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## 数学代写|期权定价理论代写Option Pricing Theory代考|THE BLACK–SCHOLES TERM STRUCTURE MODEL

The Black-Scholes model of Section $2.1$ is unrealistic, in that it does not allow for any term structure of interest rates (or volatility, for that matter). For that reason the model in (2.1) can be extended to
$$\mathrm{d} S_t=\mu_t S_t \mathrm{~d} t+\sigma_t S_t \mathrm{~d} W_t,$$
where both $\mu_t$ and $\sigma_t$ are deterministic processes. By risk neutrality, as in Section 2.7, we obtain
$$\mu_t=r_t^d-r_t^f .$$

From (2.71), we obtain
$$\mathrm{d} S_t / S_t=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t .$$
As in Section 2.4, we let $X_t=f\left(S_t\right)$, defined by $f(x)=\ln (x)$. Following (2.19a) we obtain
\begin{aligned} \mathrm{d} X_t &=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2} \ &=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t-\frac{1}{2} \sigma_t^2 \mathrm{~d} t . \end{aligned}
This integrates to give
$$X_T=X_0+\int_0^T \mu_s \mathrm{~d} s-\frac{1}{2} \int_0^T \sigma_s^2 \mathrm{~d} s+\int_0^T \sigma_s \mathrm{~d} W_s .$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|BREEDEN-LITZENBERGER ANALYSIS

Suppose that we have a continuum of prices available for call options with strike $K$ (all with the same time to expiry $T$ ). It was originally shown in Breeden and Litzenberger (1978) that this information is equivalent to an implied distribution. The argument is basically this.
We know that a call price is
\begin{aligned} C(K, T) &=\mathrm{e}^{-r^d T} \mathbf{E}^d\left[\left(S_T-K\right)^{+}\right] \ &=\mathrm{e}^{-r^d T} \int_0^{\infty}(s-K)^{+} f_{S_T}^d(s) \mathrm{d} s \ &=\mathrm{e}^{-r^d T} \int_K^{\infty}(s-K) f_{S_T}^d(s) \mathrm{d} s, \end{aligned}
where $f_{S_T}^d(s)$ is the probability distribution function for spot $S_T$ under the domestic risk-neutral measure.

Taking the first derivative with respect to $K$ follows by differentiating under the integral sign. The result we use is the following. Let $F(x)$ be defined by the following:
$$F(x)=\int_{a(x)}^{b(x)} f(x, s) \mathrm{d} s .$$
We then have
$$\frac{\mathrm{d}}{\mathrm{d} x} F(x)=f(x, b(x)) b^{\prime}(x)-f(x, a(x)) a^{\prime}(x)+\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, s) \mathrm{d} s .$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|THE BLACK-SCHOLES TERM STRUCTURE MODEL

The Black-Scholes model of Section $2.1$ is unrealistic, in that it does not allow for any term structure of interest rates (or volatility, for that matter). For that reason the model in ( $2.1$ ) can be extended to
$$\mathrm{d} S_t=\mu_t S_t \mathrm{~d} t+\sigma_t S_t \mathrm{~d} W_t,$$
where both $\mu_t$ and $\sigma_t$ are deterministic processes. By risk neutrality, as in Section $2.7$, we obtain
$$\mu_t=r_t^d-r_t^f .$$
From (2.71), we obtain
$$\mathrm{d} S_t / S_t=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t .$$
As in Section 2.4, we let $X_t=f\left(S_t\right)$, defined by $f(x)=\ln (x)$. Following (2.19a) we obtain
$$\mathrm{d} X_t=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2} \quad=\mu_t \mathrm{~d} t+\sigma_t \mathrm{~d} W_t-\frac{1}{2} \sigma_t^2 \mathrm{~d} t$$
This integrates to give
$$X_T=X_0+\int_0^T \mu_s \mathrm{~d} s-\frac{1}{2} \int_0^T \sigma_s^2 \mathrm{~d} s+\int_0^T \sigma_s \mathrm{~d} W_s$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|BREEDENLITZENBERGER ANALYSIS

