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

#### Doug I. Jones

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## 金融代写|利率建模代写Interest Rate Modeling代考|Principal Component Analysis

Having constructed the time series data of forward rates of the seven maturities, shown in Figure 4.4, we now proceed to the estimation of covariance among those forward rates. We then perform PCA with the covariance matrix. The results will shed light on the proper number of random factors that drive the evolution of the forward-rate curve, so that we can determine $n$ and subsequently $\boldsymbol{\sigma}(t, T)$ for the HJM equation.

Let $f\left(n \Delta \tau, n \Delta \tau+T_{i}\right), n=0,1, \ldots, N$ be the forward rates for the seven maturities, $T_{i}, i=1,2, \ldots, 7$, where $\Delta \tau=1 / 12$ represents the observation interval of one month, and $N$ the total number of months. For forward rates of each maturity, $T_{i}, i=1,2, \ldots, 7$, we calculate the change over $\Delta \tau$ :
\begin{aligned} \Delta f_{n, i}=& f\left((n+1) \Delta \tau,(n+1) \Delta \tau+T_{i}\right) \ &-f\left(n \Delta \tau, n \Delta \tau+T_{i}\right), \quad n=0,1, \ldots, N-1 \end{aligned}
The empirical covariance between $\Delta f_{\cdot, i}$ and $\Delta f_{\cdot, j}$ is, straightforwardly,
$$\hat{c}{i j}=\frac{1}{N} \sum{n=0}^{N-1}\left(\Delta f_{n, i}-\overline{\Delta f_{i}}\right)\left(\Delta f_{n, j}-\overline{\Delta f_{j}}\right)$$
where
$$\overline{\Delta f_{i}}=\frac{1}{N} \sum_{n=0}^{N-1} \Delta f_{n, i}$$
By performing eigenvalue decomposition on the covariance matrix, $\hat{C}=$ $\left(\hat{c}{i j}\right)$, we obtain $$\hat{C}=V \Lambda V^{T}=\sum{k=1}^{7} \lambda_{k} \Delta \tau \mathbf{v}{k} \mathbf{v}{k}^{T},$$
where $\Lambda=\Delta \tau \operatorname{diag}\left(\lambda_{1}, \lambda_{2}, \ldots, \lambda_{7}\right)$ and $V=\left(\mathbf{v}{1}, \mathbf{v}{2}, \ldots, \mathbf{v}{7}\right)$ are eigenvalue and eigenvector matrices, respectively, the $\lambda . s$ are put in descending order, that is, $\lambda{1} \geq \lambda_{2} \geq \cdots \geq \lambda_{7}$, and the $\mathbf{v}{k}$ s are normalized, $\left|\mathbf{v}{k}\right|=1$, $k=1, \ldots, 7$. These eigenvectors $\left{\mathbf{v}{1}, \mathbf{v}{2}, \ldots, \mathbf{v}{7}\right}$ are also called principal components of $\hat{C}$. In terms of components, Equation $4.49$ reads as $$\hat{c}{i j}=\sum_{k=1}^{7} \lambda_{k} \Delta \tau v_{i k} v_{j k} .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|A CASE STUDY WITH A TWO-FACTOR MODEL

Yields or forward rates of different maturities are not perfectly correlated, which is made evident in the last section with historical data. In this section, we demonstrate how to parameterize the forward-rate volatility to capture the stylized features of the forward-rate curves that are shaped by principal components.

We consider a two-factor HJM model $(v=2)$ with the following forward-rate volatility components:
\begin{aligned} &\sigma_{1}(T)=a \mathrm{e}^{-k_{1} T} \ &\sigma_{2}(T)=b\left(1-2 \mathrm{e}^{-k_{2} T}\right) \end{aligned}
where $a, b, k_{1}$, and $k_{2}$ are constants. To get a “flat” $\sigma_{1}(T)$ and a “tilted” $\sigma_{2}(T)$, we choose $k_{1}$ and $k_{2}$ such that $0 \leq k_{1} \ll 1, k_{1} \ll k_{2}$. Similar to Avellaneda and Laurence (1999), we consider the following choice of parameters
$$a=0.008, \quad b=0.003, \quad k_{1}=0.0, \quad \text { and } \quad k_{2}=0.35 .$$

