### 统计代写|风险建模代写Financial risk modeling代考|Valuing the economic benefit by simulations

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## 统计代写|风险建模代写Financial risk modeling代考|Valuing the economic benefit by simulations

In the following sections we show several numerical experiments to assess the gains offered by the Fourier estimator over other estimators in terms of in-sample and out-of-sample properties and from the

perspective of an asset-allocation decision problem. In Section 1.4.1 our attention is focused mainly on covariance estimation, since in this respect effects due to both nonsynchronicity and microstructure noise become effective. As for the finite sample variance analysis of the Fourier method, we refer the reader to Mancino and Sanfelici (2008a) for in-sample statistics and to Barucci et al. (2008) for the forecasting performance. Nevertheless, the results in Sections 1.4.2, 1.4.3 and 1.4.4 can be fully justified only by considering the properties of the different estimators for both the variance and the covariance measures. Following a large literature, we simulate discrete data from the continuous time bivariate Heston model
\begin{aligned} &d p^{1}(t)=\left(\mu_{1}-\sigma_{1}^{2}(t) / 2\right) d t+\sigma_{1}(t) d W_{1} \ &d p^{2}(t)=\left(\mu_{2}-\sigma_{2}^{2}(t) / 2\right) d t+\sigma_{2}(t) d W_{2} \ &d \sigma_{1}^{2}(t)=k_{1}\left(\alpha_{1}-\sigma_{1}^{2}(t)\right) d t+\gamma_{1} \sigma_{1}(t) d W_{3} \ &d \sigma_{2}^{2}(t)=k_{2}\left(\alpha_{2}-\sigma_{2}^{2}(t)\right) d t+\gamma_{2} \sigma_{2}(t) d W_{4} \end{aligned}
where $\operatorname{corr}\left(W_{1}, W_{2}\right)=0.35, \operatorname{corr}\left(W_{1}, W_{3}\right)=-0.5$ and $\operatorname{corr}\left(W_{2}, W_{4}\right)=$ $-0.55$. The other parameters of the model are as in Zhang et al. (2005): $\mu_{1}=0.05, \mu_{2}=0.055, k_{1}=5, k_{2}=5.5, \alpha_{1}=0.05, \alpha_{2}=0.045, \gamma_{1}=0.5$, $\gamma_{2}=0.5$. The volatility parameters satisfy Feller’s condition $2 k \alpha \geq \gamma^{2}$, which makes the zero boundary unattainable by the volatility process. Moreover, we assume that the additive logarithmic noises $\eta_{l}^{1}=\eta^{1}\left(t_{l}^{1}\right)$, $\eta_{l}^{2}=\eta^{2}\left(t_{l}^{2}\right)$ are i.i.d. Gaussian, contemporaneously correlated and independent from $p$. The correlation is set to $0.5$ and we assume $\omega_{i i}^{1 / 2}=$ $\left(E\left[\left(\eta^{i}\right)^{2}\right)^{1 / 2}=0,0.002,0.004\right.$, that is, we consider the case of no contamination and two different levels for the standard deviation of the noise. We also consider the case of dependent noise, assuming for simplicity $\eta_{l}^{i}=\alpha\left[p^{i}\left(t_{l}^{i}\right)-p^{i}\left(t_{l-1}^{i}\right)\right]+\bar{\eta}{l}^{i}$, for $i=1,2$ and $\bar{\eta}{l}^{i}$ i.i.d. Gaussian. We set $\alpha=0.1$. From the simulated data, integrated covariance estimates can be compared to the value of the true covariance quantities.

