### 金融代写|利率建模代写Interest Rate Modeling代考|MATH4513

Vasicek利率模型一词是指一种对利率的运动和演变进行建模的数学方法。它是一种基于市场风险的单因素短利率模型。瓦西克利率模型常用于经济学中，以确定利率在未来的移动方向。

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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|Yield to Maturity

In a free market, the price of a bond is determined by supply and demand. Due to discounting, the full price is normally smaller than the total notional value of coupons plus the principal. Denote the full price of a bullet bond as $B^{c}$. Suppose that all cash flows are discounted by a uniform rate, $y$, of compounding frequency $\omega$. Then $y$ should satisfy the following equation:
$$B^{c}=\operatorname{Pr} \cdot\left(\sum_{i=1}^{n} \frac{c \Delta T}{(1+y \Delta t)^{i \Delta T / \Delta t}}+\frac{1}{(1+y \Delta t)^{n \Delta T / \Delta t}}\right)$$
where $n$ is the number of coupons and $\Delta t=1 / \omega$. In bond mathematics, the compounding frequency is taken to be $\omega=1 / \Delta T$ by default, when there is $\Delta t=\Delta T$. This discount rate, which can be easily solved by a trial-and-error procedure using Equation 3.12, is defined to be the yield to maturity (YTM), as well as the internal rate of return (IRR) of the bond, and it is often simply called the bond yield.

As the function of the yield (for $\omega=1 / \Delta T$ ), the formula for a general time, $t \leq T$, is
$$B_{t}^{c}=\operatorname{Pr} \cdot\left(\sum_{i ; i \Delta T>t}^{n} \frac{c \Delta T}{(1+y \Delta T)^{(i \Delta T-t) / \Delta T}}+\frac{1}{(1+y \Delta T)^{(n \Delta T-t) / \Delta T}}\right)_{(3.13)}$$

Assuming that $t \in\left(T_{j}, T_{j+1}\right]$, and introducing
$$q=\frac{t-T_{j}}{T_{j+1}-T_{j}}=\frac{t-T_{j}}{\Delta T}$$
we then can write
$$t=T_{j}+\Delta T q=(j+q) \Delta T \text { and } i \Delta T-t=(i-j-q) \Delta T, \quad \forall i .$$
It follows that
\begin{aligned} B_{t}^{c} &=\operatorname{Pr} \cdot\left(\sum_{i=j+1}^{n} \frac{c \Delta T}{(1+y \Delta T)^{i-j-q}}+\frac{1}{(1+y \Delta T)^{n-j-q}}\right) \ &=\operatorname{Pr} \cdot(1+y \Delta T)^{q}\left(\sum_{i=1}^{n-j} \frac{c \Delta T}{(1+y \Delta T)^{i}}+\frac{1}{(1+y \Delta T)^{n-j}}\right) \end{aligned}
Given the bond price at any time, $t$, the bond yield is implied by Equation 3.14. A rough way to compare the relative cheapness/richness of two bonds with the same coupon frequency is to compare their yields. Intuitively, a bond with a higher yield is cheaper and thus may be more attractive.

There is a one-to-one price-yield relationship, as shown in Figure 3.2. Because of this relationship, a bond price is also quoted using its yield in the industry. As we can see in Figure $3.2$, a bond price is a convex function of the yield. Such a feature will be used later for convexity adjustment related to futures trading.
The price-yield relationship of a zero-coupon bond simplifies to
$$P=\operatorname{Pr} \cdot(1+y \Delta T)^{-(T-t) / \Delta T} .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Par Bonds, Par Yields, and the Par Yield Curve

The summation in Equation $3.12$ can be worked out so that
\begin{aligned} B^{c} &=\Delta T \cdot c \cdot \operatorname{Pr} \sum_{i=1}^{n}(1+y \Delta T)^{-i}+\operatorname{Pr}(1+y \Delta T)^{-n} \ &=\operatorname{Pr}\left[1-\left(1-\frac{c}{y}\right)\left(1-\frac{1}{(1+y \Delta T)^{n}}\right)\right] \end{aligned}
From the above expression, we can tell when the price is smaller, equal to, or larger than the principal value.

1. When $c<y, B^{c}<$ Pr. In such a case, we say that the bond is sold at discount (of the par value).
2. When the coupon rate is $c=y$, then $B^{c}=\operatorname{Pr}$, that is, the bond price equals the par value of the bond. In such a case, we call the bond a par bond, and the corresponding coupon rate a par yield.
3. When $c>y, B^{c}>\operatorname{Pr}$. In such a case, we call the bond a premium bond (it is traded at a premium to par).

Par yields play an important role in today’s interest-rate derivatives market. As we shall see later, there are many derivatives based on the par yields.

## 金融代写|利率建模代写Interest Rate Modeling代考|Yield Curves for U.S. Treasuries

A bond issuer may routinely issue bonds of various maturities, and, in a market, there can be many bonds of the same issuer being traded. For various reasons, some bonds are more liquid than others. The most liquid ones are often called benchmark bonds for the issuer. Their yields reflect the level of borrowing costs the market demands from the issuer. Moreover, the prices of the benchmark bonds imply a discount curve for cash flows from the issuer, and the discount curve can be used to gauge the relative cheapness/expensiveness of the issuer’s other bonds. If a relatively cheaper or more expensive bond is found, one may trade against this bond using the benchmark bonds and thus take an arbitrage profit. Hence, the prices or yields of the benchmark bonds carry essential information for the arbitrage pricing of the issuer’s other bonds, and they are treated as a summary of the status quo of all bonds offered by the same issuer.

In the U.S. Treasury market, newly issued bills and notes/bonds are called on-the-run Treasury securities. Traditionally, the on-the-run issues enjoy higher liquidity and are thus treated as benchmarks. Table $3.1$ provides the closing price quotes of the on-the-run issues for July 3 , 2008. As can be seen in the table, the on-the-run issues have maturities of 3 months, 6 months, 2 years, 3 years, 5 years, 10 years, and 30 years. When we connect the yields of the benchmark bonds through interpolation, we obtain a so-called yield curve. Since bond yields vary from day-to-day so does the yield curve. Figure $3.3$ shows the yield curves for the U.S. Treasuries constructed by linear interpolation for April 28 and May 1, 2006 , two consecutive trading days.
A yield curve is constructed based on yields of on-the-run issues using the interpolation technique. It provides a rough idea of the level of yields for various maturities. Further, the Treasury yield curve implies a discount curve, namely, the collection of prices of all zero-coupon bonds. The discount curve is used for pricing off-the-run Treasury securities, or marking to market Treasury portfolios. Moreover, the discount curve is also essential for pricing future cash flows of any security, either deterministic or stochastic. To price a portfolio of interest-rate derivatives, we may model the dynamics of the entire yield curve, in contrast to modeling the dynamics of a stock price for stock options. In the next section, we describe the technique for “backing” out the discount curve from the yield curve.

## 金融代写|利率建模代写Interest Rate Modeling代考|Yield to Maturity

q=吨−吨j吨j+1−吨j=吨−吨jΔ吨

## 金融代写|利率建模代写Interest Rate Modeling代考|Par Bonds, Par Yields, and the Par Yield Curve

1. 什么时候C<是,乙C<公关。在这种情况下，我们说债券以（面值的）折扣价出售。
2. 当票面利率为C=是， 然后乙C=公关，即债券价格等于债券面值。在这种情况下，我们称该债券为面值债券，相应的票面利率为面值收益率。
3. 什么时候C>是,乙C>公关. 在这种情况下，我们称该债券为溢价债券（以面值溢价交易）。

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## MATLAB代写

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