Suppose that we have a continuum of prices available for call options with strike $K$ (all with the same time to expiry $T$ ). It was originally shown in Breeden and Litzenberger (1978) that this information is equivalent to an implied distribution. The argument is basically this.
We know that a call price is
$$C(K, T)=\mathrm{e}^{-r^d T} \mathbf{E}^d\left[\left(S_T-K\right)^{+}\right] \quad=\mathrm{e}^{-r^d T} \int_0^{\infty}(s-K)^{+} f_{S_T}^d(s) \mathrm{d} s=\mathrm{e}^{-r^d T} \int_K^{\infty}(s-K) f_{S_T}^d(s) \mathrm{d} s,$$
where $f_{S_T}^d(s)$ is the probability distribution function for spot $S_T$ under the domestic risk-neutral measure.
Taking the first derivative with respect to $K$ follows by differentiating under the integral sign. The result we use is the following. Let $F(x)$ be defined by the following:
$$F(x)=\int_{a(x)}^{b(x)} f(x, s) \mathrm{d} s .$$
We then have
$$\frac{\mathrm{d}}{\mathrm{d} x} F(x)=f(x, b(x)) b^{\prime}(x)-f(x, a(x)) a^{\prime}(x)+\int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x, s) \mathrm{d} s .$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|期权定价理论代写Option Pricing Theory代考|INTEGRATING THE SDE FOR ST

From (2.1), let $\mathrm{d} S_t=\mu S_t \mathrm{~d} t+\sigma S_t \mathrm{~d} W_t$. We can write this more simply as
$$\frac{\mathrm{d} S_t}{S_t}=\mu \mathrm{d} t+\sigma \mathrm{d} W_t, \text { noting that } \frac{\mathrm{d} S_t^2}{S_t^2}=\sigma^2 \mathrm{~d} t .$$
Consider the process $X_t=f\left(S_t\right)$ defined by $f(x)=\ln (x)$. We have $f^{\prime}(x)=1 / x$ and $f^{\prime \prime}(x)=$ $-x^{-2}$. A simple application of Itô’s lemma gives
\begin{aligned} \mathrm{d} X_t &=f^{\prime}\left(S_t\right) \mathrm{d} S_t+\frac{1}{2} f^{\prime \prime}\left(S_t\right) \mathrm{d} S_t^2 \ &=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2} \ &=\mu \mathrm{d} t+\sigma \mathrm{d} W_t-\frac{1}{2} \sigma^2 \mathrm{~d} t \end{aligned}
This can be immediately integrated to give
$$X_t=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) t+\sigma\left[W_t-W_0\right] .$$
Since $W_t$ is assumed to be a standardised Brownian motion with $W_0=0$, one obtains
$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T$$

and since $X_t=\ln \left(S_t\right) \Leftrightarrow S_t=\exp \left(X_t\right)$, one obtains the desired result
$$S_T=S_0 \exp \left(\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T\right) .$$
Note that $(2.21)$ can be written as
$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma \sqrt{T} \xi,$$
where $\xi \sim N(0,1)$.

## 数学代写|期权定价理论代写Option Pricing Theory代考|BLACK–SCHOLES PDEs EXPRESSED IN LOGSPOT

The algebra of Section $2.4$ shows that, under the assumption of geometric Brownian motion for the traded asset, it is easier to deal with the stochastic differential equation for logspot $X_t$ than the equivalent stochastic differential equation for spot $S_t$, as the drift and volatility terms for $X_t$ are homogeneous while those for $S_t$ depend on the level of the traded asset. In the same manner, the Black-Scholes PDEs (2.10) and (2.17) are simpler when expressed in terms of spatial derivatives with respect to logspot $x$, as opposed to derivatives with respect to spot $S$. We obtain, for $V=V\left(X_t, t\right)$, with a slight abuse of notation as this should be $\hat{V}=\hat{V}\left(X_t, t\right)$ :
$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial x^2}+\left(r^d-r^f-\frac{1}{2} \sigma^2\right) \frac{\partial V}{\partial x}-r^d V=0 .$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|NTEGRATING THE SDE FOR ST

$$\frac{\mathrm{d} S_t}{S_t}=\mu \mathrm{d} t+\sigma \mathrm{d} W_t, \text { noting that } \frac{\mathrm{d} S_t^2}{S_t^2}=\sigma^2 \mathrm{~d} t .$$

$$\mathrm{d} X_t=f^{\prime}\left(S_t\right) \mathrm{d} S_t+\frac{1}{2} f^{\prime \prime}\left(S_t\right) \mathrm{d} S_t^2 \quad=\frac{\mathrm{d} S_t}{S_t}-\frac{1}{2} \frac{\mathrm{d} S_t^2}{S_t^2}=\mu \mathrm{d} t+\sigma \mathrm{d} W_t-\frac{1}{2} \sigma^2 \mathrm{~d} t$$

$$X_t=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) t+\sigma\left[W_t-W_0\right] .$$

