Posted on Categories:数学代写, 期权定价理论

# 数学代写|期权定价理论代写Option Pricing Theory代考|FINM33000 Statistical Properties and Advantages of the Model

avatest™

## 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！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 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
\eft 的分隔符缺失或无法识别

## 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。