It is now 2018, and the global quant community is realizing that size does matter: big data, big models, big computing grids, big computations – and a big regulatory rulebook to go along with it all. Not to speak of the big headaches that all this has induced across Wall Street.

The era of “big finance” has been creeping up on the banking industry gradually since the late 1990s, and got a boost when the Financial Crisis of 2007–2009 exposed a variety of deep complexities in the workings of financial markets, especially in periods of stress. Not only did this lead to more complicated models and richer market data with an explosion of basis adjustments, it also emphasized the need for more sophisticated governance as well as quoting and risk managements practices. One poster child for these developments is the new market practice of incorporating portfolio-level funding, margin, liquidity, capital, and credit effects (collectively known as “XVAs”) into the prices of even the simplest of options, turning the previously trivial exercise of pricing, say, a plain-vanilla swap into a cross-asset high-dimensional modeling problem that requires PhD-level expertise in computing and model building. Regulators have contributed to the trend as well, with credit capital calculation requirements under Basel 3 rules that are at the same level of complexity as XVA calculations, and with the transparency requirements of MiFID II requiring banks to collect and disclose vast amounts of trade data.

To get a quick sense of the computational effort involved in a basic XVA calculation, consider that such a computation typically involves path-wise Monte Carlo simulation of option trade prices through time, from today's date to the final maturity of the trades. Let us say that 10,000 simulations are used, running on a monthly grid for 10 years. As a good-sized bank probably has in the neighborhood of 1,000,000 options on its books, calculating a single XVA adjustment on the bank's derivatives holding will involve in the order of option re-pricings, on top of the often highly significant effort of executing the Monte Carlo simulation of market data required for pricing in the first place. Making matters significantly worse is then the fact that the traders and risk managers looking after the XVA positions will always require that sensitivities (i.e., partial derivatives) with respect to key risk factors in the market data are returned along with the XVA number itself. For complex portfolios, the number of sensitivities that one needs to compute can easily be in the order of ; if these are computed naively (e.g., by finite difference approximations), the number of option re-pricings needed will then grow to a truly unmanageable order of .

There are many interesting ways of chipping away at the practical problems of XVA calculations, but let us focus on the burdens associated with the computation of sensitivities, for several reasons. First, sensitivities constitute a perennial problem in the quant world: whenever one computes some quantity, odds are that somebody in a trading or governance function will want to see sensitivities of said quantity to the inputs that are used in the computation, for limit monitoring, hedging, allocation, sanity checking, and so forth. Second, having input sensitivities available can be very powerful in an optimization setting. One rapidly growing area of “big finance” where optimization problems are especially pervasive is in the machine learning space, an area that is subject to enormous interest at the moment. And third, it just happens that there exists a very powerful technique to reduce the computational burden of sensitivity calculations to almost magically low levels.

To expand on the last point above, let us note the following quote by Phil Wolfe ([1]):

There is a common misconception that calculating a function of variables and its gradient is about times as expensive as just calculating the function. This will only be true if the gradient is evaluated by differencing function values or by some other emergency procedure. If care is taken in handling quantities, which are common to the function and its derivatives, the ratio is usually 1.5, not , whether the quantities are defined explicitly or implicitly, for example, the solutions of differential equations...

The “care” in “handling quantities” that Wolfe somewhat cryptically refers to is now known as Algorithmic Adjoint Differentiation (AAD), also known as reverse automatic differentiation or, in machine learning circles, as backward propagation (or simply backprop). Translated into our XVA example, the promise of the “cheap gradient” principle underpinning AAD is that computation of all sensitivities to the XVA metric – no matter how many thousands of sensitivities this might be – may be computed at a cost that is order times the cost of computing the basic XVA metric itself. It can be shown (see [2]) that the constant in the term is bounded from above by 5. To paraphrase [3], this remarkable result can be seen as somewhat of a “holy grail” of sensitivity computation.

The history of AAD is an interesting one, marked by numerous discoveries and re-discoveries of the same basic idea which, despite its profoundness,¹ has had a tendency of sliding into oblivion; see [3] for an entertaining and illuminating account. The first descriptions of AAD date back to the 1960s, if not earlier, but did not take firm hold in the computer science community before the late 1980s. In Finance, the first published account took another 20 years to arrive, in the form of the award-winning paper [5].

As one starts reading the literature, it soon becomes clear why AAD originally had a hard time getting a foothold: the technique is hard to comprehend; is often hidden behind thick computer science lingo or is buried inside applications that have little general interest.² Besides, even if one manages to understand the ideas behind the method, there are often formidable challenges in actually implementing AAD in code, especially with management of memory or retro-fitting AAD into an existing code library.

The book you hold in your hands addresses the above challenges of AAD head-on. Written by a long-time derivatives quant, Antoine Savine, the exposition is done at a level, and in an applications setting, that is ideal for a Finance audience. The conceptual, mathematical, and computational ideas behind AAD are patiently developed in a step-by-step manner, where the many brain-twisting aspects of AAD are demystified. For real-life application projects, the book is loaded with modern C++ code and battle-tested advice on how to get AAD to run for real.

Select topics include: parallel C++ programming, operator overloading, tapes, check-pointing, model calibration, and much more. For both newcomers and those quaint exotics quants among us who need an upgrade to our coding skills and to our understanding of AAD, my advice is this: start reading!