### 统计代写|风险建模代写Financial risk modeling代考| The timeline of a limit order history

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## 统计代写|风险建模代写Financial risk modeling代考|The timeline of a limit order history

A stylized timeline of the limit order history is shown on the next page. The first row marks the clock time $T_{i}$ (measured in seconds since start of the trading day), the second row marks the time $t_{k i}$ since the moment $t_{k 0}$ of $k$ th limit order arrival, the third row marks $i(k)$ duration episodes $\Delta t_{k i}$ between consecutive changes of covariates $x_{k i}$ and the fourth row shows the values of time-varying covariates immediately before the beginning of each new episode of the limit order history. Row five shows the hazard rate $v\left(x_{k 0}\right)$ of cancellation for fleeting orders, row six shows the hazard rate $\xi\left(x_{k i}\right)$ of cancellation for regular (non-fleeting) limit orders and row seven shows the hazard rate $\mu\left(\boldsymbol{x}{k i}\right) R\left(\boldsymbol{x}{k i}\right)$ of order execution in each of the durations prior to the $k$ th limit order termination.

## 统计代写|风险建模代写Financial risk modeling代考|The likelihood function

To derive the expression for the likelihood function $L_{k}$ of $k$ th limit order, we start with the model where the risks of execution and cancellation are conditionally independent given the values of covariates. Then we show how the derived log-likelihood function can be maximized by standard methods of survival analysis, since the likelihood function $L_{k}$ can be decomposed into the product of two terms, $L_{c k}$ and $L_{e k}$, corresponding to the likelihood terms of cancellation and execution risks. The likelihood function of cancellation can be written as follows:
\begin{aligned} L_{c k}(\pi, v, \xi)=& \frac{e^{-\pi^{\prime} x_{k 0}}}{1+e^{-\pi^{\prime} x_{k 0}}} \exp \left[\sum_{i=1}^{i(k)}\left(\delta_{k i} \ln v\left(x_{k 0}\right)-v\left(x_{k 0}\right) \Delta t_{k i}\right)\right] \ &+\frac{1}{1+e^{-\pi^{\prime} x_{k 0}}} \exp \left[\sum_{i=1}^{i(k)}\left(\delta_{k i} \ln \xi\left(x_{k i}\right)-\xi\left(x_{k i}\right) \Delta t_{k i}\right)\right] \end{aligned}
where $\delta_{k i}$ is the indicator of the event that ith duration episode is terminated by cancellation. The likelihood function of execution is written similarly as:
$$L_{e k}(\mu)=\exp \left[\sum_{i=1}^{i(k)}\left(d_{k i} \ln \mu\left(x_{k i}\right)-R\left(x_{k i}\right) \mu\left(x_{k i}\right) \Delta t_{k i}\right)\right]$$
where $d_{k i}$ is the indicator of the event that ith duration episode is terminated by execution.

The log-likelihood function corresponding to the cancellation risk can be written in the additive form as follows:
\begin{aligned} \ln L_{c k}(\pi, v, \xi)=&-\left(\pi^{\prime} x_{k 0}+\ln \left(1+e^{-\pi^{\prime} x_{k 0}}\right)+\ln \left(1+Z_{c k}(\pi, v, \xi)\right)\right) \ &+\sum_{i=1}^{i(k)}\left(\delta_{k i} \ln v\left(x_{k 0}\right)-v\left(x_{k 0}\right) \Delta t_{k i}\right) \end{aligned}
where
$$Z_{c k}(\pi, v, \xi)=\pi^{\prime} x_{k 0}+\sum_{i=1}^{i(k)}\left(\delta_{k i} y_{k i}-v\left(x_{k 0}\right)\left(1-e^{y_{k i}}\right)\right)$$
and
$$y_{k i}=\ln \xi\left(x_{k i}\right)-\ln v\left(x_{k 0}\right) .$$
The log-likelihood function corresponding to execution risk is written similarly as follows:
$$\ln L_{e k}(\mu)=\sum_{i=1}^{i(k)}\left(d_{k i} \ln \mu\left(x_{k i}\right)-R\left(x_{k i}\right) \mu\left(x_{k i}\right) \Delta t_{k i}\right)$$

## 统计代写|风险建模代写Financial risk modeling代考|Estimation results

Panel A of Table $2.7$ shows the results for the model of intensity of limit order arrival at best quotes for ask orders for 13 randomly chosen stocks and Panel B shows the same for the mixture model for cancellation of limit orders arriving at best quotes. Estimates that have the same sign at least 90 percent of the days are boldfaced.

Panel $B$ shows that the probability of a fleeting order at best ask quotes depends:

• positively on recent buyer-initiated trading volume (in the last 5 seconds),
• positively (but not as strongly) on recent (last 5 seconds) executions of hidden bid orders,
• negatively on LOB depth at and near the best quote on the same side,
• negatively (except for ISRG) on LOB depth at best quote on the opposite side,
• negatively for small relative spread stocks (AAPL, CMCSA) and positively for larger relative spread stocks (AKAM, GOOG, ISRG) on recent (in the last five seconds) seller-initiated trading volume.

The intensity of limit order arrival at best ask quotes (shown in Panel A)

• depends positively on recent (in the last five seconds) buyer- and seller-initiated trading volume, although more strongly on sellerinitiated trading volume,
• depends negatively on LOB depth at and near the best quote on the opposite side,
• exhibits positive dependence on LOB depth near the same side best quote for AMGN and ISRG and negative dependence on LOB depth near the same side best quote for CMCSA.

## 统计代写|风险建模代写Financial risk modeling代考|The likelihood function

ln⁡大号Cķ(圆周率,在,X)=−(圆周率′Xķ0+ln⁡(1+和−圆周率′Xķ0)+ln⁡(1+从Cķ(圆周率,在,X))) +∑一世=1一世(ķ)(dķ一世ln⁡在(Xķ0)−在(Xķ0)Δ吨ķ一世)

ln⁡大号和ķ(μ)=∑一世=1一世(ķ)(dķ一世ln⁡μ(Xķ一世)−R(Xķ一世)μ(Xķ一世)Δ吨ķ一世)

## 统计代写|风险建模代写Financial risk modeling代考|Estimation results

• 积极的买卖差价，
• 对近期买家发起的交易量（在最后 5 秒内）持积极态度，
• 对最近（最后 5 秒）隐藏的投标订单执行积极（但不那么强烈），
• 在同一侧的最佳报价处和附近对 LOB 深度产生负面影响，
• 负面（ISRG 除外）对 LOB 深度的最佳报价在另一侧，
• 在最近（过去 5 秒内）卖方发起的交易量中，对相对价差较小的股票（AAPL、CMCSA）不利，对价差较大的股票（AKAM、GOOG、ISRG）有利。

• 正依赖于买卖差价，
• 积极地取决于最近（在最后 5 秒内）买方和卖方发起的交易量，尽管更强烈地取决于卖方发起的交易量，
• 负面取决于对面最佳报价处和附近的 LOB 深度，
• 对 AMGN 和 ISRG 的同侧最佳报价附近的 LOB 深度表现出正相关性，对 CMCSA 的同侧最佳报价附近的 LOB 深度表现出负相关性。

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