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

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## 金融代写|利率建模代写Interest Rate Modeling代考|CHANGE OF MEASURES UNDER BROWNIAN FILTRATION

2.2.1 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}{\mathbb{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
$$\prod_{i=1}^{n} \operatorname{Prob}{\mathbb{P}}\left(A{i}\right)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}\right)^{2}}$$
Next, suppose that the same path is realized under a different marginal probability,
$$f_{\mathbb{Q}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}+\gamma \Delta t\right)^{2} / \Delta t\right]} \sim N\left(x_{i}-\gamma \Delta t, \Delta t\right), \quad \forall i$$

where $\gamma$ is taken to be constant for simplicity. Then the corresponding corridor probability can be similarly obtained to be
$$\prod_{i=1}^{n} \operatorname{Prob}{\mathrm{Q}}\left(A{i}\right)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}+\gamma \Delta t\right)^{2}}$$

## 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 socalled 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 2.2 (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 $\int_{0}^{T} \varphi_{t}^{2} \sigma_{t}^{2} \mathrm{~d} t<\infty$ almost surely, and
$$N_{t}=N_{0}+\int_{0}^{t} \varphi_{s} \mathrm{~d} M_{s}, \quad t \leq T$$
or, in differential form,
$$\mathrm{d} N_{t}=\varphi_{t} \mathrm{~d} M_{t} .$$
A sketchy proof along the lines of Steele (2000) is provided at the end 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, $\mathbb{Q}$, according to Equation 2.36, we have $$\tilde{W}{t}=W_{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s$$
which is a $\mathbb{Q}$-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 $\mathbb{Q}$-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|CHANGE OF MEASURES UNDER BROWNIAN FILTRATION

$2.2 .1$ 布朗路径的 Radon-Nikodym 导数

$$W(0)=0, W(\Delta t), W(2 \Delta t), \ldots, W(n \Delta t)$$

$A_{i}=\left(x_{i}-(\Delta x / 2), x_{i}+(\Delta x / 2)\right), i=1,2, \ldots, n$ ， 在哪里 $x_{i}=W(i \Delta t)$ 和 $\Delta x>0$ 是一个数字。

$$f_{\mathrm{P}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}\right)^{2} / \Delta t\right]} \sim N\left(x_{i}, \Delta t\right)$$

$$\operatorname{Prob} \mathbb{P}(A i+1)=\int_{x_{i+1}-\Delta x / 2}^{x_{i+1}+\Delta x / 2} f_{\mathrm{P}}(x) \mathrm{d} x \quad \approx f_{\mathrm{P}}\left(x_{i+1}\right) \Delta x=\frac{\Delta x}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x_{i+1}-x_{i}\right)^{2} / \Delta t\right]} .$$

$$\prod_{i=1}^{n} \operatorname{Prob} \mathbb{P}(A i)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}\right)^{2}}$$

$$f_{\mathbb{Q}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}+\gamma \Delta t\right)^{2} / \Delta t\right]} \sim N\left(x_{i}-\gamma \Delta t, \Delta t\right), \quad \forall i$$

$$\prod_{i=1}^{n} \operatorname{Prob} \mathrm{Q}(A i)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}+\gamma \Delta t\right)^{2}}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|THE MARTINGALE REPRESENTATION THEOREM

$$\mathrm{d} M_{t}=\sigma_{t} \mathrm{~d} W_{t},$$

$$N_{t}=N_{0}+\int_{0}^{t} \varphi_{s} \mathrm{~d} M_{s}, \quad t \leq T$$

$$\mathrm{d} N_{t}=\varphi_{t} \mathrm{~d} M_{t}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|A COMPLETE MARKET WITH TWO SECURITIES

$$\mathrm{d} B_{t}=r_{t} B_{t} \mathrm{~d} t, \quad B_{0}=1, \mathrm{~d} S_{t}=S_{t}\left(\mu_{t} \mathrm{~d} t+\sigma_{t} \mathrm{~d} W_{t}\right), S_{0}=S_{0}$$

$$\mathrm{d} Z_{t}=Z_{t}\left(\left(\mu_{t}-r_{t}\right) \mathrm{d} t+\sigma_{t} \mathrm{~d} W_{t}\right) \quad=Z_{t} \sigma_{t} \mathrm{~d}\left(W_{t}+\int_{0}^{t} \frac{\left(\mu_{s}-r_{s}\right)}{\sigma_{s}} \mathrm{~d} s\right)$$

$$\gamma_{t}=\frac{\mu_{t}-r_{t}}{\sigma_{t}}$$

$$\tilde{W} t=W_{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s$$

$$\mathrm{d} Z_{t}=\sigma_{t} Z_{t} \mathrm{~d} \tilde{W}_{t}$$

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