### 金融代写|利率建模代写Interest Rate Modeling代考|MATH5985

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|A Motivating Example

Consider the simplest option-pricing model with an underlying asset following a one-period binomial process, as depicted in Figure 2.1. In Figure $2.1,0 \leq p \leq 1$ and $\bar{p}=1-p$. The option’s payoffs at time $1, f\left(S_{u}\right)$ and $f\left(S_{d}\right)$, are given explicitly, and we want to determine $f(S)$, the value of the option at time 0 . Without loss of generality, we assume that there is a zero interest rate in the model. To avoid arbitrage, we must impose the order $S_{d} \leq S \leq S_{u}$. We call $\mathbb{P}={p, \bar{p}}$ the objective measure of the underlying process.

It may be tempted to price the option by expectation under $\mathbb{P}$ :
\begin{aligned} f(S) &=E^{\mathbb{P}}\left[f\left(S_{1}\right)\right] \ &=p f\left(S_{u}\right)+\bar{p} f\left(S_{d}\right) \end{aligned}
However, except for a special $p$, the above price generates arbitrage and thus is wrong. To see that, we replicate the payoff of the option at time 1 using a portfolio of the underlying asset and a cash bond, with respective numbers of units, $\alpha$ and $\beta$, such that, at time 1 ,
\begin{aligned} &\alpha S_{u}+\beta=f\left(S_{u}\right) \ &\alpha S_{d}+\beta=f\left(S_{d}\right) \end{aligned}
Solving for $\alpha$ and $\beta$, we obtain
\begin{aligned} \alpha &=\frac{f\left(S_{u}\right)-f\left(S_{d}\right)}{S_{u}-S_{d}} \ \beta &=\frac{S_{u} f\left(S_{d}\right)-S_{d} f\left(S_{u}\right)}{S_{u}-S_{d}} \end{aligned}
Equation $2.2$ implies that the time-1 values of the portfolio and option are identical. To avoid arbitrage, their values at time 0 must be identical as well ${ }^{1}$ which yields the arbitrage price of the option at time 0 :
\begin{aligned} f(S) &=\alpha S+\beta \ &=q f\left(S_{u}\right)+\bar{q} f\left(S_{d}\right) \ &=E^{Q}\left[f\left(S_{1}\right)\right] \end{aligned}
where $\mathbb{Q}={q, \bar{q}}$, and
$$q=\frac{S-S_{d}}{S_{u}-S_{d}}, \quad \bar{q}=1-q$$

is a different set of probabilities. Note that Equation $2.4$ gives the no-arbitrage price of the option. Any other price will induce arbitrage to the market. Hence, the expectation price, in Equation 2.1, is correct only if $p=q$. In fact, ${q, \bar{q}}$ is the only set of probabilities that satisfies
$$S=q S_{u}+\bar{q} S_{d}=E^{\mathbb{Q}}\left(S_{1}\right)$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Binomial Trees and Path Probabilities

Let us move one step forward and consider the binomial tree model up to two time steps, as shown in Figure 2.2, where each pair of numbers represents a state (which can be associated with the price of an asset if necessary). Out of each state at time $j$, two possible states are generated at time $j+1$. Hence, we have $2^{j}$ states at time $j$, starting with a single state at time 0 . The branching probabilities for reaching the next two states from one state, $(i, j)$, are $p_{i, j} \in[0,1]$ and $\bar{p}{i, j}=1-p{i, j}$, respectively. The collection of branching probabilities, $\mathbb{P}=\left{p_{i, j}, \bar{p}{i, j}\right}$, is again called a measure. As is shown in Figure 2.2, there are two paths over the time horizon from 0 to 1 , whereas there are four paths over the time horizon from 0 to 2 . The corresponding path probabilities for the horizon from 0 to 1 are $$\pi{0,1}=\bar{p}{0,0} \quad \text { and } \quad \pi{1,1}=p_{0,0}$$
whereas for the horizon from 0 to 2 , they are
$$\pi_{0,2}=\bar{p}{0,0} \bar{p}{0,1}, \pi_{1,2}=\bar{p}{0,0} p{0,1}, \pi_{2,2}=p_{0,0} \bar{p}{1,1}, \text { and } \pi{3,2}=p_{0,0} p_{1,1}$$

Consider now another set of branching probabilities, $\mathbb{Q}=\left{q_{i, j}, \bar{q}{i, j}=\right.$ $\left.1-q{i, j}\right}$, for the same tree. The corresponding path probabilities are
$$\pi_{0,1}^{\prime}=\bar{q}{0,0} \quad \text { and } \quad \pi{1,1}^{\prime}=q_{0,0}$$
up to time 1 , and
$$\pi_{0,2}^{\prime}=\bar{q}{0,0} \bar{q}{0,1}, \pi_{1,2}^{\prime}=\bar{q}{0,0} q{0,1}, \pi_{2,2}^{\prime}=q_{0,0} \bar{q}{1,1}, \text { and } \pi{3,2}^{\prime}=q_{0,0} q_{1,1}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|The Radon-Nikodym Derivative of a Brownian Path

