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

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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|Discretization of Forward Rate Process

We recall the forward rate process in the H.JM model as
\begin{aligned} f(t, T)=& f(0, T)+\int_{0}^{t}\left{-\sigma(s, T) v(s, T)+\sigma(s, T) \varphi_{s}\right} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} \end{aligned}
where $W_{t}$ is a $d$-dimensional P-Brownian motion. The HJM model is called Gaussian if $\sigma(t, T)$ is a deterministic function of $t$ and $T$. In this book, we always assume that volatility $\sigma(t, T)$ is deterministic and continuous with respect to $t$ and $T$. Then, $v(t, T)$ is also deterministic and continuous. In the following, the market price of risk is assumed to be constant. The validity of the constancy assumption will be examined in the context of risk management in Chapter $7 .$
Data observation
Denoting the market price of risk as a constant $\varphi$, the above forward rate process is expressed by
\begin{aligned} f(t, T)=& f(0, T)+\int_{0}^{t}{-\sigma(s, T) v(s, T)+\sigma(s, T) \varphi} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} . \end{aligned}
Next, we specify a historical dataset as follows. Let a time interval $\Delta t>0$ be fixed, and $\left{t_{k}\right}_{k=1, \cdots, J+1}$ be a sequence of observation dates such that $t_{1}=0$ and $t_{k+1}-t_{k}=\Delta t$, where $J+1$ is the number of observation times. We denote the time length to a maturity $T$ from $t$ by $x=T-t$. For an integer $n \geq d$, $x_{1}, \cdots, x_{n}$ denotes a sequence of time lengths to maturity.

Typically, we observe the instantaneous forward rate $F\left(t_{k}, x_{i}\right)$ with respect to fixed $x_{i}$. Fig. $6.1$ illustrates an example of forward rate curves observed at $t_{k}, t_{k+1}$, and $t_{k+2}$, showing $F\left(t_{k}, x_{i}\right), F\left(t_{k+1}, x_{i}-\Delta t\right)$, and so on. We assume that the dynamics of these observations follow equation (6.2).

## 金融代写|利率建模代写Interest Rate Modeling代考|Estimation of Market Price of Risk

We recall the volatility structure associated with PCA in Section 4.3. A sample covariance matrix $V$ is defined by
\begin{aligned} V_{i j}=& \frac{1}{\Delta t} \operatorname{Cov}\left(F\left(t_{k}+\Delta t, x_{i}-\Delta t\right)-F\left(t_{k}, x_{i}\right)\right.\ &\left.F\left(t_{k}+\Delta t, x_{j}-\Delta t\right)-F\left(t_{k}, x_{j}\right)\right) ; \quad i, j=1, \cdots, n . \end{aligned}
We assume that $V$ has rank $d \leq n$. By the argument in Appendix $\mathrm{B}$, the covariance matrix is decomposed into $V_{i j}=\sum_{l=1}^{d} e_{i}^{l} \rho_{l}^{2} e_{j}^{l}$ for $i, j \leq n$, where $\rho_{l}^{2}$ is the lth eigenvalue, and $e^{l}=\left(e_{1}^{l}, \cdots, e_{n}^{l}\right)^{T}$ is the $l$ th principal component of the covariance for $l=1, \cdots, d$.
We always assume that
$$e_{1}^{l}>0, \quad \rho_{l}>0 ; \quad l=1, \cdots, d$$
This assumption is significant in the interpretation of the meaning of the market price of risk in Sections $6.4$ and 6.5. Recall the equation (B.4), that is, that the principal components $e^{1}, \cdots, e^{d}$ form an orthonormal set. Thus,
$$\sum_{i=1}^{n} e_{i}^{l} e_{i}^{h}=\delta_{l h} \quad ; \quad l, h=1, \cdots, d$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Market Price of Risk: State Space Setup

This section introduces another method to estimate the market price of risk: working in a state space.