$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T$$

$$S_T=S_0 \exp \left(\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma W_T\right) .$$

$$X_T=X_0+\left(\mu-\frac{1}{2} \sigma^2\right) T+\sigma \sqrt{T} \xi,$$

## 数学代写|期权定价理论代写Option Pricing Theory代考|BLACK-SCHOLES PDEs EXPRESSED IN LOGSPOT

$$\frac{\partial V}{\partial t}+\frac{1}{2} \sigma^2 \frac{\partial^2 V}{\partial x^2}+\left(r^d-r^f-\frac{1}{2} \sigma^2\right) \frac{\partial V}{\partial x}-r^d V=0$$

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## 数学代写|期权定价理论代写Option Pricing Theory代考|RISK CONSIDERATIONS

In equities, it should be pretty clear whether one is considering upside or downside risk. Downside risk is if the value of the stock goes down and upside risk is if the stock appreciates dramatically. Further, if one is long the stock, then upside risk is positive to the stockholder and downside risk is negative.

In FX, the complexity of having two currencies to consider makes this more subtle. Perhaps one is a euro based investor who is long USD dollars. In that case the investor, regarding this particular exposure, is long US dollars and commensurately short euros. More complicated situations arise when options are introduced: if the euro based investor above attempts to hedge his or her long USD/short EUR position by buying a USD put/EUR call option. Since the premium for such an option is generally quoted in USD (see Section 3.3.1), purchasing the FX option hedge will make the option holder’s position somewhat shorter US dollars (as intended) and longer euros from holding the option – and additionally, short the number of USD required to purchase the option.

## 数学代写|期权定价理论代写Option Pricing Theory代考|SPOT SETTLEMENT RULES

A foreign currency spot transaction, if entered into today with spot reference $S_0$, will involve the exchange of $N_{\mathrm{d}}$ units of domestic currency for $N_f$ units of foreign currency, where the two notionals are related by the spot rate $S_0$, i.e. $N_d=S_0 \cdot N_f$. However, these two payments are in general not made on the day the transaction is agreed. The day on which the two payments, in domestic and foreign currency, are made is known as the value date. For conventional spot trades, which are almost always the case, it is known as the spot date. So we have two dates of special interest: today and spot.

It is often believed that FX trades settle two business days (2bd) after the trade date, known as $\mathrm{T}+2$ settlement, so that the spot date is obtained by counting forward to the second good business day after today. This is mostly correct, but with some notable exceptions and with the specific handling of holidays requiring some further explanation.

The first point to make is that not all currency pairs trade with $\mathrm{T}+2$ settlement. The most commonly cited exception is USDCAD which trades with $\mathrm{T}+1$ settlement; i.e. the currencies are exchanged one good business day after today. Eventually we shall probably see some currency pairs trading with $\mathrm{T}+0$ settlement. In the meantime, at the time of compilation of this work, the exceptions to $\mathrm{T}+2$ settlement $\mathrm{I}$ could find are listed in Table 1.2. For notational ease, we shall refer to $\mathrm{T}+x$ settlement. Now we need to define the concept of a good business day for currency pairs.

Definition: A day is a good business day for a currency ccy if it is not a weekend (Saturday and Sunday for most currencies, but not always for Islamic countries). For currency pairs ccy1ccy2, a day is only a good business day if it is a good business day for both ccy1 and ccy2.
The situation with respect to currencies in the Islamic world is complicated, and appears to vary by institution and country. The reason for this is that Friday is a particularly holy day for Jumu’ah prayers in the Muslim faith, influencing trading calendars similarly to Sunday in the Christian faith.

Weekends in Islamic countries are generally constructed with this in mind. For some countries such as Saudi Arabia, the weekend is taken on Thursday and Friday. It is, however, becoming more common for Islamic countries to change to observing weekends on Friday and Saturday – such as is the case in Algeria, Bahrain, Egypt, Iraq, Jordan, Kuwait, Oman, Qatar, Syria and the UAE – in an effort to harmonise business arrangements with the rest of the world and their neighbours.