Consider a path of $\mathbb{P}$-Brownian motion over $(0, t)$ with discrete time stepping,
$${W(0)=0, W(\Delta t), W(2 \Delta t), \ldots, W(n \Delta t)}$$
where $\Delta t=t / n$. With the probability ratio in mind, our immediate question is what the path probability is. The answer, unfortunately, is zero. The implication that we cannot define the notion of the probability ratio given that the same path is realized under two different probability measures. To circumvent this problem, we first seek to calculate the probability for the Brownian motion to travel in a corridor (the so-called corridor probability), as is shown in Figure $2.5$, and then we define the ratio of the corridor probabilities. The ratio of the path probabilities is finally defined through a limiting procedure. The corridor can be represented by the intervals $A_{i}=\left(x_{i}-(\Delta x / 2), x_{i}+(\Delta x / 2)\right), i=1,2, \ldots, n$, where $x_{i}=W(i \Delta t)$ and $\Delta x>0$ is a small number.

For a Brownian motion, the marginal distribution at $t_{i}=i \Delta t$ is known to be
$$f_{\mathrm{P}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}\right)^{2} / \Delta t\right]} \sim N\left(x_{i}, \Delta t\right)$$

Hence, the probability for the next step to fall in $A_{i+1}$ is
\begin{aligned} \operatorname{Prob}{\mathrm{P}}\left(A{i+1}\right) &=\int_{x_{i+1}-\Delta x / 2}^{x_{i+1}+\Delta x / 2} f_{\mathrm{P}}(x) \mathrm{d} x \ & \approx f_{\mathrm{P}}\left(x_{i+1}\right) \Delta x=\frac{\Delta x}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x_{i+1}-x_{i}\right)^{2} / \Delta t\right]} . \end{aligned}
Approximately, we can define the corridor probability to be
$$\prod_{i=1}^{n} \operatorname{Prob}{2}\left(A{i}\right)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}\right)^{2}} .$$
Next, suppose that the same path is realized under a different marginal probability,
$$f_{\mathrm{Q}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}+\gamma \Delta t\right)^{2} / \Delta t\right]} \sim N\left(x_{i}-\gamma \Delta t, \Delta t\right), \quad \forall i,$$
where $\gamma$ is taken to be constant for simplicity. Then the corresponding corridor probability can be similarly obtained to be
$$\prod_{i=1}^{n} \operatorname{Prob}{\mathrm{Q}}\left(A{i}\right)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}+\gamma \Delta t\right)^{2}}$$
It follows that the ratio of the two corridor probabilities is
\begin{aligned} \zeta_{t} &=\exp \left(-\frac{1}{2 \Delta t} \sum_{i=0}^{n-1}\left[\left(x_{i+1}-x_{i}+\gamma \Delta t\right)^{2}-\left(x_{i+1}-x_{i}\right)^{2}\right]\right) \ &=\exp \left(-\frac{1}{2 \Delta t} \sum_{i=0}^{n-1}\left[2\left(x_{i+1}-x_{i}\right) \cdot \gamma \Delta t+\gamma^{2} \Delta t^{2}\right]\right) \ &=\exp \left(-\gamma \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}\right)-\frac{1}{2} \gamma^{2} \Delta t \cdot n\right) \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|A Motivating Example

F(小号)=和磷[F(小号1)] =pF(小号在)+p¯F(小号d)

F(小号)=一个小号+b =qF(小号在)+q¯F(小号d) =和问[F(小号1)]

q=小号−小号d小号在−小号d,q¯=1−q

## 金融代写|利率建模代写Interest Rate Modeling代考|The Radon-Nikodym Derivative of a Brownian Path

F磷(X)=12圆周率Δ吨和−(1/2)[(X−X一世)2/Δ吨]∼ñ(X一世,Δ吨)

∏一世=1n概率⁡2(一个一世)=(ΔX2圆周率Δ吨)n和−(1/2Δ吨)∑一世=0n−1(X一世+1−X一世)2.

F问(X)=12圆周率Δ吨和−(1/2)[(X−X一世+CΔ吨)2/Δ吨]∼ñ(X一世−CΔ吨,Δ吨),∀一世,

∏一世=1n概率⁡问(一个一世)=(ΔX2圆周率Δ吨)n和−(1/2Δ吨)∑一世=0n−1(X一世+1−X一世+CΔ吨)2

G吨=经验⁡(−12Δ吨∑一世=0n−1[(X一世+1−X一世+CΔ吨)2−(X一世+1−X一世)2]) =经验⁡(−12Δ吨∑一世=0n−1[2(X一世+1−X一世)⋅CΔ吨+C2Δ吨2]) =经验⁡(−C∑一世=0n−1(X一世+1−X一世)−12C2Δ吨⋅n)

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