### 金融代写|利率建模代写Interest Rate Modeling代考|ACTL40004

Vasicek利率模型一词是指一种对利率的运动和演变进行建模的数学方法。它是一种基于市场风险的单因素短利率模型。瓦西克利率模型常用于经济学中，以确定利率在未来的移动方向。

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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|Correlated Brownian Motions

Let $W(t)$ and $\tilde{W}(t)$ be two Brownian motions under the probability space $(\Omega, \mathcal{F}, \mathbb{P})$. We say that $W(t)$ and $\tilde{W}(t)$ are correlated if
$$\operatorname{Cov}[\Delta W(t), \Delta \bar{W}(t)]=\rho \Delta t, \quad \rho \neq 0$$
Equivalently, we can write
\begin{aligned} &W(t)=W_{1}(t) \ &\tilde{W}(t)=\rho W_{1}(t)+\sqrt{1-\rho^{2}} W_{2}(t) \end{aligned}

where $W_{1}(t)$ and $W_{2}(t)$ are independent Brownian motions. We have the following additional operation rule for correlated Brownian motions:
$$\mathrm{d} W(t) \mathrm{d} \tilde{W}(t)=\rho \mathrm{d} t .$$
With the help of the above operation rule, we can derive the processes of the product and quotient of two Ito’s processes. The following results are very useful for financial modeling, and for this reason we call them the product rule and the quotient rule, respectively.
Product rule: Let $X(t)$ and $Y(t)$ be two Ito’s processes such that
\begin{aligned} &\mathrm{d} X(t)=\sigma_{X}(t) \mathrm{d} W(t)+u_{X}(t) \mathrm{d} t \ &\mathrm{~d} Y(t)=\sigma_{Y}(t) \mathrm{d} W(t)+u_{Y}(t) \mathrm{d} t \end{aligned}
where $\mathrm{d} W(t) \mathrm{d} \tilde{W}(t)=\rho \mathrm{d} t$. Then,
\begin{aligned} \mathrm{d}(X(t) Y(t)) &=X(t) \mathrm{d} Y(t)+Y(t) \mathrm{d} X(t)+\mathrm{d} X(t) \mathrm{d} Y(t) \ &=X(t) \mathrm{d} Y(t)+Y(t) \mathrm{d} X(t)+\sigma_{X}(t) \sigma_{Y}(t) \rho \mathrm{d} t \end{aligned}
Quotient rule: Let $X(t)$ and $Y(t)$ be two Ito’s processes. Then,
$$\mathrm{d}\left(\frac{X(t)}{Y(t)}\right)=\frac{\mathrm{d} X(t)}{Y(t)}-\frac{X(t) \mathrm{d} Y(t)}{(Y(t))^{2}}-\frac{\mathrm{d} X(t) \mathrm{d} Y(t)}{(Y(t))^{2}}+\frac{X(t)(\mathrm{d} Y(t))^{2}}{(Y(t))^{3}}$$
The proofs for both rules are left as exercises.

## 金融代写|利率建模代写Interest Rate Modeling代考|The Multi-Factor Lognormal Model

As an important area of application for the multi-factor Ito’s lemma, we now introduce the classic model of a financial market with multiple assets. This financial market consists of a money market account (also called a savings account), $B_{t}$, and $n$ risky assets, $\left{S_{t}^{i}\right}_{i=1}^{n}$. The price evolutions of these $n+1$ assets are governed by the following equations:
\begin{aligned} \mathrm{d} B_{t} &=r_{t} B_{t} \mathrm{~d} t \ \mathrm{~d} S_{t}^{i} &=S_{t}^{i}\left(\mu_{t}^{i} \mathrm{~d} t+\sigma_{i}^{\mathrm{T}}(s) \mathrm{d} \mathbf{W}{t}\right), \quad i=1,2, \ldots, n . \end{aligned} Here $r{t}$ is the risk-free interest rate, $\mu_{t}^{i}$ and $\sigma_{i}(s)$ the rate of return and volatility of the $i$ th asset, and
$$\boldsymbol{\sigma}{i}(t)=\left(\begin{array}{c} \sigma{i 1}(t) \ \sigma_{i 2}(t) \ \vdots \ \sigma_{i n}(t) \end{array}\right) \quad \text { and } \quad \mathbf{W}{t}=\left(\begin{array}{c} W{1}(t) \ W_{2}(t) \ \vdots \ W_{n}(t) \end{array}\right)$$

Driving the market are $n$ independent Brownian motions. We therefore call the above model an $n$-factor model. Note that the savings account is considered a riskless asset so that it is not driven by any Brownian motion.

