### 金融代写|金融工程作业代写Financial Engineering代考|Estimation Errors

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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|金融工程作业代写Financial Engineering代考|Estimation Errors

Once the maximum likelihood estimations have been computed, we should concern ourselves with the properties of the estimation errors, i.e., find the

asymptotic behavior of $\sqrt{n}\left(\hat{\mu}{n}-\mu\right)$ and $\sqrt{n}\left(\hat{\sigma}{n}-\sigma\right)$. In general, the limiting distribution will be a multivariate Gaussian distribution (see Appendix A.6.18).

The next proposition follows from the results of Appendix B.5.1. To denote “convergence in law,” we use the symbol $\rightsquigarrow$, as defined in Appendix B.

Proposition 1.4.1 As $n$ tends to $\infty,\left(\begin{array}{c}\sqrt{n}\left(\hat{\mu}{n}-\mu\right) \ \sqrt{n}\left(\hat{\sigma}{n}-\sigma\right)\end{array}\right)$ converges in law to a centered bivariate Gaussian distribution with covariance matrix $V$, denoted
$$\left(\begin{array}{l} \sqrt{n}\left(\hat{\mu}{n}-\mu\right) \ \sqrt{n}\left(\hat{\sigma}{n}-\sigma\right) \end{array}\right) \leadsto N_{2}(0, V),$$
where $V=\frac{\sigma^{2}}{2}\left(\begin{array}{cc}\frac{2}{h}+\sigma^{2} & \sigma \ \sigma & 1\end{array}\right)$. A consistent estimator of $V$ is given by
$$\hat{V}=\frac{\hat{\sigma}{n}^{2}}{2}\left(\begin{array}{cc} \frac{2}{h}+\hat{\sigma}{n}^{2} & \hat{\sigma}{n} \ \hat{\sigma}{n} & 1 \end{array}\right) .$$
In particular, we obtain that $\sqrt{n}\left(\hat{\mu}{n}-\mu\right) / \sqrt{\hat{V}{11}} \leadsto N(0,1)$ and $\sqrt{n}\left(\hat{\sigma}{n}-\right.$ $\sigma) / \sqrt{\hat{V}{22}} \leftrightarrow N(0,1)$.
The proof is given in Appendix 1.C.2.

## 金融代写|金融工程作业代写Financial Engineering代考|Estimation of Parameters for Apple

For an example of application of the previous results, consider the MATLAB file ApplePrices ${ }^{3}$ containing the adjusted values of Apple stock on Nasdaq (aapl), from January $13^{\text {th }}, 2009$, to January $14^{\text {th }}, 2011$.

The results of the estimation of parameters $\mu$ and $\sigma$ on an annual time scale, are given in Table 1.1. They have been obtained with the MATLAB function EstBS1dExp, using the results in Propositions $1.3 .1$ and 1.4.1. According to our comments on time scale and trading days at the end of Section 1.2.2, we have taken $h=1 / 252$.

Remark 1.4.1 When using a transformation like $\sigma=\exp (\alpha)$ to work with unconstrained parameters, we obtain an estimation $V_{0}$ of the limiting covariance matrix for the estimation error on the parameter $\theta=(\mu, \alpha)$. It follows from the delta method (Theorem B.3.4.1) that the estimation $V$ of the limiting covariance matrix for the estimation error on the parameter $(\mu, \sigma)$ is
$$\begin{gathered} \hat{V}=\hat{J} V_{0} \hat{J}, \ \text { where } \hat{J}=\operatorname{diag}\left(1, \hat{\sigma}{n}\right) \text {, i.e., } \hat{J}=\left(\begin{array}{cc} 1 & 0 \ 0 & \hat{\sigma}{n} \end{array}\right) \text {. This is because }(\mu, \sigma)= \end{gathered}$$

$H(\mu, \alpha)=\left(\mu, e^{\alpha}\right)$, so the Jacobian matrix of $H$ is $J=\left(\begin{array}{ll}1 & 0 \ 0 & \sigma\end{array}\right)$, which is estimated by $\hat{J}$.

This approach is implemented in the MATLAB function EstBS1dNum. For more details on the transformation of parameters and the corresponding limiting covariance matrix, see Remark B.3.3.

## 金融代写|金融工程作业代写Financial Engineering代考|European Call Option

Suppose that the price $S$ of the underlying asset is modeled by a geometric Brownian motion (1.1), and that $r$ is the risk-free rate, assumed to be constant on the period $[0, T]$. One also assumes that there is no dividend on that period. It can be shown [Black and Scholes, 1973] that the value at time $t$, of a European call option of strike price $K$ and maturity $T$, depends only on $S(t)=$ $s$ and on the volatility $\sigma$, and is given by
$$C(t, s)=s \mathcal{N}\left(d_{1}\right)-K e^{-r(T-t)} \mathcal{N}\left(d_{2}\right)$$
where $\mathcal{N}^{5}$ is the distribution function of a standard Gaussian variable, and
\begin{aligned} &d_{1}=\frac{\ln (s / K)+r \tau+\sigma^{2} \tau / 2}{\sigma \sqrt{\tau}} \ &d_{2}=d_{1}-\sigma \sqrt{\tau}=\frac{\ln (s / K)+r \tau-\sigma^{2} \tau / 2}{\sigma \sqrt{\tau}} \end{aligned}
where $\tau=T-t$ is the time to maturity. Note that as $t \rightarrow T, \tau \rightarrow 0$, and one can check that $\lim _{t \rightarrow T} C(t, s)=\max (s-K, 0)$. Furthermore, as expected, $C(t, s) / s \rightarrow 0$, as $s \rightarrow 0$, and $C(t, s) / s \rightarrow 1$, as $s \rightarrow \infty$ Remark 1.5.1 It is remarkable that the value of the option does not depend on the average return $\mu$, only on the volatility $\sigma$. It does not mean however that $\mu$ is not important. For example, if one wants to characterize the behavior (mean, variance, $V a R$, etc.) of the future value at time $t$ of a portfolio containing call options based on asset $S$, one needs to consider $C{t, S(t)}$, which in turn depends on both parameters $\mu$ and $\sigma$.

## 金融代写|金融工程作业代写Financial Engineering代考|Estimation Errors

(n(μ^n−μ) n(σ^n−σ))⇝ñ2(0,在),

## 金融代写|金融工程作业代写Financial Engineering代考|Estimation of Parameters for Apple

H(μ,一种)=(μ,和一种), 所以雅可比矩阵H是Ĵ=(10 0σ), 估计为Ĵ^.

## 金融代写|金融工程作业代写Financial Engineering代考|European Call Option

C(吨,s)=sñ(d1)−ķ和−r(吨−吨)ñ(d2)

d1=ln⁡(s/ķ)+rτ+σ2τ/2στ d2=d1−στ=ln⁡(s/ķ)+rτ−σ2τ/2στ

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