Pricing and risk-managing derivatives is both complex and expensive. Now, two quants have developed a method that speeds up the process and, therefore, cuts computational costs.

They do so by exploiting the differentials provided by the adjoint algorithmic differentiation (AAD) method to train a machine learning algorithm to price a derivatives book.

The pair – Brian Huge, senior specialist quant at Saxo Bank, and Antoine Savine, chief quantitative analyst with Superfly Analytics at Danske Bank – expand on the technique in a recent article entitled *Axes that matter: PCA with a difference*. They talk about their latest work in this month’s Quantcast.

The basis of the method is a supervised version of principal component analysis (PCA), which reduces problems represented by complex functions into dependency on a small number of factors.

The work is part of an overarching methodology dubbed differential machine learning, whose purpose is to provide a fast and accurate way to calculate derivative prices, as well as their sensitivities to market factors, risk measures required by the new FRTB market risk regime, valuation adjustments and others. “If we can learn a value function pricing and replicate it via machine learning in real time, basically we can solve all bottlenecks in the calculation,” says Huge. With differential machine learning they can “run them on our workstation, even run them on a slim laptop”, he says.

The value of financial products depends on different risk factors. For some, those are easily identified. For example, options depend on volatility and European swaptions depend on the underlying swap rate.

But if the question is what factors determine the value of exotic products or an entire training book, things become more complex.

The answer is encoded in derivatives sensitivities. Huge and Savine’s technique – which they call differential PCA – extracts this information from differential data. “Differential PCA uses pathwise derivatives to identify the risk factors of arbitrary transactions or trading books,” says Huge. These differentials allow the pricing model to be designed in a simplified and parsimonious way, because they zero in on the most important factors to take into account, and conversely discard those that lack significant explanatory power.

“If you can reduce dimensionality safely, reliably and for little cost, then you should definitely do it,” says Savine.

A simpler method means a quicker method – which is critical in derivatives pricing. Real-time risk measures allow better hedging and avoid the need to cut corners. And speed ultimately saves money. Valuation adjustments and other regulatory risk measures traditionally require a large number of computer cores running overnight.

“With differential machine learning we can perform the same computations in minutes,” says Huge.

**Index**

00:00 Intro

02:04 Differential machine learning and fast derivatives pricing

04:30 The combined power of AAD and ML

10:40 Differential PCA

16:00 Why do we need to reduce the problem’s dimensionality?

20:10 Key difference between differential PCA and standard PCA

25:15 Further applications of differential PCA

26:36 Is there a ‘black-box’ risk in the use of machine learning for differential PCA?

31:22 Why is speed so valuable in derivatives pricing?

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