## 统计代写|风险建模代写Financial risk modeling代考|Covariance estimation and forecast

As a first application we perform an in-sample analysis in order to shed light on the properties of the different estimators in terms of different statistics of the covariance estimates, such as bias, MSE and others. More precisely, we consider the following relative error statistics:
$$\mu=E\left[\frac{\hat{C}^{12}-\int_{0}^{2 \pi} \Sigma^{12}(t) d t}{\int_{0}^{2 \pi} \Sigma^{12}(t) d t}\right], \quad s t d=\left{\operatorname{Var}\left[\frac{\hat{C}^{12}-\int_{0}^{2 \pi} \Sigma^{12}(t) d t}{\int_{0}^{2 \pi} \Sigma^{12}(t) d t}\right]\right}^{1 / 2}$$
which can be interpreted as relative bias and standard deviation of an estimator $\hat{C}^{12}$ for the covariance. The estimators have been optimized by choosing the cutting frequency $N$ of the Fourier expansion, the parameters $H$ and $S$ and the sampling interval for $R C^{o p t}$ on the basis of their MSE. The results are reported in Tables $1.1$ and $1.2$. Within each table, entries are the values of $\mu, s t d$, MSE and bias, using 750 Monte Carlo replications, which roughly correspond to three years. Rows correspond to the different estimators. The sampling interval for the realized covariance-type estimators is indicated as a superscript. The optimal sampling frequency for $R C$ opt is obtained by direct minimization of the true MSE of the covariance estimates and corresponds to $1 \mathrm{~min}$ in the absence of noise, to $1.33$ min when $\omega_{i i}^{1 / 2}=0.002$, to $1.67$ min when $\omega_{i i}^{1 / 2}=0.004$ and to $1.5 \mathrm{~min}$ when $\omega_{i i}^{1 / 2}=0.004$ and the noise is dependent on the price. The other optimal MSE-based parameter values are listed in the tables.

When we consider covariance estimates, the most important effect to deal with is the “Epps effect.” The presence of other microstructure effects represents a minor aspect in this respect. On the contrary, it may in some sense even compensate the effects due to nonsynchronicity, as we can see from the smaller MSE of the 1-min realized covariance estimator with respect to the 5 -min estimator in the cases with noise. We remark that the corresponding 1-min estimator for variances is more affected by the presence of noise, since it is not compensated for by nonsynchronicity. Moreover, in the absence of noise the Epps effect hampers consistency of the realized covariance estimates, yielding an optimal MSE-based frequency of $1 \mathrm{~min}$.

In fact, as with any estimator based on interpolated prices, the realized covariance-type estimators suffer from the Epps effect when trading is nonsynchronous. The lead-lag correction reduces such an effect, at least in terms of bias, to the disadvantage of a slightly larger MSE. Note that the lead-lag correction contrasts with the Epps effect, thus producing occasionally positive biases. In the absence of noise the best performance is achieved by the unbiased $\mathrm{AO}$ estimator and this justifies the optimal $S$ value for its subsampled version which is set to 1 , that is, no subsampling is needed. We remark that the optimal $H$ value for the kernel estimator $(K)$ is set to 4 , that is, the use of some weighted autocovariance is needed to contrast with the Epps effect, differently from the variance estimation, where the optimal MSE-based $H$ value is equal to 0 , which corresponds to the realized variance. On the other hand, the presence of noise strongly affects the AO estimator. This is due to the “Poisson trading scheme” with correlated noise. In fact, the $\mathrm{AO}$ remains unbiased under independent noise whenever the probability of trades occurring at the same time is zero, which is not the case for Poisson arrivals. In the same fashion, the Kernel estimator provides an acceptable estimate in the absence of noise but is rapidly swamped by the presence of noise. This is quite striking, because the corresponding variance estimator provides the best estimates at the highest frequencies in the presence of noise, as already discussed in Mancino and Sanfelici (2008a). Nevertheless, Barndorff-Nielsen et al.