By the multi-factor Ito’s lemma, we can derive the equations for the log of asset prices:
$$\mathrm{d} \ln S_{t}^{i}=\left(\mu_{t}^{i}-\frac{1}{2}\left|\sigma_{i}(t)\right|^{2}\right) \mathrm{d} t+\sigma_{i}^{\mathrm{T}}(t) \mathrm{d} \mathbf{W}{t}$$ The above equation readily allows us to solve for the asset price: $$S{t}^{i}=S_{0}^{i} \exp \left(\int_{0}^{t} \sigma_{i}^{\mathrm{T}}(s) \mathrm{d} \mathbf{W}{s}+\left(\mu{s}^{i}-\frac{1}{2}\left|\sigma_{i}(s)\right|^{2}\right) \mathrm{d} s\right)$$
for $i=1,2, \ldots, n$. The value of the money market account, meanwhile, is simply
$$B_{t}=\exp \left(\int_{0}^{t} r_{s} \mathrm{~d} s\right)$$

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

We finish this chapter with the introduction of martingales, which is a key concept in derivatives modeling. The definition is given below.

Definition 1.5.1. A stochastic process, $M_{t}$, is called a $\mathbb{P}$-martingale if and only if it has the following properties:

1. $E^{\mathbb{P}}\left[\mid M_{t} |<\infty, \quad \forall t\right.$.
2. $E^{\mathbb{P}}\left[M_{t} \mid \mathcal{F}{s}\right]=M{s}, \quad \forall s \leq t$.
The martingale properties are associated with fair games in investments or speculations. Let us think of $M_{t}-M_{s}$ as the profit or loss $(\mathrm{P} \& \mathrm{~L})$ of a gamble between two parties over the time period $(s, t)$. Then the game is considered fair if the expected P\&L is zero. Daily life examples of fair games include the coin tossing game and futures investments in financial markets. In mathematics, there are plenty of examples as well. In fact, we have already seen several of them so far, of which we remind readers below.
Example 1.4
3. The simple random walk, $X_{n}$, is a martingale because $E\left[\left|X_{n}\right|\right]<n \sqrt{\Delta t}$ and $E\left[X_{n} \mid \mathcal{F}{m}\right]=X{m}, m \leq n$.
4. A P-Brownian motion, $W_{t}$, is a martingale by definition.
1. The stochastic integral $X_{t}=\int_{0}^{t} f(u) \mathrm{d} W_{u}$ is a martingale, since
\begin{aligned} E^{\mathrm{P}}\left[X_{t} \mid \mathcal{F}{s}\right] &=E^{\mathrm{P}}\left[\int{0}^{s}+\int_{s}^{t} f(u) \mathrm{d} W_{u} \mid \mathcal{F}{s}\right] \ &=\int{0}^{s} f(u) \mathrm{d} W_{u}=X_{s}, \quad \forall s \leq t \end{aligned}
Here, we have applied the first property of stochastic integrals (see page 11).
2. The process $M_{t}=\exp \left(\int_{0}^{t} \sigma_{s} \mathrm{~d} W_{s}-\frac{1}{2} \sigma_{s}^{2} \mathrm{~d} s\right)$ is an exponential martingale. In fact, using the Ito’s lemma, we can show that
$$\mathrm{d} M_{t}=\sigma_{t} M_{t} \mathrm{~d} W_{t}$$
which is an Ito’s process without drift. It follows that
$$M_{t}=M_{s}+\int_{s}^{t} M_{u} \sigma_{u} \mathrm{~d} W_{u} .$$
Based on the conclusion of the last example, we know that $M_{t}$ is a martingale.