Note that if we take $b=0$, this two-factor model reduces to the Hull-White model, under which forward rates of all maturities are perfectly correlated.
Let us examine the correlation between the three-month (i.e., shortterm) and 30-year (long-term) forward rates. In general, the covariance between forward rates of two maturities, $T$ and $T^{\prime}$, is calculated according to
$$c\left(T, T^{\prime}\right)=\sigma_{1}(T) \sigma_{1}\left(T^{\prime}\right)+\sigma_{2}(T) \sigma_{2}\left(T^{\prime}\right)$$
The correlation between forward rates of two maturities is thus
$$\rho\left(T, T^{\prime}\right)=\frac{c\left(T, T^{\prime}\right)}{\sqrt{c(T, T)} \sqrt{c\left(T^{\prime}, T^{\prime}\right)}} .$$
Taking $T=0.25$ (three-month) and $T^{\prime}=30$, we have
\begin{aligned} c(0.25,0.25) &=7.024 \times 10^{-5} \ c(0.25,30) &=5.651 \times 10^{-5} \ c(30,30) &=7.300 \times 10^{-5} \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|MONTE CARLO IMPLEMENTATIONS

We now consider the application of the HJM model to derivatives pricing. As a demonstration, we consider the pricing of a bond option that matures at $T_{0}$ with payoff
$$X_{T_{0}}=\left(\sum_{i=1}^{n} \Delta T \cdot c \cdot P_{T_{0}}^{T_{i}}+P_{T_{0}}^{T_{n}}-K\right)^{+}$$
Here $c$ is the coupon rate of the bond, $K$ the strike price of the option, and $T_{i}=T_{0}+i \Delta T$ the cash flow date of the $i$ th coupon of the underlying bond. We call $T_{n}-T_{0}$, the life of the underlying bond beyond $T_{0}$, the tenor of the bond. The value of the option is given by
\begin{aligned} V_{0} &=E^{\mathrm{Q}}\left[\frac{1}{B_{T_{0}}}\left(\sum_{i=1}^{n} \Delta T \cdot c \cdot P_{T_{0}}^{T_{i}}+P_{T_{0}}^{T_{n}}-K\right)^{+} \mid \mathcal{F}{0}\right] \ &=E^{\mathrm{Q}}\left[\left(\sum{i=1}^{n} \Delta T c \cdot \frac{P_{T_{0}}^{T_{i}}}{B_{T_{0}}}+\frac{P_{T_{0}}^{T_{n}}}{B_{T_{0}}}-\frac{K}{B_{T_{0}}}\right)^{+} \mid \mathcal{F}{0}\right] \end{aligned} where $\mathbb{Q}$ is the risk-neutral measure. Based on Equation 4.25, we have the following expression for the discounted value of zero-coupon bonds: $$\frac{P{T_{0}}^{T_{i}}}{B_{T_{0}}}=P_{0}^{T_{i}} \exp \left(\int_{0}^{T_{0}}-\frac{1}{2}\left|\boldsymbol{\Sigma}\left(t, T_{i}\right)\right|^{2} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}\left(t, T_{i}\right) \mathrm{d} \tilde{\mathbf{W}}_{t}\right),$$ for $i=0,1, \ldots, n$. Taking $i=0$, in particular, we obtain the expression for the reciprocal of the money market account:
$$\frac{1}{B_{T_{0}}}=P_{0}^{T_{0}} \exp \left(\int_{0}^{T_{0}}-\frac{1}{2}\left|\mathbf{\Sigma}\left(t, T_{0}\right)\right|^{2} \mathrm{~d} t+\mathbf{\Sigma}^{\mathrm{T}}\left(t, T_{0}\right) \mathrm{d} \tilde{\mathbf{W}}_{t}\right)$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Principal Component Analysis

