### 金融代写|利率建模代写Interest Rate Modeling代考|MTH5520

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|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 piecewise 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 $\left{B_{j}^{c}, T_{j}\right}_{j=1}^{7}$ be the prices and maturities of the seven on-the-run issues. Let $T_{0}=0$ and $\Delta T=0.5$. We assume that the zero-coupon yield for maturities between $\left[T_{0}, T_{7}\right]$ is a piecewise linear function. The determination of the YTMs is done sequentially. Because the first two on-the-run issues are zero-coupon bonds, we first back out $y(0.25)$ and $y(0.5)$, 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 $y(0.5)$ is found, we proceed to determining $y(i \Delta T), i=2,3,4$ from the following equation:
$$B_{3}^{c}=\frac{c_{3} \Delta T}{(1+y(\Delta T) \Delta T)^{i}}+\sum_{i=2}^{4} \frac{c_{3} \Delta T}{(1+y(i \Delta T) \Delta T)^{i}}+\frac{1}{(1+y(4 \Delta T) \Delta T)^{4}}$$
where
$$y(i \Delta T)=y(0.5)+\alpha \times(i \Delta T-0.5), \quad i=2,3,4$$
So, our zero-coupon yield is a linear function over $T \in\left[T_{2}, T_{3}\right]$. Equation $9.20$ become the equation for $\alpha$, which can be determined through a root-finding procedure. This procedure can continue all the way to $j=7$. The entire 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

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 agreement 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 $f_{\Delta T}(t, T)$. 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 $f_{\Delta T}(t, T)$. 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
$$V=1+\Delta T f_{\Delta T}(t, T)-\frac{P(t, T)}{P(t, T+\Delta T)}$$
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:
$$f_{\Delta T}(t, T)=\frac{1}{\Delta T}\left(\frac{P(t, T)}{P(t, T+\Delta T)}-1\right) .$$
Hence, the arbitrage free or fair forward lending rate is totally determined by the prices of zero-coupon bonds. We call $f_{\Delta T}(t, T)$ the simple forward rate for the period $(T, T+\Delta T)$ seen at time $t$, or simply a forward rate.

Consider now the limiting case, $\Delta T \rightarrow 0$, for the forward rate. There is
\begin{aligned} f(t, T) & \triangleq \lim {\Delta T \rightarrow 0} f{\Delta T}(t, T) \ &=\lim {\Delta T \rightarrow 0} \frac{1}{\Delta T}\left(\frac{P(t, T)}{P(t, T+\Delta T)}-1\right) \ &=\frac{-1}{P(t, T)} \frac{\partial P(t, T)}{\partial T} \ &=-\frac{\partial \ln P(t, T)}{\partial T} \end{aligned} We call $f(t, T)$ an instantaneous forward rate. According to Equation $3.23$, we can express the price of a $T$-maturity zero-coupon bond in terms of $f(t, s), t \leq$ $s \leq T$ : $$P(t, T)=\mathrm{e}^{-\int{t}^{T} f(t, s) \mathrm{d} s}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Duration and Convexity

In the bond market, bond prices change unpredictably on a daily basis. The changes in bond prices can be interpreted as the consequence of unpredictable changes in yields. The duration of a bond is a measure of risk exposure with respect to a possible change in the bond yield. It has been observed that the prices of long-maturity bonds are more sensitive to change in yields than are the prices of short-maturity bonds, and the impact of yield changes on bond prices seems proportional to the cash flow dates of the bonds. Intuitively, Macaulay (1938) introduced the weighted average of the cash flow dates as a measure of price sensitivity with respect to the bond yield:
\begin{aligned} D_{\text {mac }}=& \frac{\operatorname{Pr}}{B_{t}^{c}}\left[\sum_{i, T_{i}>t}^{n} \Delta T \cdot c(1+y \Delta T)^{-\left(T_{i}-t\right) / \Delta T}\left(T_{i}-t\right)\right.\ &\left.+(1+y \Delta T)^{-\left(T_{n}-t\right) / \Delta T}\left(T_{n}-t\right)\right] . \end{aligned}
This measure is called the Macaulay duration in the bond market. Note that, for a zero-coupon bond, the duration is simply its maturity. It was later understood that the Macaulay duration is closely related to the derivative of the bond price with respect to its yield. In fact, differentiating Equation $3.13$ with

