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## 金融代写|利率建模代写Interest Rate Modeling代考|THE HJM MODEL

Under the martingale measure, $\mathbb{Q}$, the price process of a zero-coupon bond becomes
$$\mathrm{d} P(t, T)=P(t, T)\left[r_{t} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}}_{t}\right]$$
For the purpose of derivatives pricing, $\boldsymbol{\Sigma}(t, T)$ must satisfy at least the following additional conditions: (1) $\boldsymbol{\Sigma}(t, t)=0, \forall t ;$ and (2) $P(t, t)=1, \forall t$. These two conditions reflect only one fact: at maturity, the price of the zero-coupon bond equals its par value and thus has no volatility.

The specification of $\boldsymbol{\Sigma}(t, T)$ is a difficult job if we work directly with the process of $P(t, T)$. But the job will become quite amenable if we work with the process of forward rates. By Ito’s lemma, there is
$$\mathrm{d} \ln P(t, T)=\left[r_{t}-\frac{1}{2} \boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \boldsymbol{\Sigma}(t, T)\right] \mathrm{d} t+\boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}}_{t}$$

Assume, moreover, that $\boldsymbol{\Sigma}{T}(t, T)=\partial \boldsymbol{\Sigma}(t, T) / \partial T$ exists and $\int{0}^{T}\left|\boldsymbol{\Sigma}{T}(t, T)\right|^{2}$ $\mathrm{d} t<\infty$. By differentiating Equation $4.14$ with respect to $T$ and recalling that $$f(t, T)=-\frac{\partial \ln P(t, T)}{\partial T}$$ we obtain the process of forward rates under the $\mathbb{Q}$-measure, $$\mathrm{d} f(t, T)=\mathbf{\Sigma}{T}^{\mathrm{T}} \mathbf{\Sigma} \mathrm{d} t-\mathbf{\Sigma}{T}^{\mathrm{T}} \mathrm{d} \tilde{\mathbf{W}}{t}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|SPECIAL CASES OF THE HJM MODEL

Since the publication of the HJM model in 1992, arbitrage pricing models have quickly acquired dominant status in fixed-income modeling. Arbitrage pricing models have been generated from the HJM framework by making various specifications of forward-rate volatility. In this section, we study two specifications of the forward-rate volatility that, in terms of the short-rate dynamics, reproduce the popular one-factor models of Ho and Lee (1986) and Hull and White (1989), respectively.
4.3.1 The Ho-Lee Model
The simplest specification of the HJM model is $\sigma=$ const for $n=1$, corresponding to the forward-rate equation
$$\mathrm{d} f(t, T)=\sigma \mathrm{d} \tilde{W}{t}+\sigma^{2}(T-t) \mathrm{d} t$$ By integrating the equation over $[0, t]$, we obtain $$f(t, T)-f(0, T)=\sigma \tilde{W}{t}+\frac{1}{2} \sigma^{2} t(2 T-t) .$$
By making $T=t$, we have the expression for the short rate:
$$r_{t}=f(t, t)=f(0, t)+\frac{1}{2} \sigma^{2} t^{2}+\sigma \tilde{W}{t}$$ In differential form, the last equation becomes $$\mathrm{d} r{t}=\left(f_{T}(0, t)+\sigma^{2} t\right) \mathrm{d} t+\sigma \mathrm{d} \tilde{W}_{t} .$$
Equation $4.26$ is interpreted as the continuous-time version of the so-called Ho-Lee (1986) model, which was first developed in the context of binomial trees.

## 金融代写|利率建模代写Interest Rate Modeling代考|From a Yield Curve to a Forward-Rate Curve

The inputs for estimating the HJM model are historical data on U.S. Treasury yields, demonstrated in Figure $4.1$ with monthly quotes of yields for a 10-year period, from 1996 to 2006 . There are seven curves in the figure, which are dot plots of yield-to-maturities for 3-month, 6-month, 2-year, 3-year, 5-year, 10-year, or 30-year maturity benchmark U.S. Treasury bonds, respectively.