Since readers with a special geographic interest in this trading region very likely have colleagues with local experience they can refer questions to directly, I refer the reader to the section ‘Arab currencies’ of the web page http://www. londonfx.co. uk/valdates. html for further details.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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## avatest™帮您通过考试

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 期权定价理论

## avatest™帮您通过考试

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## 数学代写|期权定价理论代写Option Pricing Theory代考|Description of the data

Summary statistics of both interest rates and stock returns are reported in Table $6.1$, a time-series plot and salient features of both data sets can be found in Figures $6.1$ and 6.2. The interest rates used in this paper as a proxy of the riskless rates are daily U.S. 3-month Treasury bill rates and the underlying stock considered in this paper is $3 \mathrm{Com}$ Corporation which is listed in NASDAQ. Both the stock and its options are actively traded. The stock claims no dividend and thus theoretically all options on the stock can be valued as European type options. The data covers the period from March 12 , 1986 to August 18, 1997 providing 2,860 observations. From Table 6.1, we can see that both the first difference of logarithmic interest rates and that of logarithmic stock prices (i.e. the daily stock returns) are skewed to the left and have positive excess kurtosis $(>>3)$ suggesting skewed and fat-tailed distributions. Similarly, the filtered interest rates $Y_{r_t}$ as well as the filtered stock returns $Y 1_{s_t}$ (with systematic effect) and $Y 2_{s_t}$ (without systematic effect) are also skewed to the left and have positive excess kurtosis. However, the logarithmic squared filtered series, as proxy of the logarithmic conditional volatility, all have negative excess kurtosis and appear to justify the Gaussian noise specified in the volatility process. As far as dynamic properties, the filtered interest rates and stock returns as well as logarithmic squared filtered series are all temporally correlated. For the logarithmic squared filtered series, the first order autocorrelations are in general low, but higher order autocorrelations are of similar magnitudes as the first order autocorrelations. This would suggest that all series are roughly $\operatorname{ARMA}(1,1)$ or equivalently $\operatorname{AR}(1)$ with measurement error, which is consistent with the first order autoregressive SV model specification. Estimates of trend parameters in the general model are reported in Table 6.2. For stock returns, interest rate has significant explanatory power, suggesting the presence of systematic effect or certain predictability of stock returns. For logarithmic interest rates, there is an insignificant linear mean-reversion, which is consistent with many findings in the literature.

## 数学代写|期权定价理论代写Option Pricing Theory代考|Structural models and Estimation Results

The general model: the model specified in Section $2.1$ assumes stochastic volatility for both the stock returns and interest rate dynamics as well as systematic effect on stock returns. This model nests the Amin and $\mathrm{Ng}$ (1993) model as a special case when $\lambda_2=0$. Following are four alternative model specifications:

• Submodel 1: No systematic effect, i.e. $\phi_s=0$ and $\alpha=0$, i.e. a bi-variate stochastic volatility model;
• Submodel 2: No stochastic interest rates, i.e. interest rate is constant, $r_t=r$, which is the Hull-White model and the Bailey and Stulz (1989) model;
• Submodel 3: Constant stock return volatility but stochastic interest rate, $\sigma_{s t}=$ $\sigma$, which is the Merton (1973), Turnbull and Milne (1991) and Amin and Jarrow (1992) models;
• Submodel 4: Constant stock return volatility and constant interest rate, $\sigma_{s t}=$ $\sigma, r_t=r$, which is the Black-Scholes model.

## 数学代写|期权定价理论代写Option Pricing Theory代考|Structural models and Estimation Results

• 子模型1: 无系统效应，即 $\phi_s=0$ 和 $\alpha=0$ ，即双变量随机波动率模型；
• 子模型2: 无随机利率，即利率是恒定的， $r_t=r$ ，即 Hull-White 模型和 Bailey 和 Stulz (1989) 模型;
• 子模型 3: 股票收益波动恒定但利率随机， $\sigma_{s t}=\sigma$ ，即 Merton (1973)、Turnbull 和 Milne (1991) 以及 Amin 和 Jarrow (1992) 模型;
• 子模型 4: 恒定的股票收益波动率和晅定的利率， $\sigma_{s t}=\sigma, r_t=r$ ，这是布莱克-斯科尔斯模型。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

