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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|A GENERAL TREE-BUILDING PROCEDURE

The procedure for tree building presented in the last section can be generalized to other diffusion models for short rates. The major disadvantage of the resulting tree is that the interest-rate span can be too wide and fall too deep in the negative territory. In this section, we introduce a more sophisticated method of tree building for short-rate models with mean reversion. Although the resulting tree does not guarantee that negative interest rates are avoided, it does provide a tree with a much narrower rate span around the level of the mean interest rate.

Without loss of generality, we consider a trinomial tree approximation of a general short-rate model with a state-dependent drift term,
$$\mathrm{d} r_t=\mu\left(r_t, t ; \theta_t\right) \mathrm{d} t+\sigma_t \mathrm{~d} W_t .$$
An immediate example of $\mu$ is the drift term for the Hull-White model, $\kappa\left(\theta_t-r_t\right)$, but the method can be applied to other models by replacing $r_t$ with a function $f\left(r_t\right)$. Literally, the discrete version of Equation $5.79$ is
$$\Delta r_t=\mu\left(r_t, t ; \theta_t\right) \Delta t+\sigma_t \Delta W_t .$$
Its first two moments are
\begin{aligned} E^{\mathbb{Q}}\left[\Delta r_t \mid r_t\right] &=\mu\left(r_t, t ; \theta_t\right) \Delta t, \ E^{\mathbb{Q}}\left[\left(\Delta r_t\right)^2 \mid r_t\right] &=\sigma_t^2 \Delta t+\mu^2\left(r_t, t ; \theta_t\right) \Delta t^2 . \end{aligned}
Consider an element with trinomial branching as depicted in Figure 5.10, where we choose the middle branch for the next time step, $r_{k, n+1}$, according to the principle that it is the one closest to the expected value of the shortrate conditional on $r_{j, n}$,
\begin{aligned} k &=k(j)=\underset{i}{\arg \min }\left|E^{\mathbb{Q}}\left[r_{\cdot, n+1} \mid r_{j, n}\right]-\left(r_{0, n+1}+i \delta r\right)\right| \ r_{0, n+1} &=r_{0, n}+\mu\left(r_{0, n}, t_n ; \theta_n\right) \Delta t \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|The Black–Karasinski Model

The discussions of tree methods for short-rate models have focused on the Ho-Lee and Hull-White models. In this section, we discuss a more general class of short-rate models and their lattice implementations. This class of models can be cast in the form
$$\mathrm{d} f\left(r_t\right)=\kappa\left(\theta_t-f\left(r_t\right)\right) \mathrm{d} t+\sigma_t \mathrm{~d} W_t,$$
for some monotonic function, $f(x)$, such that $f^{-1}(y)$ exists. For a given function of $\theta_t$, we can apply the procedure from Equations $5.82$ through $5.89$ to build a tree for $R_t=f\left(r_t\right)$. The interest-rate tree for $r_t$ then results from the one-to-one correspondence between $R_t$ and $r_t$. For the calibration procedure, however, there is a small difference from the one for the HullWhite model: due to the non-linearity of $f(x), \theta_n$ will be solved through a root-finding procedure, instead of being obtained explicitly. Specifically,

with $r_t$ taken over by $R_t$ elsewhere in the algorithm, we need to replace Equations $5.99$ and $5.100$ with a single equation
$$P(0,(j+1) \Delta t)=\sum_{i=-j}^j Q_{i, j} \mathrm{e}^{-f^{-1}\left(R_{i, j}-\kappa \theta_{j-1} \Delta t\right) \Delta t},$$
and solve for $\theta_{j-1}$ iteratively.
As a major model in the class of Equation 5.104, we now introduce the Black and Karasinski (1991) model, which corresponds to $f(x)=\ln x$. Alternatively, we can write the short rate as an exponential function,
$$r_t=\mathrm{e}^{X_t}$$
while $X_t$ follows a Vasicek process,
$$\mathrm{d} X_t=\sigma_t \mathrm{~d} W_t+\kappa_t\left(\theta_t-X_t\right) \mathrm{d} t .$$

## 金融代写|利率建模代写利率建模代考|一个通用的树的建立过程

.

$\mu$的一个直接例子是Hull-White模型的漂移项$\kappa\left(\theta_t-r_t\right)$，但该方法可以应用于其他模型，方法是将$r_t$替换为函数$f\left(r_t\right)$。从理论上讲，方程$5.79$的离散版本是
$$\Delta r_t=\mu\left(r_t, t ; \theta_t\right) \Delta t+\sigma_t \Delta W_t .$$

\begin{aligned} E^{\mathbb{Q}}\left[\Delta r_t \mid r_t\right] &=\mu\left(r_t, t ; \theta_t\right) \Delta t, \ E^{\mathbb{Q}}\left[\left(\Delta r_t\right)^2 \mid r_t\right] &=\sigma_t^2 \Delta t+\mu^2\left(r_t, t ; \theta_t\right) \Delta t^2 . \end{aligned}

\begin{aligned} k &=k(j)=\underset{i}{\arg \min }\left|E^{\mathbb{Q}}\left[r_{\cdot, n+1} \mid r_{j, n}\right]-\left(r_{0, n+1}+i \delta r\right)\right| \ r_{0, n+1} &=r_{0, n}+\mu\left(r_{0, n}, t_n ; \theta_n\right) \Delta t \end{aligned}

## 金融代写|利率建模代写利率建模代考| Black-Karasinski模型

$$\mathrm{d} f\left(r_t\right)=\kappa\left(\theta_t-f\left(r_t\right)\right) \mathrm{d} t+\sigma_t \mathrm{~d} W_t,$$

， $r_t$在算法的其他地方被$R_t$取代，我们需要用单个方程
$$P(0,(j+1) \Delta t)=\sum_{i=-j}^j Q_{i, j} \mathrm{e}^{-f^{-1}\left(R_{i, j}-\kappa \theta_{j-1} \Delta t\right) \Delta t},$$

$$r_t=\mathrm{e}^{X_t}$$
，而$X_t$遵循Vasicek过程，
$$\mathrm{d} X_t=\sigma_t \mathrm{~d} W_t+\kappa_t\left(\theta_t-X_t\right) \mathrm{d} t .$$

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assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师