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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|Zero-Coupon Bonds

Let us refer to the cash flow of the coupon bond shown in Figure $3.1$ again. When there is no coupon, $c=0$, the principal is the only cash flow, and the coupon bond is reduced to a zero-coupon bond. The corresponding yield is called a zero-coupon yield. As a convention, the time- $t$ price of a zero-coupon bond maturing at time $T$ into a par value of one dollar is denoted as $P(t, T)$ or $P_{t}^{T}$. In terms of its yield, the price of the zero-coupon bond is
$$P_{t}^{T}=\frac{1}{(1+y \Delta t)^{(T-t) / \Delta t}}$$
where the time to maturity, $T-t$, does not have to be a multiple of $\Delta t$. The collection of $P_{t}^{T}$ for $T \geq t$ is called a discount curve.

With the discount curve, one can price any bond portfolio with deterministic cash flows. This is because any such portfolio can be treated as a portfolio of zero-coupon bonds. For example, we can express the price of the coupon bond in terms of those of zero-coupon bonds:
$$B^{c}(0)=\sum_{i=1}^{n} c \cdot \Delta T \operatorname{Pr} \cdot P_{0}^{i \Delta T}+\operatorname{Pr} \cdot P_{0}^{n \Delta T}$$
In continuous-time finance, it is often favorable to work with continuous compounding, that is, by letting the term $\Delta t \rightarrow 0$ in Equation 3.18. At this limit, we have
$$P_{t}^{T}=\mathrm{e}^{-y \times(T-t)} .$$
Given $P_{t}^{T}$, the corresponding zero-coupon yield can be calculated from the last equation:
$$y_{T-t}=-\frac{1}{T-t} \ln P_{t}^{T} .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Bootstrapping the Zero-Coupon Yields

The determination of the zero-coupon yield curve (or discount curve) based on the yields of the on-the-run issues is an under-determined problem: we need to solve for infinitely many unknowns based on a few inputs. To define a meaningful solution, one must parameterize the zero-coupon yield curve. The simplest parameterization that is financially acceptable is to assume piece-wise constant functional forms for the zero-coupon yield curve. Under such a parameterization, the zero-coupon yield curve can be derived sequentially. Such a procedure is often called bootstrapping in finance. Next, we describe the bootstrapping procedure with the construction of the zero-coupon yield curve for U.S. Treasuries.

Let $\left{B_{i}^{c}, T_{i}\right}_{i=1}^{7}$ be the prices and maturities of the seven on-the-run issues. We assume that the zero-coupon yield for maturities between $\left(T_{i-1}, T_{i}\right.$ ] is a constant, $y_{i}$, with $i=1, \ldots, 7$ and $T_{0}=0$. The determination of the YTMs is done sequentially. Because the first two on-the-run issues are zero-coupon bonds, we first back out $y_{1}$ and $y_{2}$, the zero yields for $\left(0, T_{1}\right]$ and $\left(T_{1}, T_{2}\right]$, using formula $3.18$. This will require a root-finding procedure. Once $y_{2}$ is found, we proceed to determining $y_{3}$ from the following equation:
$$B_{3}^{c}=\sum_{i \Delta T \leq T_{2}} \frac{c_{3} \Delta T}{\left(1+y_{2} \Delta T\right)^{i}}+\sum_{i \Delta T>T_{2}} \frac{c_{3} \Delta T}{\left(1+y_{3} \Delta T\right)^{i}}+\frac{1}{\left(1+y_{3} \Delta T\right)^{T_{3} / \Delta T}} .$$
Here, we have used $y_{2}$ as the discount rate for all cash flows between $\left(T_{0}, T_{2}\right]$, and $y_{3}$ is the only unknown for the equation that again can be determined through a root-finding procedure. This procedure can continue all the way to $i=7$. The entire zero-coupon yield curve for maturity $T \leq 30$ so-determined is displayed in Figure 3.4.

A zero-coupon yield curve implies a discount curve. Suppose that the $y_{T}$ is the zero-coupon yield for maturity $T$. Then the corresponding zerocoupon bond price is calculated according to Equation 3.18. With discount bond prices, we can value any coupon bond using Equation 3.19.