Posted on Categories:数学代写, 期权定价理论

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 数学代写|期权定价理论代写Option Pricing Theory代考|Statistical Properties and Advantages of the Model

In the above SV model setup, the conditional volatility of both stock return and the change of logarithmic interest rate are assumed to be AR(1) processes except for the additional systematic effect in the stock return’s conditional volatility. Statistical properties of SV models are discussed in Taylor (1994) and summarized in Ghysels, Harvey, and Renault (1996), and Shephard (1996). Assume $r_t$ as given or $\alpha=0$ in the stock return volatility, the main statistical properties of the above model can be summarized as: (i) if $\left|\gamma_s\right|<1,\left|\gamma_r\right|<1$, then both $\ln \sigma_{s t}^2$ and $\ln \sigma_{r t}^2$ are stationary Gaussian autoregression with $\mathrm{E}\left[\ln \sigma_{s t}^2\right]=\omega_s /\left(1-\gamma_s\right), \operatorname{Var}\left[\ln \sigma_{s t}^2\right]=\sigma_s^2 /\left(1-\gamma_s^2\right)$ and $\mathrm{E}\left[\ln \sigma_{r t}^2\right]=\omega_r /\left(1-\gamma_r\right)$, $\operatorname{Var}\left[\ln \sigma_{r t}^2\right]=\sigma_r^2 /\left(1-\gamma_r^2\right)$; (ii) both $y_{s t}$ and $y_{r t}$ are martingale differences as $\epsilon_{s t}$ and $\epsilon_{r t}$ are iid, i.e. $\mathrm{E}\left[y_{s t} t F_{t-1}\right]=0, \mathrm{E}\left[y_{r t} \mid F_{t-1}\right]=0$ and $\operatorname{Var}\left[y_{s t} \mid F_{t-1}\right]=\sigma_{s t}^2$, $\operatorname{Var}\left[y_{r t} \mid F_{t-1}\right]=\sigma_{r t}^2$, and if $\left|\gamma_s\right|<1,\left|\gamma_r\right|<1$, both $y_{s t}$ and $y_{r t}$ are white noise; (iii) $y_{s t}$ is stationary if and only if $\ln \sigma_{s t}^2$ is stationary and $y_{r t}$ is stationary if and only if $\ln \sigma_{r t}^2$ is stationary; (iv) since $\eta_{s t}$ and $\eta_{r t}$ are assumed to be normally distributed, then $\ln \sigma_{s t}^2$ and $\ln \sigma_{r t}^2$ are also normally distributed. The moments of $y_{s t}$ and $y_{r t}$ are given by
$$\mathrm{E}\left[y_{s t}^\nu\right]=\mathrm{E}\left[\epsilon_{s t}^\nu\right] \exp \left{\nu \mathrm{E}\left[\ln \sigma_{s t}^2\right] / 2+v^2 \operatorname{Var}\left[\ln \sigma_{s t}^2\right] / 8\right}$$
and
$$\mathrm{E}\left[y_{r t}^\nu\right]=\mathrm{E}\left[\epsilon_{r t}^\nu\right] \exp \left{v \mathrm{E}\left[\ln \sigma_{r t}^2\right] / 2+v^2 \operatorname{Var}\left[\ln \sigma_{r t}^2\right] / 8\right}$$
which are zero for odd $v$. In particular, $\operatorname{Var}\left[y_{s t}\right]=\exp \left{\mathrm{E}\left[\ln \sigma_{s t}^2\right]+\operatorname{Var}\left[\ln \sigma_{s t}^2\right] / 2\right}$, $\operatorname{Var}\left[y_{r t}\right]=\exp \left{\mathrm{E}\left[\ln \sigma_{r t}^2\right]+\operatorname{Var}\left[\ln \sigma_{r t}^2\right] / 2\right}$. More interestingly, the kurtosis of $y_{s t}$ and $y_{r t}$ are given by $3 \exp \left{\operatorname{Var}\left[\ln \sigma_{s t}^2\right]\right}$ and $3 \exp \left{\operatorname{Var}\left[\ln \sigma_{r t}^2\right]\right}$ which are greater than 3, so that both $y_{s t}$ and $y_{r t}$ exhibit excess kurtosis and thus fatter tails than $\epsilon_{s t}$ and $\epsilon_{r t}$ respectively. This is true even when $\gamma_s=\gamma_r=0 ;$ (v) when $\lambda_4=0$, $\operatorname{Cor}\left(y_{s t}, y_{r t}\right)=$ $\lambda_1$; (vi) when $\lambda_2 \neq 0, \lambda_3 \neq 0$, i.e. $\epsilon_{s t}$ and $\eta_{s t}, \epsilon_{s t}$ and $\eta_{s t}$ are correlated with each other, $\ln \sigma_{s t+1}^2$ and $\ln \sigma_{r t+1}^2$ conditional on time $t$ are explicitly dependent of $\epsilon_{s t}$ and $\epsilon_{r t}$ respectively. In particular, when $\lambda_2<0$, a negative shock $\epsilon_{s t}$ to stock return will tend to increase the volatility of the next period and a positive shock will tend to decrease the volatility of the next period.

