### 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|The Martingale Representation Theorem

The martingale representation theorem plays a critical role in the so-called martingale approach to derivatives pricing. This theorem has two important

consequences. First, it leads to a general principle for derivatives pricing. Second, it implies a replication or hedging strategy of a derivative using its underlying security. We first present a simple version of the theorem based on a single Brownian filtration, $\mathcal{F}{t}=\sigma\left(W{s}, 0 \leq s \leq t\right)$. We begin with a martingale process, $M_{t}$, such that
$$\mathrm{d} M_{t}=\sigma_{t} \mathrm{~d} W_{t},$$
and we call $\sigma_{t}$ the volatility of $M_{t}$.
Theorem 2.3.1 (The Martingale Representation Theorem). Suppose that $N_{t}$ is a $\mathbb{Q}$-martingale process that is adaptive to $\mathcal{F}{t}$ and satisfies $E^{\mathbb{Q}}\left[N{T}^{2}\right]<$ $\infty$ for some T. If the volatility of $M_{t}$ is non-zero almost surely, then there exists a unique $\mathcal{F}{t}-$ adaptive process, $\varphi{t}$, such that $E^{Q}\left[\int_{0}^{T} \varphi_{t}^{2} \sigma_{t}^{2} d t\right]<\infty$ almost surely, and
$$N_{t}=N_{0}+\int_{0}^{t} \varphi_{s} d M_{s}, \quad t \leq T$$
or, in differential form,
$$d N_{t}=\varphi_{t} d M_{t}$$
The proof combining the techniques of Steele $(2000)$ and $Ø k s e n d a l ~(2003)$ is provided in the appendix of this chapter. A different proof can be found in Korn and Korn (2000).

## 金融代写|利率建模代写Interest Rate Modeling代考|A Complete Market with Two Securities

We consider the first “complete market” in continuous time, which consists of a money market account and a risky security. The price processes for the two securities, $B_{t}$ and $S_{t}$, are assumed to be
\begin{aligned} \mathrm{d} B_{t} &=r_{t} B_{t} \mathrm{~d} t, & & B_{0}=1 \ \mathrm{~d} S_{t} &=S_{t}\left(\mu_{t} \mathrm{~d} t+\sigma_{t} \mathrm{~d} W_{t}\right), & S_{0} &=S_{0} \end{aligned}
Here, the volatility of the risky asset is $\sigma_{t} \neq 0$ almost surely, and the short rate, $r_{t}$, can be stochastic. Denote the discounted price of the risky asset as $Z_{t}=B_{t}^{-1} S_{t}$, which can be shown to follow the process
\begin{aligned} \mathrm{d} Z_{t} &=Z_{t}\left(\left(\mu_{t}-r_{t}\right) \mathrm{d} t+\sigma_{t} \mathrm{~d} W_{t}\right) \ &=Z_{t} \sigma_{t} \mathrm{~d}\left(W_{t}+\int_{0}^{t} \frac{\left(\mu_{s}-r_{s}\right)}{\sigma_{s}} \mathrm{~d} s\right) \end{aligned}
By introducing
$$\gamma_{t}=\frac{\mu_{t}-r_{t}}{\sigma_{t}}$$

which is $\mathcal{F}{t}$-adaptive, and by defining a new measure, Q, according to Equation $2.36$, we have $$\tilde{W}{t}=W_{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s,$$
which is a Q-Brownian motion. In terms of $\tilde{W}{t}, Z{t}$ satisfies
$$\mathrm{d} Z_{t}=\sigma_{t} Z_{t} \mathrm{~d} \tilde{W}_{t},$$
which is a lognormal Q-martingale. Recall that in the binomial model for option pricing, we also derived the martingale measure for the underlying security.