Denoting the market price of risk by $\varphi^{\prime}=\left(\varphi_{1}^{\prime}, \cdots, \varphi_{d}^{\prime}\right)^{T}$, we return to the discretization as equation (6.13), which we reproduce below:
$$\Delta F_{i}\left(t_{k}\right)=-\sigma_{0 i} v_{0 i} \Delta t+\sigma_{0 i} \varphi^{\prime} \Delta t+\sqrt{\Delta t} \sigma_{0 i} W_{1} ; i=1, \cdots, n, k=1, \cdots, J$$
We remark that the volatility is assumed to be determined by a principal component. Our objective here is to directly obtain $\varphi^{\prime}$ from the above equations.
We denote by $\epsilon\left(\varphi^{\prime}\right)$ the sum of the squared difference between each side of equation $(6.26)$ in the time series and cross sections, neglecting the random part, such that
$$\epsilon\left(\varphi^{\prime}\right)=\frac{1}{J} \sum_{k=1}^{J} \sum_{i=1}^{n}\left{\Delta F_{i}\left(t_{k}\right)+\left(\sigma_{0 i} v_{0 i}-\sigma_{0 i} \varphi^{\prime}\right) \Delta t\right}^{2}$$
Let $\varphi^{\prime}$ be the solution that minimizes $\epsilon\left(\varphi^{\prime}\right)$. We call this setting a state space setup, and call that used in the previous section a $P C A$ setup to distinguish between the two approaches. We note the implications of both definitions below.

• $\varphi$ is the solution that minimizes $\theta_{l}\left(\varphi_{l}\right)$ in equation $(6.19)$ in the principal component space, and also is the maximum likelihood estimate.
• $\varphi^{\prime}$ is the solution that minimizes $\epsilon\left(\varphi^{\prime}\right)$ of equation $(6.27)$ in the state space.

## 金融代写|利率建模代写Interest Rate Modeling代考|Discretization of Forward Rate Process

\begin{对齐} f(t, T)=& f(0, T)+\int_{0}^{t}\left{-\sigma(s, T) v(s, T)+\sigma( s, T) \varphi_{s}\right} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} \end{aligned}\begin{对齐} f(t, T)=& f(0, T)+\int_{0}^{t}\left{-\sigma(s, T) v(s, T)+\sigma( s, T) \varphi_{s}\right} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} \end{aligned}

F(吨,吨)=F(0,吨)+∫0吨−σ(s,吨)在(s,吨)+σ(s,吨)披ds +∫0吨σ(s,吨)d在s.

## 金融代写|利率建模代写Interest Rate Modeling代考|Estimation of Market Price of Risk

∑一世=1n和一世l和一世H=dlH;l,H=1,⋯,d

## 金融代写|利率建模代写Interest Rate Modeling代考|Market Price of Risk: State Space Setup

ΔF一世(吨ķ)=−σ0一世在0一世Δ吨+σ0一世披′Δ吨+Δ吨σ0一世在1;一世=1,⋯,n,ķ=1,⋯,Ĵ

\epsilon\left(\varphi^{\prime}\right)=\frac{1}{J} \sum_{k=1}^{J} \sum_{i=1}^{n}\left{\ Delta F_{i}\left(t_{k}\right)+\left(\sigma_{0 i} v_{0 i}-\sigma_{0 i} \varphi^{\prime}\right) \Delta t \对}^{2}\epsilon\left(\varphi^{\prime}\right)=\frac{1}{J} \sum_{k=1}^{J} \sum_{i=1}^{n}\left{\ Delta F_{i}\left(t_{k}\right)+\left(\sigma_{0 i} v_{0 i}-\sigma_{0 i} \varphi^{\prime}\right) \Delta t \对}^{2}

• 披是最小化的解决方案θl(披l)在等式中(6.19)在主成分空间中，也是最大似然估计。
• 披′是最小化的解决方案ε(披′)方程的(6.27)在状态空间。

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