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# PhD Defense Antoine Savine

Title: **Modern Computational Finance: AAD and Parallel Simulations Simulations**

In the aftermaths of the Global Financial Crisis of 2008, Investment Banks were imposed massive regulations and forced to conduct frequent regulatory calculations. While these regulations made it harder for banks to conduct derivatives businesses, they also contributed to starting a new golden age for Computational Finance.

A typical example of regulatory calculation is the CVA (Counterparty Value Adjustment), an estimation of the loss subsequent to a future default of a counterparty when the value of the sum of all transactions against that counterparty (called netting set) is positive, and, therefore, lost. The CVA is actually the value of a real option a bank gives away whenever it trades with a defaultable counterparty. That option is a put on the netting set, contingent on default. It is an exotic option, and a particularly complex one, since the underlying asset is the netting set, consisting itself in thousands of transactions, some of which may be optional or exotic.

In addition, the netting set typically includes derivatives transactions in different currencies on various underlying assets belonging to different asset classes. Options on a set of heterogeneous underlying assets are known to the derivatives industry and called hybrid options. Investment banks Quantitative Research departments actively developed hybrid models and related numerical implementations in the decade 1998-2008 for the risk management of very profitable transactions like Callable Reverse Power Duals (CPRDs) in Japan.

The specification, calibration and simulation of hybrid models are now well known. What is unprecedented is the number and variety of transactions and cashflows involved, and the dimension of the simulation. With a naive implementation, the CVA on a large netting set cantake minutes or even hours to calculate.

In addition, the market and credit risk of the CVA must be hedged. The CVA is a cost that impacts the revenue and balance sheet of the bank, and its value may change by hundreds of millions when financial and credit markets move. In order to hedge the CVA, it is not enough to compute it. All its sensitivities to market variables must also be produced. And a CVA is typically sensitive to thousands of market variables: all the rate and spread curves and volatility surfaces for all the currencies involved, as well as foreign exchange rates and their volatility surfaces, and, of course, credit curves. In order to compute all these risks with classical methods, the valuation of the CVA must be repeated thousands of times with inputs bumped one by one.

This is of course not viable, so investment banks first implemented crude approximations, at the expense of accuracy, and distributed calculations over large data centres, incurring massive hardware, development and maintenance costs. Calculation speed became a question of survival, and banks had to find new methodologies and paradigms, at the junction of mathematics, numerics and computer science, in order to conduct CVA and other regulatory calculations accurately and fast on limited hardware.

That search for speed produced new, superior mathematical modeling of CVA (see Volume III), a renewed interest in the crucial technology of scripting derivatives cash-flows (to which our volume II is dedicated), the systematic incorporation of parallel computing (part I of volume I) and exiting new technologies such as Algorithmic Adjoint Differentiation (AAD, part III of volume I) that computes thousands of derivatives sensitivities in constant time.

The Quantiative Research department in Danske Bank, under the management of Jesper Andreasen, led the industry’s effort in that direction. In 2015, at a public Global Derivatives event in Amsterdam, they demonstrated the computation of the xVA and capital charge over a sizeable netting set, together with all its risk sensitivities, within a minute on an Apple laptop. That same year, Danske Bank won the In-House System of the Year Risk award. This publication summarizes the main quantitative, algorithmic and programmatic achievementsthat made it possible.

Volume I, written by Antoine Savine, focuses on AAD (part III) and discusses this technology and its implementation in C++, from theoretical foundations to advanced use in the context of derivatives risk management. While AAD is the core of that volume, we also introduce parallel programming in C++ in part I and discuss parallel simulations in part II as a realistic area of application for AAD.

Volume II, written by Jesper Andreasen and Antoine Savine, is dedicated to the scripting of derivatives cash-flows. That crucial technology is discussed in detail, and it is demonstrated that scripting should not be viewed as a convenience for the structuring of exotics, but as the most practical and effective means to represent and manipulate transactions and cash-flows in modern derivatives systems. Advanced extensions, such as the automation of fuzzy logic to stabilize risks, and the aggregation, compression and decoration of cash-flows for the purpose of xVA, are also covered.

Volume III, written by Brian Huge and Antoine Savine, explores effective algorithms for the computation and differentiation of xVA, and discusses in detail the efficient implementation, use and differentiation of the LSM algorithm, at the heart of Andreasen’s fast and accurate valuation algorithm for xVA. Evidently, Andreasen’s algorithm and its extensions, such as branching simulations for collateralized xVA, are also covered.

**Supervisor: Prof. Rolf Poulsen, Math, University of Copenhagen**

**Assessment committee:**

**Ass. Prof. David Skovmand (Chairman), MATH, University of Copenhagen**

**Prof. Natalie Packham, Berlin School of Economics**

**Co-head Leif B. Andersen, Bank of America Merril Lynch **