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

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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
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• 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|MARTINGALES

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

Definition $1.5$ A stochastic process, $M_{t}$, is called a $\mathbb{P}$-martingale if and only if it has the following properties:

1. $E^{\mathbb{P}}\left[\left|M_{t}\right|\right]<\infty, \quad \forall t$.
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 (P\&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 $E\left[X_{n} \mid \mathcal{F}{m}\right]=X{m}, m \leq n$
1. A $\mathbb{P}$-Brownian motion, $W_{t}$, is a martingale by definition.
2. The stochastic integral $X_{t}=\int_{0}^{t} f(u) \mathrm{d} W_{u}$ is a martingale, since
\begin{aligned} E^{\mathbb{P}}\left[X_{t} \mid \mathcal{F}{s}\right] &=E^{\mathbb{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).
3. 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.

## 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 tempting 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 $l$ 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, ${ }^{*}$ 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^{\mathbb{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 noarbitrage 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 further 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
$$\pi_{0,2}=\bar{p}{0,0} \bar{p}{0,1}, \pi_{1,2}=\bar{p}{0,0} p{0,1}, \pi_{2,2}=p_{0,0} \bar{p}{1,1} \text {, and } \pi{3,2}=p_{0,0} p_{1,1} \text {. }$$
The path probabilities can also be marked in a binomial tree as is shown in Figure 2.3.

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
$$\pi_{0,2}^{\prime}=\bar{q}{0,0} \bar{q}{0,1}, \pi_{1,2}^{\prime}=\bar{q}{0,0} q{0,1}, \pi_{2,2}^{\prime}=q_{0,0} \bar{q}{1,1} \text {, and } \pi{3,2}^{\prime}=q_{0,0} q_{1,1}$$
up to time 2. Suppose that the $\mathbb{P}$-probability of paths $\pi_{i, j} \neq 0$ for all $i, j$. We then can define the ratio of path probabilities as follows:
$$\zeta_{i, j}=\frac{\pi_{i, j}^{\prime}}{\pi_{i, j}} .$$

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

1. $E^{\mathbb{P}}\left[\left|M_{t}\right|\right]<\infty, \quad \forall t$
2. $E^{\mathbb{P}}\left[M_{t} \mid \mathcal{F}{s}\right]=M s, \quad \forall s \leq t$ 鞅属性与投资或投机中的公平游戏有关。让我们想想 $M{t}-M_{s}$ 作为两方在一段时间内赌博的损益 $(\mathrm{P} \backslash \& \mathrm{Q})(s, t)$. 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傝。在 数学中，也有很茤例子。事实上，到目前为止，我们已经看到了其中的几个，我们在下面提酲读者。
例子 $1.4$
3. 简单的随机游走， $X_{n}$, 是鞅，因为 $E\left[\left|X_{n}\right|\right]<n \sqrt{\Delta t}$ 和 $E\left[X_{n} \mid \mathcal{F} m\right]=X m, m \leq n$
$\mathrm{~ 2 . ~ 一 个 巴}$
4. 随机积分 $X_{t}=\int_{0}^{t} f(u) \mathrm{d} W_{u}$ 是鞅，因为
$$E^{P}\left[X_{t} \mid \mathcal{F} s\right]=E^{P}\left[\int 0^{s}+\int_{s}^{t} f(u) \mathrm{d} W_{u} \mid \mathcal{F}{s}\right] \quad=\int 0^{s} f(u) \mathrm{d} W{u}=X_{s}, \quad \forall s \leq t$$
在这里，我们应用了随机积分的第一个性质（参见第 11 页）。
5. 过程 $M_{t}=\exp \left(\int_{0}^{t} \sigma_{s} \mathrm{~d} W_{s}-\frac{1}{2} \sigma_{s}^{2} \mathrm{~d} s\right)$ 是指数鞅。事实上，使用伊藤引理，我们可以证明
$$\mathrm{d} M_{t}=\sigma_{t} M_{t} \mathrm{~d} W_{t}$$
这是一个没有漂移的伊藤工艺。它曎循
$$M_{t}=M_{s}+\int_{s}^{t} M_{u} \sigma_{u} \mathrm{~d} W_{u}$$
根据上一个例子的结论，我们知道 $M_{t}$ 是鞅。
我们在此强调，Ito 过程是鞅过程当且仅当其漂移项为零。最后，我们提出两个额外的例子。

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

$$f(S)=E^{\mathbb{P}}\left[f\left(S_{1}\right)\right] \quad=p f\left(S_{u}\right)+\bar{p} f\left(S_{d}\right)$$

$$\alpha S_{u}+\beta=f\left(S_{u}\right) \quad \alpha S_{d}+\beta=f\left(S_{d}\right)$$

$$\alpha=\frac{f\left(S_{u}\right)-f\left(S_{d}\right)}{S_{u}-S_{d}}, \beta=\frac{S_{u} f\left(S_{d}\right)-S_{d} f\left(S_{u}\right)}{S_{u}-S_{d}}$$

$$f(S)=\alpha S+\beta \quad=q f\left(S_{u}\right)+\bar{q} f\left(S_{d}\right)=E^{\mathbb{Q}}\left[f\left(S_{1}\right)\right]$$

$$q=\frac{S-S_{d}}{S_{u}-S_{d}}, \quad \bar{q}=1-q$$

$$S=q S_{u}+\bar{q} S_{d}=E^{\mathbb{Q}}\left(S_{1}\right)$$

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

Imathbb{P}=\left{p_{i, j}, Vbar{p}{,j}}rright} , 又称为测度。如图 $2.2$ 所示，在从 0 到 1 的时间范围内有两条路径，而在从 0 到 2 的时间范围内有四条路径。地平线从 0 到 1 的相应路径概率为
$$\pi 0,1=\bar{p} 0,0 \quad \text { and } \quad \pi 1,1=p_{0,0},$$

$$\pi_{0,2}=\bar{p} 0,0 \bar{p} 0,1, \pi_{1,2}=\bar{p} 0,0 p 0,1, \pi_{2,2}=p_{0,0} \bar{p} 1,1, \text { and } \pi 3,2=p_{0,0} p_{1,1} .$$

$\mathrm{~ 现 在 考 虑 另 一 组 分 支 概 率 ， ~ I m a t h b b { Q } = \ l e f t { q _ { i , j } , \ b a r { q } { i , j } =}$ 概率是
$$\pi_{0,1}^{\prime}=\bar{q} 0,0 \quad \text { and } \quad \pi 1,1^{\prime}=q_{0,0}$$

$$\pi_{0,2}^{\prime}=\bar{q} 0,0 \bar{q} 0,1, \pi_{1,2}^{\prime}=\bar{q} 0,0 q 0,1, \pi_{2,2}^{\prime}=q_{0,0} \bar{q} 1,1, \text { and } \pi 3,2^{\prime}=q_{0,0} q_{1,1}$$

$$\zeta_{i, j}=\frac{\pi_{i, j}^{\prime}}{\pi_{i, j}} .$$

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