### 金融代写|利率建模代写Interest Rate Modeling代考|MATH3075

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## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model

The LIBOR market model was introduced by Miltersen et al. (1997), Brace et al. (1997; hereinafter, BGM), Musiela and Rutkowski (1997), and Jamshidian $(1997)$. The notable points of this model are listed here:

• The model has positive LIBOR.
• The model admits an arbitrary deterministic volatility structure.
• The price formulae of a caplet and a floorlet are derived so as to be consistent with the corresponding Black’s price.
• An approximated price formula for a swaption is derived.
From these, the LIBOR market model has a usability advantage in calibration, and so it is widely applied as a standard model for derivatives pricing. As a particular example, the BGM model is the most well-known type of LIBOR market model, and is built in the HJM framework. The BGM approach requires a kind of differentiability for LIBOR volatility. It is impossible to satisfy this smoothness in practice because the volatility cannot be constructed except as a piecewise continuous, but not necessarily smooth, function. Because of this, the BGM model is not strictly supported in the HJM framework. For more advanced study of this problem, see Yasuoka (2001, 2013b).

At one end of the spectrum of models, the approaches by Musiela and Rutkowski (1997) and Jamshidian (1997) stand on a martingale pricing theory, with no theoretical imperfections. However, their models are constructed under a risk-neutral measure without referring to the real-world measure.

In this section and the next, we introduce the LIBOR market model as described by Jamshidian (1997). Because the topic of this book is risk management, pricing of derivatives is not addressed here at length. For a more advanced treatment of pricing, readers are recommended to consult Brigo and Mercurio (2007) or Gatarek et al. (2007).

Similarly to the argument for the HJM model, when the LIBOR and bond prices are represented under a risk-neutral measure, we call the resulting system a risk-neutral model. When, instead, they are represented under $\mathbf{P}$, the resulting system is referred to as a real-world model. Strict definitions of these terms will be given later.

## 金融代写|利率建模代写Interest Rate Modeling代考|Existence of LIBOR Market Model

The existence of the LIBOR model is shown in the following theorem.
Theorem 5.2.1 For arbitrary deterministic volatility $\lambda_{i}(t), i=1, \cdots, n-1$, the LIBOR market model exists.

The LIBOR model can be constructed under any of several risk-neutral measures. Applying this, we will show the existence of the LIBOR model under the real-world measure in the next section, and show how the models are implied under other measures in Sections $5.4$ and $5.5$ of this chapter. It is thought that this approach is the simplest method of constructing the LIBOR market model for practical use. Therefore, we here only sketch Jamshidian’s LIBOR market model under a forward measure, omitting the proof.

Let each of $\lambda_{i}(t)$ be an arbitrary deterministic function in $t$ for $i=1, \cdots, n-$

1. Consider the following equation:
$$\frac{d L_{i}(t)}{L_{i}(t)}=\sum_{j=i+1}^{n-1} \frac{\delta_{j} L_{j}(t) \lambda_{i}(t) \lambda_{j}(t)}{1+\delta_{j} L_{j}} d t+\lambda_{i}(t) d Z_{t}$$
Here, $Z_{t}$ is a $d$-dimensional Brownian motion with respect to a measure $\mathbf{Q}(\sim$ $\mathbf{P})$. With this setup, the following proposition is given in Jamshidian ( 1997 , Corollary 2.1).

Proposition 5.2.2 The equation (5.4) admits a unique positive solution for an arbitrary initial condition $L_{i}(0)>0$ for all i. Further, $Y_{i}(t)=(1+$ $\left.\delta_{i} L_{i}(t)\right) \cdots\left(1+\delta_{i} L_{n-1}(t)\right)$ is a $\mathbf{Q}$-martingale.
Let $B_{n}(t)$ be an arbitrary bond price process such that $B_{n}\left(T_{n}\right)=1$ and
$$B_{n}\left(T_{i}\right)=\frac{1}{\prod_{j=i}^{n-1}\left(1+\delta_{j} L_{j}\left(T_{j}\right)\right)}$$ at each $T_{i}$. Accordingly, we define $B_{i}(t)$ for $i<n$ by
$$\frac{B_{i}(t)}{B_{n}(t)}=\prod_{j=i}^{n-1}\left(1+\delta_{j} L_{j}(t)\right)$$
From these, we see that $B_{i}\left(T_{i}\right)=1$ and the relation (5.2) is satisfied for all $i$. By Proposition 5.2.2, $\prod_{j=i}^{n-1}\left(1+\delta_{j} L_{j}(t)\right)$ is a Q-martingale for every $i$. Hence $B_{i}(t) / B_{n}(t)$ is a Q-martingale for all $i$.