## 统计代写|风险建模代写Financial risk modeling代考|Dynamic portfolio choice and economic gains

In this section, we consider the benefit of using the Fourier estimator with respect to others from the perspective of the asset-allocation problem of Section 1.3. Given any time series of daily variance/covariance estimates we split our samples of 750 days into two parts: the first one containing 30 percent of total estimates is used as a “burn-in” period, while the second one is saved for out-of-sample purposes. The out-of-sample forecast is based on univariate ARMA models, as in the previous section. More precisely, following Aït-Sahalia and Mancini (2008), the estimated series of 225 in-sample covariance matrices is used to fit univariate AR(1) models for each variance/covariance estimate separately.

The total number of out-of-sample forecasts $m$ for each series is equal to 525 . Each time a new forecast is performed, the corresponding actual variance/covariance measure is moved from the forecasting horizon to the first sample and the AR(1) parameters are reestimated in real time. Given sensible choices of $R f_{1} \mu_{p}$ and $\mu_{t}$, each one-day-ahead variance/covariance forecast leads to the determination of a daily

portfolio weight $\omega_{t}$. The time series of daily portfolio weights then leads to daily portfolio returns and utility estimation.

We implement the criterion in (1.14) by setting $R f$ equal to $0.03$ and considering three target $\mu_{p}$ values, namely $0.09,0.12$ and $0.15$. In order to concentrate on volatility timing and abstract from issues related to expected stock-return predictability, for all times $t$ we set the components of the vector $\mu_{t}=E_{t}\left[R_{t+1}\right]$ equal to the sample means of the returns on the risky assets over the forecasting horizon. For all times $t$, the conditional covariance matrix is computed as an out-of-sample forecast based on the different variance/covariance estimates.

We interpret the difference $U^{\hat{C}}-U^{\text {Fourier }}$ between the average utility computed on the basis of the Fourier estimator and that based on alternative estimators $\hat{C}$, as the fee that the investor would be willing to pay to switch from covariance forecasts based on estimator $\hat{C}$ to covariance forecasts based on the Fourier estimator. Tables 1.7, $1.8$ and $1.9$ contain the results for three levels of risk aversion and three target expected returns in the different noise scenarios considered in our analysis. We remark that, in general, the optimal sampling frequencies for the realized variances and covariances are different within each scenario, due to the different effects of microstructure noise and nonsynchronicity on the volatility measures. Therefore, in the asset-allocation application we chose to use a unique sampling frequency for realized variances and covariances, given by the maximum among the optimal sampling intervals corresponding to variances and covariances.

## 统计代写|风险建模代写Financial risk modeling代考|Valuing the economic benefit by simulations

dp1(吨)=(μ1−σ12(吨)/2)d吨+σ1(吨)d在1 dp2(吨)=(μ2−σ22(吨)/2)d吨+σ2(吨)d在2 dσ12(吨)=ķ1(一种1−σ12(吨))d吨+C1σ1(吨)d在3 dσ22(吨)=ķ2(一种2−σ22(吨))d吨+C2σ2(吨)d在4

## 统计代写|风险建模代写Financial risk modeling代考|Covariance estimation and forecast

\mu=E\left[\frac{\hat{C}^{12}-\int_{0}^{2 \pi} \Sigma^{12}(t) d t}{\int_{0}^{ 2 \pi} \Sigma^{12}(t) d t}\right], \quad s t d=\left{\operatorname{Var}\left[\frac{\hat{C}^{12}-\int_{ 0}^{2 \pi} \Sigma^{12}(t) d t}{\int_{0}^{2 \pi} \Sigma^{12}(t) d t}\right]\right}^{ 1 / 2}\mu=E\left[\frac{\hat{C}^{12}-\int_{0}^{2 \pi} \Sigma^{12}(t) d t}{\int_{0}^{ 2 \pi} \Sigma^{12}(t) d t}\right], \quad s t d=\left{\operatorname{Var}\left[\frac{\hat{C}^{12}-\int_{ 0}^{2 \pi} \Sigma^{12}(t) d t}{\int_{0}^{2 \pi} \Sigma^{12}(t) d t}\right]\right}^{ 1 / 2}

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