rate therefore plays a key role in the valuation of derivatives. For most of this book we will refer to the ‘‘risk-free’’ rate without explicitly deﬁning which rate we are referring to. This is because derivatives practitioners use a number of diﬀerent proxies for the riskfree rate. Traditionally LIBOR has been used as the risk-free rate—even though LIBOR is not risk-free because there is some small chance that a AA-rated ﬁnancial institution will default on a short-term loan. However, this is changing. In Chapter 9, we will discuss the issues that practitioners currently consider when they choose the ‘‘risk-free’’ rate and some of the theoretical arguments that can be advanced.

# 4.2 Measuring Interest Rates

Suppose that an amount A is invested for n years at an interest rate of R per annum. If the rate is compounded once per annum, the terminal value of the investment is

A(1+R)^{n}

If the rate is compounded m times per annum, the terminal value of the investment is

A\left(1+\frac{R}{m}\right)^{m n}

When m ¼ 1, the rate is sometimes referred to as the equivalent annual interest rate.

# 4.2.1 Continuous Compounding

The limit as the compounding frequency, m, tends to inﬁnity is known as continuous compounding. With continuous compounding, it can be shown that an amount A invested for n years at rate R grows to

A e^{R n}

Continuous Compounded Rate与Periodical Compounded Rate之间的转换

Suppose that $R_c$ is a rate of interest with continuous compounding and $R_m$ is the equivalent rate with compounding m times per annum.

\begin{array}{c}A e^{R_{c} n}=A\left(1+\frac{R_{m}}{m}\right)^{m n} \\ e^{R_{c}}=\left(1+\frac{R_{m}}{m}\right)^{m} \\ R_{c}=m \ln \left(1+\frac{R_{m}}{m}\right) \\ R_{m}=m\left(e^{R_{c} / m}-1\right)\end{array}

# 4.3 Zero-coupon Rates

The n-year zero-coupon interest rate is sometimes also referred to as the n-year spot rate, the n-year zero rate, or just the n-year zero.

# 4.4 Bond Pricing

# 4.4.1 Present Value Method - With Spot Rate

Principal: $100
Coupon: $3

3 e^{-0.05 \times 0.5}+3 e^{-0.058 \times 1.0}+3 e^{-0.064 \times 1.5}+103 e^{-0.068 \times 2.0}=98.39

# 4.4.2 Bond Yield

3 e^{-y \times 0.5}+3 e^{-y \times 1.0}+3 e^{-y \times 1.5}+103 e^{-y \times 2.0}=98.39

# 4.4.3 Par Yield

The par yield for a certain bond maturity is the coupon rate that causes the bond price to equal its par value. (The par value is the same as the principal value.) Usually the bond is assumed to provide semiannual coupons. Suppose that the coupon on a 2-year bond in our example is c per annum (or 12c per 6 months).

\frac{c}{2} e^{-0.05 \times 0.5}+\frac{c}{2} e^{-0.058 \times 1.0}+\frac{c}{2} e^{-0.064 \times 1.5}+\left(100+\frac{c}{2}\right) e^{-0.068 \times 2.0}=100

# 4.5 Determing Treasury Zero Rates

bootstrap method

# 4.6 Forward Rates

Forward interest rates are the future rates of interest implied by current zero rates for periods of time in the future.

R_{F}=\frac{R_{2} T_{2}-R_{1} T_{1}}{T_{2}-T_{1}}

R_{F}=R_{2}+\left(R_{2}-R_{1}\right) \frac{T_{1}}{T_{2}-T_{1}}

Example:

R_{F}=R+T \frac{\partial R}{\partial T}

At this time, $R_F$ is known as the instantaneous forward rate for a maturity of T

Deﬁne $P(0,T)$ as the price of a zero-coupon bond maturing at time $T$ . Because $P(0,T)=e^{-RT}$

R_{F}=-\frac{\partial}{\partial T} \ln P(0, T)

# 4.7 Forward Rate Agreements (FRA)

A forward rate agreement (FRA) is an over-the-counter transaction designed to ﬁx the interest rate that will apply to either borrowing or lending a certain principal during a speciﬁed future period of time.
The usual assumption underlying the contract is that the borrowing or lending would normally be done at LIBOR.

\begin{array}{l}R_{K}: \text { The fixed rate of interest agreed to in the FRA } \\ R_{F}: \text { The forward LIBOR interest rate for the period between times } T_{1} \text { and } T_{2} \text { , } \\ \text { calculated today }^{5} \\ R_{M}: \text { The actual LIBOR interest rate observed in the market at time } T_{1} \text { for the } \\ \text { period between times } T_{1} \text { and } T_{2} \\ L: \text { The principal underlying the contract. }\end{array}

\pm \frac{L\left(R_{K}-R_{M}\right)\left(T_{2}-T_{1}\right)}{1+R_{M}\left(T_{2}-T_{1}\right)}

# 4.7.1 valuation

An FRA is worth zero when the ﬁxed rate R Kequals the forward rate $R_F$ .

