A new way to calculate conditional expectations
As acronyms, MGM–DCKE – Estimation of the kernel dynamically controlled by the Gaussian mixture model – is a bit complicated. Its proponents, however, regard it as the simplest expression of conditional expectations, one of the fundamental ingredients of quantitative finance.
Conditional expectations – the expected values of random variables given a set of conditions – have all sorts of applications, ranging from pricing exotic and out-of-the-money options to calculating derivative valuation adjustments and calibrating models. of volatility.
They are usually calculated using the Monte Carlo method of least squares, originally proposed by Longstaff and Schwartz in 2001 to value American options. But despite its popularity, this approach is not without drawbacks – chief among them being the need to simulate a large number of paths, especially when dealing with more complex instruments.
MGM–DCKE, on the other hand, is based on a combination of several Gaussian distributions and requires minimal input data. “All I need to apply this model is data from two time points and the Gaussian mixture model,” says Jörg Kienitz, partner and principal quant at Acadiasoft, with academic affiliations at the universities of Cape Town and de Wuppertal, who introduced the technique in an article published in Risk.net last month. He describes it as a semi-analytical expression of conditional expectations that is purely data-driven and model-agnostic.
Gaussian mixture models can replicate almost any distribution by combining enough of them. The first step in calculating conditional expectations using MGM–DCKE is to select the number of Gaussian distributions needed for replication. This can also be inferred from the data. Kienitz started with five to 10 Gaussians, which worked adequately. The maximum likelihood of the corresponding probability distribution is then used to analytically calculate the mean and the covariance, based on the estimated parameters.
A control variable — a way to smooth out estimation errors — can also be incorporated as an indirect hedge, Kienitz adds, and sensitivities can be hedged using this approximation.
So far, the results are promising. “We are currently using this method for Bermuda pricing and for calibrating local stochastic volatility models, and are researching it for solving forward-backward stochastic differential equations, where we have very promising results compared to analytical solutions,” says Kienitz.
It appears that with the Kienitz method, one can obtain a better calibration for local stochastic volatility models
Bernard Gourion, Natixis
Compared to other methods, MGM–DCKE can estimate exposure more easily by moving the means around and maintaining the shape of the distribution, he adds, although it is a bit slower due to this expectation maximization.
Kienitz’s article was well received in the industry and MGM–DCKE should quickly find its way into the banks’ model libraries.
“For a long time we used the Longstaff-Schwartz model for Bermudian options, but we looked at the MGM–DCKE model as an alternative to price our Bermudian pound faster and potentially more accurately,” says Nicholas Burgess, an independent consultant who worked with the quantitative equity derivatives team at HSBC. “Normally, banks are hesitant to modify the software infrastructure to integrate new models, but MGM–DCKE is a relatively lightweight implementation and is likely to make it into production.
Bernard Gourion, senior quant in the model validation team at Natixis in Paris, is also impressed. He see MGM–DCKE as an alternative to the particle calibration methods developed by Guyon and Henry-Labordere and to that proposed by Aitor Muguruza.
“From the preliminary results of the tests we have performed so far, it appears that with the Kienitz method we can achieve better calibration for local stochastic volatility models,” Gourion says, provided the testing process at Natixis still has a long way to go before reaching final approval.
Others are investigating whether the approach can be used to fill in missing values – for example, in incomplete time series, such as illiquid securities prices. The distribution information obtained from the data can be used to calculate the conditional expectation and then simulate the missing values. A senior quant from a European bank sees this as a promising line of research. He is watching MGM–DCKE as a lower-rank approximation, similar to other recently developed techniques that are emerging as an easy-to-use alternative to neural networks.
MGM–DCKE builds on another recent paper that Kienitz co-authored with Gordon Lee, Nikolai Nowaczyk, and Nancy Qingxin Geng, in which the estimation of the dynamically controlled kernel (DCKE) was introduced for the purpose of calculating conditional expectations.
The DCKE uses a numerical method that has limited bandwidth to approximate the conditional distribution on which the calculation is based. The Gaussian mixture model completely eliminates this local bandwidth optimization and replaces it with an analytically smoother and more stable approach.
“I had this idea in mind for a long time,” says Kienitz. “The Python code is available on GitHub and is only a few lines long.”
The limit of MGM–DCKE is the dimensionality it can handle. “I applied it to dimensions up to 20,” Kienitz explains. “Beyond that, it might not work well.” Addressing this limitation is Kienitz’s next challenge.