respect to $y$ yields
\begin{aligned} \frac{\mathrm{d} B_{t}^{c}}{\mathrm{~d} y}=&-\frac{\operatorname{Pr}}{1+y \Delta T}\left[\sum_{i ; T_{i}>t}^{n} \Delta T \cdot c(1+y \Delta T)^{-\left(T_{i}-t\right) / \Delta T}\left(T_{i}-t\right)\right.\ &\left.+(1+y \Delta T)^{-\left(T_{n}-t\right) / \Delta T}\left(T_{n}-t\right)\right] \end{aligned}
In terms of $D_{\operatorname{mac}}$, the Macaulay duration just defined, we have
$$\frac{\mathrm{d} B_{t}^{c}}{B_{t}^{c}}=-\frac{D_{\mathrm{mac}}}{1+y \Delta T} \mathrm{~d} y \quad \text { or } \quad \frac{1}{B_{t}^{c}} \frac{\mathrm{d} B_{t}^{c}}{\mathrm{~d} y}=-\frac{D_{\operatorname{mac}}}{1+y \Delta T}$$
According to Equation 3.25, the Macaulay duration is essentially the rate of change with respect to the yield for each dollar of market value of the bond. After multiplying by the change in the yield, the Macaulay duration gives the percentage change in the value of the bond. For convenience, we define
$$D_{\text {mod }}=\frac{D_{\text {mac }}}{1+y \Delta T},$$
and call it the modified duration. Then the first equation of Equation $3.25$ can be written in the following simple form:
$$\frac{\mathrm{d} B_{t}^{c}}{B_{t}^{c}}=-D_{\bmod } \mathrm{d} y$$
Both $D_{\text {mac }}$ and $D_{\text {mod }}$ are called duration measures of bonds.
By using Equation 3.17, the succinct bond formula, we can obtain the following formula for the modified duration:
$$D_{\text {mod }}=\frac{\operatorname{Pr}}{B^{c}}\left[\frac{c}{y^{2}}\left(1-\frac{1}{(1+y \Delta T)^{n}}\right)+\left(1-\frac{c}{y}\right) \frac{n \Delta T}{(1+y \Delta T)^{n+1}}\right]$$
The above expression is simplified for par bonds. When $c=y$ and $B^{c}=\operatorname{Pr}$, we have
$$D_{\mathrm{mod}}=\frac{1}{y}\left[1-(1+y \Delta T)^{-n}\right] .$$
Note that Treasury bonds are quoted in yields and that recent issues are usually traded close to par, so Equation $3.28$ gives us an approximate value of the durations for bonds being traded close to par.

The next example shows how much the dollar value of a bond changes given its duration.

## 金融代写|利率建模代写Interest Rate Modeling代考|Forward Rates and Forward-Rate Agreements

FΔ吨(吨,吨)=1Δ吨(磷(吨,吨)磷(吨,吨+Δ吨)−1).

F(吨,吨)≜林Δ吨→0FΔ吨(吨,吨) =林Δ吨→01Δ吨(磷(吨,吨)磷(吨,吨+Δ吨)−1) =−1磷(吨,吨)∂磷(吨,吨)∂吨 =−∂ln⁡磷(吨,吨)∂吨我们称之为F(吨,吨)瞬时远期汇率。根据方程3.23，我们可以表示a的价格吨-到期零息债券F(吨,s),吨≤ s≤吨 :

## 金融代写|利率建模代写Interest Rate Modeling代考|Duration and Convexity

D苹果电脑 =公关乙吨C[∑一世,吨一世>吨nΔ吨⋅C(1+是Δ吨)−(吨一世−吨)/Δ吨(吨一世−吨) +(1+是Δ吨)−(吨n−吨)/Δ吨(吨n−吨)].

d乙吨C d是=−公关1+是Δ吨[∑一世;吨一世>吨nΔ吨⋅C(1+是Δ吨)−(吨一世−吨)/Δ吨(吨一世−吨) +(1+是Δ吨)−(吨n−吨)/Δ吨(吨n−吨)]

d乙吨C乙吨C=−D米一个C1+是Δ吨 d是 或者 1乙吨Cd乙吨C d是=−D苹果电脑1+是Δ吨

D反对 =D苹果电脑 1+是Δ吨,

d乙吨C乙吨C=−D反对d是

D反对 =公关乙C[C是2(1−1(1+是Δ吨)n)+(1−C是)nΔ吨(1+是Δ吨)n+1]

D米○d=1是[1−(1+是Δ吨)−n].

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