The market value of a derivative at a particular time is referred to as its mark-to-market, or MTM, value.

The value of the ﬁrst FRA, where $R_K$ is received, must be

V_{\mathrm{FRA}}=L\left(R_{K}-R_{F}\right)\left(T_{2}-T_{1}\right) e^{-R_{2} T_{2}}

FRA可定价的条件

Calculate the payoff on the assumption that forward rates are realized (that is, on the assumption that $R_M=R_F$ ).
Discount this payoff at the risk-free rate.

# 4.8 Duration

The duration of a bond, as its name implies, is a measure of how long on average the holder of the bond has to wait before receiving cash payments.

A zero-coupon bond that lasts n years has a duration of n years. However, a coupon-bearing bond lasting n years has a duration of less than n years, because the holder receives some of the cash payments prior to year n.

Suppose that a bond provides the holder with cash ﬂows $c_i$ at time $t_i (1\leq i \leq n)$ . The bond price $B$ and bond yield $y$ (continuously compounded) are related by

B=\sum_{i=1}^{n} c_{i} e^{-y t_{i}}

The duration of the bond, $D$ , is deﬁned as

D=\frac{\sum_{i=1}^{n} t_{i} c_{i} e^{-y t_{i}}}{B}

Also can be written as:

D=\sum_{i=1}^{n} t_{i}\left[\frac{c_{i} e^{-y t_{i}}}{B}\right]

关于久期的重要公式：到期收益率(ytm)与债券价格变化率的关系

\frac{\Delta B}{B}=-D \Delta y

久期计算表格

# 4.8.1 Modified Duration

The preceding analysis is based on the assumption that $y$ is expressed with continuous compounding. If $y$ is expressed with annual compounding, it can be shown that the approximate relationship:

\Delta B=-\frac{B D \Delta y}{1+y}

More generally, if $y$ is expressed with a compounding frequency of $m$ times per year, then

\Delta B=-\frac{B D \Delta y}{1+y / m}

A variable $D^*$ , deﬁned by

D^{*}=\frac{D}{1+y / m}

is sometimes referred to as the bond's modified duration. It allows the duration relationship to be simpliﬁed to

\Delta B=-B D^{*} \Delta y

# 4.8.2 Bond Portfolios

The duration, $D$ , of a bond portfolio can be deﬁned as a weighted average of the durations of the individual bonds in the portfolio, with the weights being proportional to the bond prices. They estimate the change in the value of the bond portfolio for a small change Áy in the yields of all the bonds.

It is important to realize that, when duration is used for bond portfolios, there is an implicit assumption that the yields of all bonds will change by approximately the same amount. When the bonds have widely diﬀering maturities, this happens only when there is a parallel shift in the zero-coupon yield curve.

金融机构常通过确保其资产平均久期等于其负债平均久期来对冲其面临的利率风险（即净久期为0），但资产组合对于利率较大的平行移动和非平行移动仍有风险暴露

# 4.9 Convexity

久期是一阶导，因此：The duration relationship applies only to small changes in yields.

曲率，久期相同

C=\frac{1}{B} \frac{d^{2} B}{d y^{2}}=\frac{\sum_{i=1}^{n} c_{i} t_{i}^{2} e^{-y t_{i}}}{B}

根据泰勒展开，

\Delta B=\frac{d B}{d y} \Delta y+\frac{1}{2} \frac{d^{2} B}{d y^{2}} \Delta y^{2}

\frac{\Delta B}{B}=-D \Delta y+\frac{1}{2} C(\Delta y)^{2}

对于给定的久期，当债券组合提供的收入均匀地分布在很长时间区间上时，组合的曲率一般是最大的（思考：为什么？）
当收支都集中在某一个时间附近时，曲率会较小

# 4.10 利率期限结构理论

预期理论：假设长期利率应该反映所期望的将来短期利率
市场分割理论：认为短期、中期以及长期理论之间没有任何关系
流动性偏好理论：基本假设投资者喜欢保持资金的流动性，并因此将资金投资于较短的期限；但另一方面，借贷人一般喜欢借较长期限的固定利率（与收益率曲线向上倾斜的实证结果一致）