多维高斯统计特性

## 随机模拟 (Stochastic Simulation)

A well-known market phenomenon in the futures market is that the futures prices may deviate from the spot price of the underlying asset. As shown in an earlier article, the differential between **futures** and **spot** prices, called the **basis**, can be positive or negative, but are expected to converge to zero or near-zero at the expiration of the futures contract.

期货市场中一个众所周知的市场现象是，期货价格可能会与基础资产的现货价格有所不同。 如前一篇文章中所示， **期货**和**现货**价格之间的差(称为**基准** )可以为正或负，但在期货合约到期时，预计将收敛至零或接近零。

For each underlying asset, there are multiple futures with different maturities (ranging from 1 month to over a year). And for each futures contract, there is one basis process. Therefore, when we consider all different spot assets and their associated futures, there are a high number of basis processes.

对于每种基础资产，都有多个到期日不同的期货(从1个月到一年以上不等)。 对于每个期货合约，都有一个基础流程。 因此，当我们考虑所有不同的现货资产及其关联的期货时，存在大量的基础流程。

Moreover, these stochastic processes are clearly **dependent**. First, the spot assets, such as silver and gold, can be (possibly highly) correlated. Second, the futures written on the same underlying asset are clearly driven by a common source of randomness, among other factors. For anyone trading futures (on the same underlying or different assets), it is crucial to understand the **dependence** among these processes.

而且，这些随机过程显然是**依赖的** 。 首先，可以(可能是高度)关联现货资产，例如白银和黄金。 其次，写在同一基础资产上的期货显然是由共同的随机性来源以及其他因素驱动的。 对于任何交易期货(使用相同基础资产或不同资产)的人来说，了解这些流程之间的**依赖性**至关重要。

# 多维比例布朗桥 (Multidimensional Scaled Brownian Bridge)

This motivates us to develop a novel model to capture the joint dynamics of stochastic basis from different underlyings and different futures contracts. Once this model is built, we apply it to dynamic futures trading, as studied in this paper.

这激励我们开发一种新颖的模型，以捕获来自不同底层证券和不同期货合约的随机基础的联合动态。 建立此模型后，我们将其应用于动态期货交易，如本文所述 。

The **Multidimensional Scaled Brownian Bridge (MSBB)** is a continuous-time stochastic model described by the following multidimensional stochastic differential equation (SDE):

**多维比例布朗桥(MSBB)**是一种连续时间随机模型，由以下多维随机微分方程(SDE)描述：

where

哪里

and **W** consists of Brownian motions.

**W**由布朗运动组成。

In fact, **Z** is a N-dimensional process where each component is a 1-dimensional scaled Brownian bridge:

实际上， **Z**是一个N维过程，其中每个分量都是一维缩放的布朗桥 ：

The SDE for the multidimensional scaled Brownian bridge has a unique solution

多维比例布朗桥的SDE具有独特的解决方案

Here, we used the shorthand notation for the diagonal matrix: diag (aᵢ ) = diag(a₁ , . . . , a_N).

在这里，我们使用对角矩阵的简写形式：diag(aᵢ)= diag(a₁，。。，a_N)。

The **mean function **of **Z** is given by

**Z**的**平均函数**为

where

哪里

And the **covariance function** is

**协方差函数**是

Figure 1 illustrates the simulated paths of **Z** , **S **, and **F** for two pairs of futures and underlying assets (i.e. N = 2). Here, each Z is the log basis (i.e. log(F/S)) for the corresponding asset and futures contract. The plots for (Zt,1) and (Zt,2) also show the 95% confidence intervals of the log-bases.

图1说明了两对期货和基础资产(即N = 2)的**Z** ， **S**和**F**的模拟路径。 在此，每个Z是相应资产和期货合约的对数基础(即log(F / S))。 (Zt，1)和(Zt，2)的图还显示了对数基准的95％置信区间。

These plots showcase two characteristics of the log-bases. Firstly, they are **mean-reverting** in that any deviation from their mean is corrected. Secondly, they partially converge to zero at the end of the trading horizon (T = 0.25) as evident by narrowing of the confidence intervals. Indeed, (Zt,1) and (Zt,2) are Brownian bridges that converge to zero at T₁ = 0.27 and T₂ = 0.254, respectively. This convergence is not realized since trading stops at T = 0.25 in this particular example.

这些图展示了对数库的两个特征。 首先，它们是**均值回归** ，因为任何与均值的偏差都可以得到校正。 其次，在置信区间变窄的情况下，它们在交易期限结束时(T = 0.25)部分收敛于零。 实际上，(Zt，1)和(Zt，2)是布朗桥，它们分别在T 1 ＝ 0.27和T 2 ＝ 0.254处收敛到零。 由于在此特定示例中交易在T = 0.25处停止，因此无法实现这种收敛。

Since, Zt |Z₀ is a multivariate normal random variable, this means that

由于Zt |Z₀是多元正态随机变量，因此这意味着

has **chi-squared distribution** with *N* degrees of freedom. This relationship is utilized for obtaining the 95% confidence regions of Zt represented by dashed blue ellipses. These plots also illustrate partial convergence of log-bases at the end of time horizon.

具有*N个*自由度的**卡方分布** 。 利用该关系来获得由虚线蓝色椭圆表示的Zt的95％置信区域。 这些图还说明了时间范围结束时对数基准的部分收敛。

# 模拟 (Simulation)

The SDE solution for the multidimensional scaled Brownian bridge actually lends itself to a simulation algorithm. We refer to the paper for details, but the main idea is to discretize the time horizon in M time steps, simulate independent Gaussian random variables, and put them in the right places as follows:

多维比例布朗桥的SDE解决方案实际上适用于仿真算法。 我们参考本文以获取详细信息，但是主要思想是在M个时间步中离散时间范围，模拟独立的高斯随机变量，并将其放在正确的位置，如下所示：

For any notations not explained herein, please refer to the paper below.

对于此处未解释的任何符号，请参阅下面的论文。

# 外卖 (Takeaways)

For trading systems that involve multiple futures and assets, it is imperative to properly capture the dependence among price processes. **MSBB **described herein is designed for modeling the joint dynamics among futures and their spot assets. Through the solution to the stochastic differential question, this model is straightforward to simulate. With simulated sample paths, one can test the performance of trading strategies.

对于涉及多个期货和资产的交易系统，必须正确地捕捉价格过程之间的依赖关系。 **本文**描述的**MSBB**旨在模拟期货及其现货资产之间的联合动态。 通过解决随机微分问题，该模型很容易模拟。 通过模拟的样本路径，可以测试交易策略的性能。

多维高斯统计特性