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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|Duration and Convexity

In the bond markct, 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.+\operatorname{Pr} \cdot(1+y \Delta T)^{-\left(T_{n}-t\right) / \Delta T}\left(T_{n}-t\right)\right] \end{aligned}
In terms of $D_{\text {mac }}$, the Macaulay duration just defined, we have
$$\frac{\mathrm{d} B_{t}^{c}}{B_{t}^{c}}=-\frac{D_{\text {mac }}}{1+y \Delta T} \mathrm{~d} y \quad \text { or } \quad \frac{1}{B_{t}^{c}} \frac{\mathrm{d} B_{t}^{c}}{\mathrm{~d} y}=-\frac{D_{\text {mac }}}{1+y \Delta T} .$$

金融代写|利率建模代写Interest Rate Modeling代考|Portfolio Risk Management

We can also calculate the duration and convexity of a portfolio of fixedincome instruments. Consider a portfolio of $N$ instruments, with $n_{i}$ units and price $B_{i}^{c}$ for the $i$ th instrument. Then, the absolute change in the portfolio value upon a parallel yield shift is given by
$$\mathrm{d} V=\sum_{i} n_{i} \mathrm{~d} B_{i}^{c}=\sum_{i} n_{i} B_{i}^{c} \cdot\left(-D_{\mathrm{mod}}^{i} \mathrm{~d} y+\frac{1}{2} C^{i} \mathrm{~d} y^{2}\right)$$
The percentage change is then
$$\frac{\mathrm{d} V}{V}=-\left(\sum_{i} x_{i} D_{\bmod }^{t}\right) \mathrm{d} y+\frac{1}{2}\left(\sum_{i} x_{i} C^{t}\right) \mathrm{d} y^{2}$$
where $x_{i}=n_{i} B_{i}^{c} / V$ is the percentage of the value in the $i$ th instrument. Equation $3.32$ indicates that the duration and convexity of a portfolio are the weighted average of the duration and convexity of its components, respectively.

In classical risk management, a portfolio manager can limit his/her exposure to interest-rate risk by reducing the duration while increasing the convexity of the portfolio. To avoid possible losses in case of large yield moves, the manager usually will not tolerate negative net convexity. A portfolio with very small duration is called a duration-neutral portfolio. Practically, interest-rate futures and swaps are often used as hedging instruments for duration management.

The basic premise of the duration and convexity technology is that the yield curves shift in parallel, either upward or downward by the same amount. This is, however, a very crude assumption about the yield curve movement, as, in reality, points in a yield curve do not often shift by the same amount and sometimes they do not even move in the same direction. For a more elaborate model of yield curve dynamics, we will have to resort to stochastic calculus in a multi-factor setting.

金融代写|利率建模代写Interest Rate Modeling代考|LOGNORMAL MODEL: THE STARTING POINT

The theoretical basis of this chapter starts from the usual assumption of lognormal asset dynamics for zero-coupon bonds of all maturities:
$$\mathrm{d} P(t, T)=P(t, T)\left[\mu(t, T) \mathrm{d} t+\mathbf{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \mathbf{W}{t}\right]$$ under the physical measure, $\mathbb{P}$. Here $\mu(t, T)$ is a scalar function of $t$ and $T$, $\Sigma(t, T)$ is a column vector, $$\Sigma(t, T)=\left(\Sigma{1}(t, T), \Sigma_{2}(t, T), \ldots, \Sigma_{n}(t, T)\right)^{\mathrm{T}}$$
and $\mathbf{W}{t}$ is an $n$-dimensional $\mathbb{P}$-Brownian motion, $$\mathbf{W}{t}=\left(W_{1}(t), W_{2}(t), \ldots, W_{n}(t)\right)^{\mathrm{T}}$$
In principle, the coefficients in Equation 4.1 can be estimated from time series data of zero-coupon bonds, yet it is not guaranteed that Equation $4.1$ with estimated drift and volatility functions can exclude arbitrage. For the time being, we assume that both $\mu(t, T)$ and $\Sigma(t, T)$ are sufficiently regular deterministic functions on $t$, so that the SDE (Equation 4.1) admits a unique strong solution.