## 数学代写|期权定价理论代写Option Pricing Theory代考|Estimation and Volatility Reprojection

SV models cannot be estimated using standard maximum likelihood method due to the fact that the time varying volatility is modeled as a latent or unobserved variable which has to be integrated out of the likelihood. This is not a standard problem since the dimension of this integral equals the number of observations, which is typically large in financial time series. Standard Kalman filter techniques cannot be applied due to the fact that either the latent process is non-Gaussian or the resulting state-space form does not have a conjugate filter. Therefore, the SV processes were viewed as an unattractive class of models in comparison to other time-varying volatility models, such as $\mathrm{ARCH} / \mathrm{GARCH}$. Over the past few years, however, remarkable progress has been made in the field of statistics and econometrics regarding the estimation of nonlinear latent variable models in general and SV models in particular. Earlier papers such as Wiggins (1987), Scott (1987), Chesney and Scott (1987), Melino and Turnbull (1990) and Andersen and Sørensen (1996) applied the inefficient GMM technique to SV models and Harvey, Ruiz and Shephard (1994) applied the inefficient QML technique. Recently, more sophisticated estimation techniques have been proposed: Kalman filter-based techniques of Fridman and Harris (1997) and Sandmann and Koopman (1997), Bayesian MCMC methods of Jacquier, Polson and Rossi (1994) and Kim, Shephard and Chib (1998), Simulated Maximum Likelihood (SML) by Danielsson (1994), and EMM of Gallant and Tauchen (1996). These recent techniques have made tremendous improvements in the estimation of SV models compared to the early GMM and QML.

In this paper we employ EMM of Gallant and Tauchen (1996). The main practical advantage of this technique is its flexibility, a property it inherits of other momentbased techniques. Once the moments are chosen one may estimate a whole class of SV models. In addition, the method provides information for the diagnostics of the underlying model specification. Theoretically this method is first-order asymptotically efficient. Recent Monte Carlo studies for SV models in Andersen, Chung and Sørensen (1997) and van der Sluis (1998) confirm the efficiency for SV models for sample sizes larger than 1,000 , which is rather reasonable for financial time-series. For lower sample sizes there is a small loss of efficiency compared to the likelihood based techniques such as Kim, Shephard and Chib (1998), Sandmann and Koopman (1997) and Fridman and Harris (1996). This is mainly due to the imprecise estimate of the weighting matrix for sample sizes smaller than 1,000 . The same phenomenon occurs in ordinary GMM estimation.

## 数学代写|期权定价理论代写Option Pricing Theory代考|Statistical Properties and Advantages of the Model

$\mathrm{E}\left[y_{s t} t F_{t-1}^2\right]=0, \mathrm{E}\left[y_{r t} \mid F_{t-1}\right]=0 \sigma_{\text {和 } \operatorname{Var}}^2\left[y_{s t} \mid F_{t-1}\right]=\sigma_{s t}^2 \operatorname{Var}\left[y_{r t} \mid F_{t-1}\right]=\sigma_{r t}^2$, 㳑如果 $\left|\gamma_s\right|<1,\left|\gamma_r\right|<1$, 两 设为正态分布，则 $\ln \sigma_{s t}^2$ 和柇n $\sigma_{r t}^2$ 也也是正忩分布的。的时刻 $y_{s t}$ 和 $y_{r t}$ 由
〈left 的分隔符缺失或无法识别
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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。