## 金融代写|利率建模代写Interest Rate Modeling代考|FORWARD RATES AND FORWARD-RATE AGREEMENTS

A forward-rate agreement (FRA) is a contract between two parties to lend and borrow a certain amount of money for some future period of time with a pre-specified interest rate. The agreement is so structured that neither party needs to make an upfront payment. This is equivalent to saying that, as a financial instrument, the value of the contract is zero when the agreenent is entered. The key to such a contract lies in the lending rate that should be fair to both parties. Fortunately, this fair rate can be determined through arbitrage arguments.

Let the time now be $t$ and the fair lending rate for a future period, $[T, T+\Delta T]$, be $f(t ; T, \Delta T)$. To finance the lending, the lender may short $P(t, T) / P(t, T+\Delta T)$ units of the $(T+\Delta T)$ maturity zero-coupon bond, and then long one unit of the $T$-maturity zero-coupon bond. At time $T$, the proceeds from the $T$-maturity zero are lent out for a period of $\Delta T$ with the interest rate $f(t ; T, \Delta T)$. At time $T+\Delta T$, the loan is paid back from the borrower and the short position of $(T+\Delta T)$ maturity zero-coupon bond (which just matures) is covered, yielding a net cash flow of
$$V=1+\Delta T f(t ; T, \Delta T)-\frac{P(t, T)}{P(t, T+\Delta T)}$$
Because this is a set of zero net transactions initially, in the absence of arbitrage, $V$ must be zero, which leads to the following expression of the fair lending rate:
$$f(t ; T, \Delta T)=\frac{1}{\Delta T}\left(\frac{P(t, T)}{P(t, T+\Delta T)}-1\right) .$$
Hence, the arbitrage free or fair forward lending rate is totally determined by the prices of zero soupon bouds. We sull $f(t ; T, \Delta T)$ the simple for ward rate for the period $(T, T+\Delta T)$ seen at time $t$, or simply a forward rate.

## 金融代写|利率建模代写Interest Rate Modeling代考|Zero-Coupon Bonds

$$P_{t}^{T}=\frac{1}{(1+y \Delta t)^{(T-t) / \Delta t}}$$

$$B^{c}(0)=\sum_{i=1}^{n} c \cdot \Delta T \operatorname{Pr} \cdot P_{0}^{i \Delta T}+\operatorname{Pr} \cdot P_{0}^{n \Delta T}$$

$$P_{t}^{T}=\mathrm{e}^{-y \times(T-t)} .$$

$$y_{T-t}=-\frac{1}{T-t} \ln P_{t}^{T} .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Bootstrapping the Zero-Coupon Yields

$$B_{3}^{c}=\sum_{i \Delta T \leq T_{2}} \frac{c_{3} \Delta T}{\left(1+y_{2} \Delta T\right)^{i}}+\sum_{i \Delta T>T_{2}} \frac{c_{3} \Delta T}{\left(1+y_{3} \Delta T\right)^{i}}+\frac{1}{\left(1+y_{3} \Delta T\right)^{T_{3} / \Delta T}} .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|FORWARD RATES AND FORWARD-RATE AGREEMENTS

$P(t, T) / P(t, T+\Delta T)$ 的单位 $(T+\Delta T)$ 到期零息债券，然后做多一个单位 $T$ – 到期零息债券。当时 $T$ ，收益来自 $T$-到期时间为零被借出 一段时间 $\Delta T$ 与利率 $f(t ; T, \Delta T)$. 当时 $T+\Delta T$ ，贷款由借款人偿还，空头头寸 $(T+\Delta T)$ 到期零息债券（刚刚到期) 被覆盖，产生的净现 金流为
$$V=1+\Delta T f(t ; T, \Delta T)-\frac{P(t, T)}{P(t, T+\Delta T)}$$

$$f(t ; T, \Delta T)=\frac{1}{\Delta T}\left(\frac{P(t, T)}{P(t, T+\Delta T)}-1\right)$$

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