The purpose of a model like Equation $4.1$ is to price derivatives depending on (a portfolio of) $P(t, T), \forall T$ and $t \leq T$. For this purpose, we need to find a martingale measure for zero-coupon bonds of all maturities. Similar to our discussions on the multiple-asset market, we define an $\mathcal{F}{t}$-adaptive process, $\boldsymbol{\gamma}{t}$, that satisfies the following equation:
$$\boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \gamma_{t}=\mu(t, T)-r_{t} \mathbf{I}$$
Suppose that such a $\gamma_{t}$ exists, is independent of $T$, and satisfies the Novikov condition. We can define a measure, Q, as
$$\left.\frac{\mathrm{d} \mathbb{Q}}{\mathrm{dP}}\right|{\mathcal{F}{t}}=\exp \left(\int_{0}^{t}-\gamma_{s}^{\mathrm{T}} \mathrm{d} \mathbf{W}{s}-\frac{1}{2}\left|\boldsymbol{\gamma}{s}\right|^{2} \mathrm{~d} s\right)$$
Then, by the CMG theorem, the process
$$\tilde{\mathbf{W}} t=\mathbf{W}{t}+\int{0}^{t} \gamma_{s} d s$$ is a $\mathbb{Q}$-Brownian motion, and, in terms of $\tilde{\mathbf{W}}{t}$, we can rewrite Equation $4.1$ as \begin{aligned} \mathrm{d} P(t, T) &=P(t, T)\left[r{t} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}(t, T)\left(\mathrm{d} \mathbf{W}{t}+\gamma{t} \mathrm{~d} t\right)\right] \ &=P(t, T)\left[r_{t} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}}_{t}\right] \end{aligned}

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

$$D_{\operatorname{mac}}=\frac{\operatorname{Pr}}{B_{t}^{c}}\left[\sum_{i, T_{i}>t}^{n} \Delta T \cdot c(1+y \Delta T)^{-\left(T_{i}-t\right) / \Delta T}\left(T_{i}-t\right) \quad+(1+y \Delta T)^{-\left(T_{n}-t\right) / \Delta T}\left(T_{n}-t\right)\right]$$

$$\frac{\mathrm{d} B_{t}^{c}}{\mathrm{~d} y}=-\frac{\operatorname{Pr}}{1+y \Delta T}\left[\sum_{\left.i ; T_{i}\right\rangle t}^{n} \Delta T \cdot c(1+y \Delta T)^{-\left(T_{i}-t\right) / \Delta T}\left(T_{i}-t\right) \quad+\operatorname{Pr} \cdot(1+y \Delta T)^{-\left(T_{n}-t\right) / \Delta T}\left(T_{n}-t\right)\right]$$

$$\frac{\mathrm{d} B_{t}^{c}}{B_{t}^{c}}=-\frac{D_{\mathrm{mac}}}{1+y \Delta T} \mathrm{~d} y \quad \text { or } \quad \frac{1}{B_{t}^{c}} \frac{\mathrm{d} B_{t}^{c}}{\mathrm{~d} y}=-\frac{D_{\mathrm{mac}}}{1+y \Delta T}$$

金融代写|利率建模代写Interest Rate Modeling代考|Portfolio Risk Management

$$\mathrm{d} V=\sum_{i} n_{i} \mathrm{~d} B_{i}^{c}=\sum_{i} n_{i} B_{i}^{c} \cdot\left(-D_{\mathrm{mod}}^{i} \mathrm{~d} y+\frac{1}{2} C^{i} \mathrm{~d} y^{2}\right)$$

$$\frac{\mathrm{d} V}{V}=-\left(\sum_{i} x_{i} D_{\mathrm{mod}}^{t}\right) \mathrm{d} y+\frac{1}{2}\left(\sum_{i} x_{i} C^{t}\right) \mathrm{d} y^{2}$$

金融代写|利率建模代写Interest Rate Modeling代考|LOGNORMAL MODEL: THE STARTING POINT

$$\mathrm{d} P(t, T)=P(t, T)\left[\mu(t, T) \mathrm{d} t+\mathbf{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \mathbf{W} t\right]$$

$$\Sigma(t, T)=\left(\Sigma 1(t, T), \Sigma_{2}(t, T), \ldots, \Sigma_{n}(t, T)\right)^{\mathrm{T}}$$

$$\mathbf{W} t=\left(W_{1}(t), W_{2}(t), \ldots, W_{n}(t)\right)^{\mathrm{T}}$$

$$\mathbf{\Sigma}^{\mathrm{T}}(t, T) \gamma_{t}=\mu(t, T)-r_{t} \mathbf{I}$$

$$\frac{\mathrm{d} \mathbb{Q}}{\mathrm{dP}} \mid \mathcal{F} t=\exp \left(\int_{0}^{t}-\gamma_{s}^{\mathrm{T}} \mathrm{d} \mathbf{W} s-\frac{1}{2}|\gamma s|^{2} \mathrm{~d} s\right)$$

$$\tilde{\mathbf{W}} t=\mathbf{W} t+\int 0^{t} \gamma_{s} d s$$

$$\mathrm{d} P(t, T)=P(t, T)\left[r t \mathrm{~d} t+\mathbf{\Sigma}^{\mathrm{T}}(t, T)(\mathrm{d} \mathbf{W} t+\gamma t \mathrm{~d} t)\right] \quad=P(t, T)\left[r_{t} \mathrm{~d} t+\boldsymbol{\Sigma}^{\mathrm{T}}(t, T) \mathrm{d} \tilde{\mathbf{W}}_{t}\right]$$

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