Along these lines, $\mathbf{Q}$ is a $B_{n}$ numéraire measure and is referred to as a forward measure. As a result, the bond market $\mathcal{B}$ is arbitrage-free from Theorem $3.2 .2$

## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model under a Real-world Measure

Within the same setting as in Sections $5.1$ and 5.2, we give a definition of the LMRW and show the existence of the model, following Yasuoka (2013a).

Definition 5.3 The bond market $\mathcal{B}$ is called the $L M R W$ when the following conditions are satisfied.

1. The LIBOR processes $L_{i}, i=1, \cdots, n$, with $L_{i}(t)>0$, are represented under the real-world measure $\mathbf{P}$ such that each volatility $\lambda_{i}(t)$ and the market price of risk $\varphi_{t}$ are deterministic in $t$.
1. The bond market $\mathcal{B}$ is arbitrage-free; here this means that $B_{i}(t) \in \mathcal{B}, i=$ $1, \cdots, n$ and the state price deflator $\xi_{t}$ are positive Ito processes represented under $\mathbf{P}$.

For this, we define a left-continuous function $m(t)$ by $m(t)=j$, while $t \in$ $\left(T_{j-1}, T_{j}\right]$. Succinctly, $m(t)$ represents the index of the next maturity date $.$ Examination of Fig. $5.1$ may help to see the features of $m(t)$.

To show the existence of the LMRW, it is sufficient to give the simplest example for arbitrarily given volatility $\lambda$ and market price of risk $\varphi$. For this, we define a process $\bar{\mu}(t)$ by $\bar{\mu}(t)=\bar{\mu}\left(T_{m(t)}\right)$ such that
$$\bar{\mu}\left(T_{i}\right)=\frac{1}{\delta_{i-1}} \log \left{1+\delta_{i-1} L_{i-1}\left(T_{i-1}\right)\right}$$
at each time $T_{i}$. Specifically, $\bar{\mu}(t)$ represents the yield for the shortest maturity bond, with the next maturity $T_{m(t)}$. As a consequence, $\bar{\mu}(t)$ is constant on each period $\left(T_{i-1}, T_{i}\right], i=1, \cdots, n$.

Let $\varphi_{t}$ be an arbitrarily given market price of risk such that $\varphi_{l}$ is an $\mathbf{R}^{d}-$ valued deterministic function with
$$\int_{0}^{T}\left|\varphi_{t}\right|^{2} d s<\infty$$ Let $\lambda_{i}(t), i=1, \cdots, n$ be deterministic volatilities. We set $\chi_{i}(t)$ as $$\chi_{i}(t)=\frac{\lambda_{i}(t) \delta_{i} L_{i}(t)}{1+\delta_{i} L_{i}(t)} ; i=1, \cdots, n .$$ Consider the following equation with the initial LIBOR $L_{i}(0)>0$,
$$\frac{d L_{i}(t)}{L_{i}(t)}=\left{\lambda_{i}(t) \sum_{j=m(t)}^{i} \chi_{j}(t)+\lambda_{i}(t) \varphi_{t}\right} d t+\lambda_{i}(t) d W_{t}$$
for $i=1, \cdots, n$. It is known that the solution $L_{i}(t)$ exists uniquely and $L_{i}(t)>$ 0 . We assume that bond price processes $B_{i}(t), i=1, \cdots, n$ are Ito processes with initial values $B_{0}(0)=1$ and
$$B_{i}(0)=\prod_{j=0}^{i-1}\left(1+\delta_{j} L_{j}(0)\right)^{-1}$$ such that
$$\frac{d B_{i}(t)}{B_{i}(t)}=\left{\bar{\mu}(t)-\sum_{j=m(t)}^{i-1} \chi_{j}(t) \varphi_{t}\right} d t-\sum_{j=m(t)}^{i-1} \chi_{j}(t) d W_{t} .$$
Under this setup, we give the following theorem, which shows the existence of the LMRW.

## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model

LIBOR 市场模型由 Miltersen 等人引入。（1997 年），布雷斯等人。（1997 年；以下称为 BGM）、Musiela 和 Rutkowski（1997 年）和 Jamshidian(1997). 此处列出了此模型的值得注意的点：

• 该模型具有正 LIBOR。
• 该模型允许任意确定的波动率结构。
• 导出caplet和floorlet的价格公式以与对应的布莱克价格一致。
• 导出了互换期权的近似价格公式。
由此可见，LIBOR 市场模型在校准方面具有可用性优势，因此被广泛用作衍生品定价的标准模型。作为一个具体的例子，BGM 模型是最知名的 LIBOR 市场模型类型，并且构建在 HJM 框架中。BGM 方法需要一种 LIBOR 波动性的可微性。在实践中不可能满足这种平滑性，因为波动率只能构建为分段连续但不一定平滑的函数。因此，HJM 框架并不严格支持 BGM 模型。有关此问题的更高级研究，请参阅 Yasuoka (2001, 2013b)。

## 金融代写|利率建模代写Interest Rate Modeling代考|Existence of LIBOR Market Model

LIBOR 模型的存在如下定理所示。

LIBOR 模型可以在多种风险中性措施中的任何一种下构建。应用这一点，我们将在下一节中展示 LIBOR 模型在真实世界度量下的存在，并展示模型在其他度量下的隐含部分5.4和5.5本章的。这种方法被认为是构建 LIBOR 市场模型以供实际使用的最简单方法。因此，我们这里只勾勒出Jamshidian 的LIBOR 市场模型在前向测度下，省略了证明。

1. 考虑以下等式：
d大号一世(吨)大号一世(吨)=∑j=一世+1n−1dj大号j(吨)λ一世(吨)λj(吨)1+dj大号jd吨+λ一世(吨)d从吨
这里，从吨是一个d关于测度的一维布朗运动问(∼ 磷). 通过这种设置，Jamshidian (1997, Corollary 2.1) 给出了以下命题。

## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model under a Real-world Measure

1. LIBOR 流程大号一世,一世=1,⋯,n， 和大号一世(吨)>0, 在真实世界的度量下表示磷这样每个波动率λ一世(吨)和风险的市场价格披吨是确定性的吨.
2. 债券市场乙无套利；这意味着乙一世(吨)∈乙,一世= 1,⋯,n和国家价格平减指数X吨是正 Ito 过程表示下磷.

\bar{\mu}\left(T_{i}\right)=\frac{1}{\delta_{i-1}} \log \left{1+\delta_{i-1} L_{i-1 }\left(T_{i-1}\right)\right}\bar{\mu}\left(T_{i}\right)=\frac{1}{\delta_{i-1}} \log \left{1+\delta_{i-1} L_{i-1 }\left(T_{i-1}\right)\right}

∫0吨|披吨|2ds<∞让λ一世(吨),一世=1,⋯,n是确定性的波动率。我们设置χ一世(吨)作为

χ一世(吨)=λ一世(吨)d一世大号一世(吨)1+d一世大号一世(吨);一世=1,⋯,n.考虑以下具有初始 LIBOR 的等式大号一世(0)>0,

\frac{d L_{i}(t)}{L_{i}(t)}=\left{\lambda_{i}(t) \sum_{j=m(t)}^{i} \chi_{ j}(t)+\lambda_{i}(t) \varphi_{t}\right} d t+\lambda_{i}(t) d W_{t}\frac{d L_{i}(t)}{L_{i}(t)}=\left{\lambda_{i}(t) \sum_{j=m(t)}^{i} \chi_{ j}(t)+\lambda_{i}(t) \varphi_{t}\right} d t+\lambda_{i}(t) d W_{t}

\frac{d B_{i}(t)}{B_{i}(t)}=\left{\bar{\mu}(t)-\sum_{j=m(t)}^{i-1 } \chi_{j}(t) \varphi_{t}\right} d t-\sum_{j=m(t)}^{i-1} \chi_{j}(t) d W_{t} 。\frac{d B_{i}(t)}{B_{i}(t)}=\left{\bar{\mu}(t)-\sum_{j=m(t)}^{i-1 } \chi_{j}(t) \varphi_{t}\right} d t-\sum_{j=m(t)}^{i-1} \chi_{j}(t) d W_{t} 。

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