The first step of our model specification is to estimate the entire forward-rate curve, $f(\tau, \tau+T)$, for each day, $\tau$, in the data set and for the 30-year horizon, $0 \leq T \leq 30$. Since we do not have any detailed information about the benchmark Treasury bonds over this 10-year period, we treat the yields in the input data set for the last five maturities, $T_{3}=2$, $T_{4}=3, T_{5}=5, T_{6}=10$, and $T_{7}=30$, as par yiclds. ${ }^{*}$ The instantancous forward-rate curve for the day is determined by reproducing the value of the Treasury bills,
$$\frac{1}{\left(1+y_{i} \Delta T\right)^{1 / 2}}=P\left(\tau, \tau+T_{i}\right), \quad \text { for } i=1,2 \text {, }$$
and the value of the par bonds,
$$1=\sum_{j=1}^{n_{i}} y_{i}(\tau) \cdot \Delta T P(\tau, \tau+j \Delta T)+P\left(\tau, \tau+n_{i} \Delta T\right)$$
for $i=3, \ldots, 7$. Here $y_{i}(\tau)$ is the yield or par yield shown in Figure 4.1, $n_{i}=T_{i} / \Delta T, i=1,2, \ldots, 7, \Delta T=0.5$, and
$$P(\tau, \tau+T)=\mathrm{e}^{-\int_{\tau}^{\tau+T} f(\tau, s) \mathrm{ds}} .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|THE HJM MODEL

$$\mathrm{d} P(t, T)=P(t, T)\left[r_{t} \mathrm{~d} t+\mathbf{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}}{t}\right]$$ 就衍生品定价而言， $\boldsymbol{\Sigma}(t, T)$ 必须至少满足以下附加条件： (1) $\boldsymbol{\Sigma}(t, t)=0, \forall t ;(2) P(t, t)=1, \forall t$. 这两个条件只反映了一个事实: 在到期 时，零息债券的价格等于其面值，因此没有波动性。 规格 $\boldsymbol{\Sigma}(t, T)$ 如果我们直接处理 $P(t, T)$. 但是，如果我们使用远期利率的过程，这项工作将变得相当容易。根据伊藤引理，有 $$\mathrm{d} \ln P(t, T)=\left[r{t}-\frac{1}{2} \boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \boldsymbol{\Sigma}(t, T)\right] \mathrm{d} t+\boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}}_{t}$$

$$f(t, T)=-\frac{\partial \ln P(t, T)}{\partial T}$$

$$\mathrm{d} f(t, T)=\mathbf{\Sigma} T^{\mathrm{T}} \boldsymbol{\Sigma} \mathrm{d} t-\mathbf{\Sigma} T^{\mathrm{T}} \mathrm{d} \tilde{\mathbf{W}} t$$

## 金融代写|利率建模代写Interest Rate Modeling代考|SPECIAL CASES OF THE HJM MODEL

4.3.1 Ho-Lee 模型
$\mathrm{HJM}$ 模型最简单的说明是 $\sigma=$ 常量为 $n=1$ ，对应于远期利率方程
$$\mathrm{d} f(t, T)=\sigma \mathrm{d} \tilde{W} t+\sigma^{2}(T-t) \mathrm{d} t$$

$$f(t, T)-f(0, T)=\sigma \tilde{W} t+\frac{1}{2} \sigma^{2} t(2 T-t)$$

$$r_{t}=f(t, t)=f(0, t)+\frac{1}{2} \sigma^{2} t^{2}+\sigma \tilde{W} t$$

$$\mathrm{d} r t=\left(f_{T}(0, t)+\sigma^{2} t\right) \mathrm{d} t+\sigma \mathrm{d} \tilde{W}_{t}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|From a Yield Curve to a Forward-Rate Curve

$$\frac{1}{\left(1+y_{i} \Delta T\right)^{1 / 2}}=P\left(\tau, \tau+T_{i}\right), \quad \text { for } i=1,2$$

$$1=\sum_{j=1}^{n_{i}} y_{i}(\tau) \cdot \Delta T P(\tau, \tau+j \Delta T)+P\left(\tau, \tau+n_{i} \Delta T\right)$$

$$P(\tau, \tau+T)=\mathrm{e}^{-\int_{\tau}^{\tau+T} f(\tau, s) \mathrm{ds}}$$

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