the gaussian comeback

a new analysis indicates that gaussian mixture models (gmms), a machine learning technique employed for decades to fit complex financial distributions, may outperform the latest ai models in generating yield curves and volatility surfaces.

gmms excel at capturing nearly any continuous probability distribution by using a mixture of gaussian distributions. the model progressively identifies the densest parts of the probability distribution, assigning gaussians to capture the shape as closely as possible, and then fills in the tails and other parts until the entire distribution is captured to the desired level of accuracy.

a combination of gaussians has the advantage of using tractable and well-understood objects.

marco bianchetti, intesa sanpaolo

the training of the model relies on well-established statistical methods, such as expectation maximization, and is remarkably fast. once the training is done, the simulation is very easy because it’s just using simple distributions – namely the uniform and gaussian distributions, according to experts.

early results and comparisons

early results have been promising. for overnight rates like €str, sofr, and sonia, a mixture of about seven distributions was sufficient to capture them properly. for equity volatility surfaces, only three to five distributions were needed.

when compared to more complex techniques, gans struggled with a dataset of four years of daily market prices, deemed insufficient for proper training. autoencoders fared better, providing reasonable results, but gmms emerged as the superior method.

the gaussian mixtures were always realy fine, experts noted. they fitted also to multi-dimensional settings and allowed for real conditional distributions, making it possible to account for dependency in modelling and simulation, whereas autoencoders or gans just add auxiliary variables but do not use true conditional distributions.

reducing the dataset to only one year of daily data further highlighted the strengths of gmms,as neither gans nor autoencoder models could capture the salient features of the distributions,while gmms continued to perform well.

explainability and advantages

gmms offer significant advantages over more complex techniques, notably in terms of explainability.

a combination of gaussians can, like more complex machine learning algorithms, approximate any distribution, but differently from those complex algos, it has the advantage of using tractable and well-understood objects, also reducing the number of model parameters and possible overfitting problems, so that you can find statistical quantities in an analytical way.

marco bianchetti, intesa sanpaolo

this makes the model more transparent than gans or autoencoders. network-based methods are frequently enough opaque regarding how they do their job, experts explained. [this] method is not. the function is a mixture of gaussians whose parameters have a clear financial interpretation. that has clear benefits also for model validation.

the interpretation is similar to that of principal component analysis, but with a probabilistic twist. in this case, it’s a probabilistic interpretation, as a gaussian principal component and the weights are equivalent to the eigenvectors accounting for the importance of the components, experts added.

use cases and further research

gmms can be used to rectify incomplete or sparse datasets when calculating risk measures. in particular, within frtb [the basic review of the trading book] this can be an important contribution, because it could provide a tool for dealing with illiquid risk factors, experts stated.

researchers are also exploring other applications, such as using gmms to manipulate volatility surfaces to produce desired features, like smoothing them out for greater stability.

known methods are based on optimal transport solutions. since it is possible to optimally transport one gmm into another very efficiently [without] leaving the class of gmm distributions, we expect some nice results that increase speed and adaptability of the method, experts noted.

limitations and the bigger picture

gmms are not always preferable to gans and autoencoders.while gmms work well with daily price data, they struggle with larger datasets. fitting gmms to tick data, which can be enormous, is unfeasible as too many gaussians would be required, and tractability may be lost.

the research suggests that traditional modeling techniques still have a significant role to play in solving cutting-edge financial problems.

everybody seems so excited about deep learning – that you need to try deep learning instead of exhausting the more traditional algorithms like gaussian mixtures, experts concluded.

this analysis may encourage more quants to revisit and re-evaluate their existing models.

faq

• what are gaussian mixture models (gmms)? gmms are machine learning techniques that use a mixture of gaussian distributions to capture complex probability distributions.
• how do gmms compare to gans and autoencoders? in certain specific cases,particularly with smaller datasets,gmms can outperform gans and autoencoders in generating synthetic market data.
• what are the limitations of gmms? gmms can struggle with very large datasets, such as tick data, due to the high number of gaussians required.
• what are the use cases for gmms in finance? gmms can be used to rectify incomplete datasets, calculate risk measures, and manipulate volatility surfaces.