The financial industry can be tricky when it comes to risk prediction, especially when we speak about correlated financial instruments—assets with interconnected price fluctuations. In this case, the risks associated with one asset can influence the performance of others. While statistical modelling uses Gaussian processes (GP) for such cases, sometimes they are not enough. In real-life financial environments, for example, data can show non-Gaussian characteristics like skewness and heavy tails. I will explain how to apply non-Gaussian Gaussian processes (NGGP) to improve risk forecasting for correlated financial instruments.
What is the difference between Gaussian processes and non-Gaussian processes?
A Gaussian process is a collection of random variables, any finite number of which has a joint multivariate normal distribution.
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space) such that every finite collection of those random variables has a multivariate normal distribution.
A Gaussian process includes a mean function and a covariance function; the first defines the expected value of the process at each point, and the second describes the correlation between different points in the process.
Here’s the formula: f(x)∼GP(μ(x),k(x,x′))
● Where μ(x) is the mean function.
● And k(x,x′) k(x, x’) k(x,x′) is the covariance function (or kernel).
By choosing kernel and mean functions, GPs can incorporate prior knowledge, making them effective in statistical modeling and machine learning. The key difference between non-Gaussian processes is that they can handle data that does not follow a normal distribution.
What are the key characteristics of non-Gaussian processes?
In the finance industry, non-normal distributions are common because data can show the following characteristics: skewness (asymmetry), kurtosis (peakedness), and heavy tails (extreme values). Let’s take a closer look at each of them:
Skewness measures the asymmetry of the probability distribution of a real-valued random variable about its mean. Positive skew indicates a distribution with a long right tail, while negative skew indicates a long left tail.
Kurtosis describes the “tailedness” of the probability distribution: higher values show more variance due to infrequent extreme deviations.
Heavy tails in distributions demonstrate a higher likelihood of extreme values. They can be used to characterize and predict more precisely extreme events like market crashes, which occur more frequently than a Gaussian distribution can predict.
With the ability to work with deviations from normality in data, non-Gaussian Gaussian processes (NGGPs) have an upper hand over Gaussian processes.
First of all, NGGPs are more precise in capturing the behavior of financial data, including extreme events and asymmetries. This allows for more reliable predictions.
Second, NGGPs give a clearer assessment of potential risks thanks to accurate modeling of the heavy tails and skewness in financial data. Having a complete understanding of risks is a must for making informed decisions that will reduce the risk of losses.
Third, NGGPs are more versatile compared to GPs: they can be tailored to different market conditions and types of data. Additionally, they are more adaptable to market changes.
At last, NGGPs give more nuances to the uncertainty associated with financial predictions than GPs.
How to apply NGGP to financial risk forecasting
As I already mentioned, the price change of one asset affects other prices in correlated financial instruments. And correlations, in turn, can be caused by various factors, including market conditions or financial events.
There’s an algorithm that defines the implementation process of non-Gaussian Gaussian processes (NGGP) in correlated financial instruments.
- The first step is gathering historical price data for the correlated financial instruments. You need to clean it and ensure its consistency. You also have to calculate returns, or log-returns, because they are often used in risk modeling.
- Choose a kernel function that can handle non-Gaussian features. Incorporate a copula function if modeling dependencies between non-Gaussian marginals.
- Use transformations or extended models to incorporate skewness and kurtosis into the Gaussian process.
- Use a training dataset to fit the NGGP model, estimating the parameters of the kernel function and other model components. Optimize the model parameters using techniques such as maximum likelihood estimation (MLE) or Bayesian inference.
- Use the model to make predictions about future prices or returns and generate risk forecasts by computing metrics like value at risk (VaR) or expected shortfall (ES) based on the predictive distribution obtained from the NGGP.
- Validate the model using a separate test dataset.
Before applying NGGP for risk forecasts, you need to make sure that you have the right software and sufficient computing resources. The former, for example, includes Python libraries like GPy, scikit-learn, and TensorFlow Probability. When it comes to the latter, you have to keep in mind that NGGP can be computationally intensive, and parallel computing or GPUs may be required.