$T_{i}, i=1,2, \ldots, 7$ ，在哪里 $\Delta \tau=1 / 12$ 表示一个月的观尓间隔，并且 $N$ 总月数。对于每 个期限的远期利率, $T_{i}, i=1,2, \ldots, 7$ ，我们计算变化 $\Delta \tau$ :
$$\Delta f_{n, i}=f\left((n+1) \Delta \tau,(n+1) \Delta \tau+T_{i}\right) \quad-f\left(n \Delta \tau, n \Delta \tau+T_{i}\right), \quad n=0,1, \ldots, N-1$$
$$\hat{c} i j=\frac{1}{N} \sum n=0^{N-1}\left(\Delta f_{n, i}-\overline{\Delta f_{i}}\right)\left(\Delta f_{n, j}-\overline{\Delta f_{j}}\right)$$

$$\overline{\Delta f_{i}}=\frac{1}{N} \sum_{n=0}^{N-1} \Delta f_{n, i}$$

$$\hat{C}=V \Lambda V^{T}=\sum k=1^{7} \lambda_{k} \Delta \tau \mathbf{v} k \mathbf{v} k^{T},$$

$|\mathbf{v} k|=1, k=1, \ldots, 7$. 这些特征向量
$\mathrm{~ l l e f t : U m a t h b f { v } { 1 } , ~ I m a t h b f { v } { 2 } , 1 d o t 5 ,}$

$$\hat{c} i j=\sum_{k=1}^{7} \lambda_{k} \Delta \tau v_{i k} v_{j k}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|A CASE STUDY WITH A TWO-FACTOR MODEL

$$\sigma_{1}(T)=a \mathrm{e}^{-k_{1} T} \quad \sigma_{2}(T)=b\left(1-2 \mathrm{e}^{-k_{2} T}\right)$$

$$a=0.008, \quad b=0.003, \quad k_{1}=0.0, \quad \text { and } \quad k_{2}=0.35 .$$

$$c\left(T, T^{\prime}\right)=\sigma_{1}(T) \sigma_{1}\left(T^{\prime}\right)+\sigma_{2}(T) \sigma_{2}\left(T^{\prime}\right)$$

$$\rho\left(T, T^{\prime}\right)=\frac{c\left(T, T^{\prime}\right)}{\sqrt{c(T, T)} \sqrt{c\left(T^{\prime}, T^{\prime}\right)}}$$

$$c(0.25,0.25)=7.024 \times 10^{-5} c(0.25,30) \quad=5.651 \times 10^{-5} c(30,30)=7.300 \times 10^{-5}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|MONTE CARLO IMPLEMENTATIONS

$$X_{T_{0}}=\left(\sum_{i=1}^{n} \Delta T \cdot c \cdot P_{T_{0}}^{T_{i}}+P_{T_{0}}^{T_{n}}-K\right)^{+}$$

$$V_{0}=E^{\mathrm{Q}}\left[\frac{1}{B_{T_{0}}}\left(\sum_{i=1}^{n} \Delta T \cdot c \cdot P_{T_{0}}^{T_{i}}+P_{T_{0}}^{T_{n}}-K\right)^{+} \mid \mathcal{F} 0\right] \quad=E^{Q}\left[\left(\sum i=1^{n} \Delta T c \cdot \frac{P_{T_{0}}^{T_{i}}}{B_{T_{0}}}+\frac{P_{T_{0}}^{T_{n}}}{B_{T_{0}}}-\frac{K}{B_{T_{0}}}\right)\right.$$

$$\frac{P T_{0}^{T_{i}}}{B_{T_{0}}}=P_{0}^{T_{i}} \exp \left(\int_{0}^{T_{0}}-\frac{1}{2}\left|\boldsymbol{\Sigma}\left(t, T_{i}\right)\right|^{2} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}\left(t, T_{i}\right) \mathrm{d} \tilde{\mathbf{W}}{t}\right),$$ 为了 $i=0,1, \ldots, n$. 服用 $i=0$ ，特别是，我们得到货币市场账户倒数的表达式: $$\frac{1}{B{T_{0}}}=P_{0}^{T_{0}} \exp \left(\int_{0}^{T_{0}}-\frac{1}{2}\left|\boldsymbol{\Sigma}\left(t, T_{0}\right)\right|^{2} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}\left(t, T_{0}\right) \mathrm{d} \tilde{\mathbf{W}}_{t}\right)$$

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