## 金融代写|利率建模代写Interest Rate Modeling代考|Replicating and Pricing of Contingent Claims

Let $X_{T}$ be a contingent claim (or option) with payoff day or maturity $T$. The claim is an $\mathcal{F}{T}$-adaptive function whose value depends on $\left{S{t}, 0 \leq t \leq T\right}$. Define first a $\mathbb{Q}$-martingale with the discounted payoff:
$$N_{t}=E^{Q}\left(B_{T}^{-1} X_{T} \mid \mathcal{F}{t}\right) .$$ Without loss of generality, we assume that $E^{\mathrm{Q}}\left[N{t}^{2}\right]<\infty$. According to the martingale representation theorem, there exists an $\mathcal{F}{t}$-adaptive function, $\varphi{t}$, such that
$$\mathrm{d} N_{t}=\varphi_{t} \mathrm{~d} Z_{t},$$
where $Z_{t}$, defined in the last section, is the discounted price of $S_{t}$. Next, we define
$$\psi_{t}=N_{t}-\varphi_{t} Z_{t} .$$
Consider now the portfolio with $\varphi_{t}$ units of the stock and $\psi_{t}$ units of the money market account, denoted as $\left(\varphi_{t}, \psi_{t}\right)$. According to the definition of $\psi_{t}$, the discount value of the replication portfolio is
$$\tilde{V}{t}=\varphi{t} Z_{t}+\psi_{t}=N_{t}$$
This portfolio has two important properties. First, at time $T$, when the option matures,
$$\tilde{V}{T}=N{T}=B_{T}^{-1} X_{T}$$
which suggests that the (discounted) value of the portfolio equals that of the option. In other words, the portfolio replicates the payoff of the contingent claim. Second, the replicating portfolio is a self-financing one, meaning that it can track the asset allocation, $\left(\varphi_{t}, \psi_{t}\right)$, without the need for either capital

infusion or capital withdrawal. In fact, based on Equations $2.45$ and $2.47$, we have
$$\mathrm{d} \tilde{V}{t}=\mathrm{d} N{t}=\varphi_{t} \mathrm{~d} Z_{t}$$
In terms of the spot value, $B_{t}$ and $S_{t}$, Equation $2.49$ becomes
\begin{aligned} \mathrm{d} V_{t} &=\mathrm{d}\left(\tilde{V}{t} B{t}\right) \ &=B_{t} \mathrm{~d} \tilde{V}{t}+\tilde{V}{t} \mathrm{~d} B_{t} \ &=B_{t} \varphi_{t} \mathrm{~d} Z_{t}+\left(\varphi_{t} Z_{t}+\psi_{t}\right) \mathrm{d} B_{t} \ &=\varphi_{t}\left(B_{t} \mathrm{~d} Z_{t}+Z_{t} \mathrm{~d} B_{t}\right)+\psi_{t} \mathrm{~d} B_{t} \ &=\varphi_{t} \mathrm{~d}\left(B_{t} Z_{t}\right)+\psi_{t} \mathrm{~d} B_{t} \ &=\varphi_{t} \mathrm{~d} S_{t}+\psi_{t} \mathrm{~d} B_{t} \end{aligned}
A direct consequence of the above equation is the equality
\begin{aligned} \varphi_{t+d t} S_{t+d t}+\psi_{t+d t} B_{t+d t} &=\varphi_{t} S_{t}+\psi_{t} B_{t}+\varphi_{t} \mathrm{~d} S_{t}+\psi_{t} \mathrm{~d} B_{t} \ &=\varphi_{t} S_{t+d t}+\psi_{t} B_{t+d t}, \end{aligned}

## 金融代写|利率建模代写Interest Rate Modeling代考|The Martingale Representation Theorem

d米吨=σ吨 d在吨,

ñ吨=ñ0+∫0吨披sd米s,吨≤吨

dñ吨=披吨d米吨

## 金融代写|利率建模代写Interest Rate Modeling代考|A Complete Market with Two Securities

d乙吨=r吨乙吨 d吨,乙0=1  d小号吨=小号吨(μ吨 d吨+σ吨 d在吨),小号0=小号0

d从吨=从吨((μ吨−r吨)d吨+σ吨 d在吨) =从吨σ吨 d(在吨+∫0吨(μs−rs)σs ds)

C吨=μ吨−r吨σ吨

d从吨=σ吨从吨 d在~吨,

## 金融代写|利率建模代写Interest Rate Modeling代考|Replicating and Pricing of Contingent Claims

ñ吨=和问(乙吨−1X吨∣F吨).不失一般性，我们假设和问[ñ吨2]<∞. 根据鞅表示定理，存在一个F吨-自适应功能，披吨, 这样

dñ吨=披吨 d从吨,

ψ吨=ñ吨−披吨从吨.

$$\tilde{V} {T}=N {T}=B_{T}^{-1} X_{T}$$

d在~吨=dñ吨=披吨 d从吨

d在吨=d(在~吨乙吨) =乙吨 d在~吨+在~吨 d乙吨 =乙吨披吨 d从吨+(披吨从吨+ψ吨)d乙吨 =披吨(乙吨 d从吨+从吨 d乙吨)+ψ吨 d乙吨 =披吨 d(乙吨从吨)+ψ吨 d乙吨 =披吨 d小号吨+ψ吨 d乙吨

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