We emphasize here that an Ito’s process is a martingale process if and only if its drift term is zero. Finally, we present two additional examples.

1. $M_{t}=W_{t}^{2}-t$ is a martingale. Here is the verification: for $s \leq t$,
\begin{aligned} E^{\mathbb{P}}\left[W_{t}^{2}-t \mid \mathcal{F}{s}\right]=& E^{\mathbb{P}}\left[\left(W{t}-W_{s}+W_{s}\right)^{2}-t \mid \mathcal{F}{s}\right] \ =& E^{P}\left[\left(W{t}-W_{s}\right)^{2}+2 W_{s}\left(W_{t}-W_{s}\right)\right.\ &\left.+W_{s}^{2}-t \mid \mathcal{F}{s}\right] \ =&(t-s)+0+W{s}^{2}-t=W_{s}^{2}-s \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|Correlated Brownian Motions

d在(吨)d在~(吨)=ρd吨.

dX(吨)=σX(吨)d在(吨)+在X(吨)d吨  d是(吨)=σ是(吨)d在(吨)+在是(吨)d吨

d(X(吨)是(吨))=X(吨)d是(吨)+是(吨)dX(吨)+dX(吨)d是(吨) =X(吨)d是(吨)+是(吨)dX(吨)+σX(吨)σ是(吨)ρd吨

d(X(吨)是(吨))=dX(吨)是(吨)−X(吨)d是(吨)(是(吨))2−dX(吨)d是(吨)(是(吨))2+X(吨)(d是(吨))2(是(吨))3

## 金融代写|利率建模代写Interest Rate Modeling代考|The Multi-Factor Lognormal Model

d乙吨=r吨乙吨 d吨  d小号吨一世=小号吨一世(μ吨一世 d吨+σ一世吨(s)d在吨),一世=1,2,…,n.这里r吨是无风险利率，μ吨一世和σ一世(s)收益率和波动率一世资产，以及

σ一世(吨)=(σ一世1(吨) σ一世2(吨) ⋮ σ一世n(吨)) 和 在吨=(在1(吨) 在2(吨) ⋮ 在n(吨))

dln⁡小号吨一世=(μ吨一世−12|σ一世(吨)|2)d吨+σ一世吨(吨)d在吨上面的等式很容易让我们求解资产价格：

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

1. 和磷[∣米吨|<∞,∀吨.
2. 和磷[米吨∣Fs]=米s,∀s≤吨.
鞅属性与投资或投机中的公平游戏有关。让我们想想米吨−米s作为损益(磷& 大号)在一段时间内两方之间的博弈(s,吨). 如果预期盈亏为零，则认为该游戏是公平的。日常生活中公平游戏的例子包括抛硬币游戏和金融市场的期货投资。在数学中，也有很多例子。事实上，到目前为止，我们已经看到了其中的几个，我们在下面提醒读者。
示例 1.4
3. 简单的随机游走，Xn, 是鞅，因为和[|Xn|]<nΔ吨和和[Xn∣F米]=X米,米≤n.
4. P-布朗运动，在吨, 根据定义是鞅。
1. 随机积分X吨=∫0吨F(在)d在在是鞅，因为
和磷[X吨∣Fs]=和磷[∫0s+∫s吨F(在)d在在∣Fs] =∫0sF(在)d在在=Xs,∀s≤吨
在这里，我们应用了随机积分的第一个性质（参见第 11 页）。
2. 过程米吨=经验⁡(∫0吨σs d在s−12σs2 ds)是指数鞅。事实上，使用伊藤引理，我们可以证明
d米吨=σ吨米吨 d在吨
这是一个没有漂移的伊藤工艺。它遵循
米吨=米s+∫s吨米在σ在 d在在.
根据上一个例子的结论，我们知道米吨是鞅。

1. 米吨=在吨2−吨是鞅。这是验证：对于s≤吨,
和磷[在吨2−吨∣Fs]=和磷[(在吨−在s+在s)2−吨∣Fs] =和磷[(在吨−在s)2+2在s(在吨−在s) +在s2−吨∣Fs] =(吨−s)+0+在s2−吨=在s2−s

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