Papers of interest

This paper presents how to use Chebyshev Tensors to compute dynamic sensitivities of financial instruments within a Monte Carlo simulation. Dynamic sensitivities are then used to compute Dynamic Initial Margin as defined by ISDA (SIMM). The technique is benchmarked against the computation of dynamic sensitivities obtained by using pricing functions like the ones found in risk engines. We obtain high accuracy and computational gains for FX swaps and Spread Options.

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Published in Risk Magazine, Cutting Edge Section, April 2021.

In this paper we introduce a new technique based on high-dimensional Chebyshev Tensors that we call Orthogonal Chebyshev Sliding Technique. We implemented this technique inside the systems of a tier-one bank, and used it to approximate Front Office pricing functions in order to reduce the substantial computational burden associated with the capital calculation as specified by FRTB IMA. In all cases, the computational burden reductions obtained were of more than 90%, while keeping high degrees of accuracy, the latter obtained as a result of the mathematical properties enjoyed by Chebyshev Tensors.

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Published in Wilmott Magazine, January 2021.

Inspired by a series of remarkable papers in recent years that use Deep Neural Nets to substantially speed up the calibration of pricing models, we investigate the use of Chebyshev Tensors instead of Deep Neural Nets. Given that Chebyshev Tensors can be, under certain circumstances, more efficient than Deep Neural Nets at exploring the input space of the function to be approximated — due to their exponential convergence —  the problem of calibration of pricing models seems, a priori, a good case where Chebyshev Tensors can excel.

In this piece of research, we built Chebyshev Tensors — either directly or with the help of the Tensor Extension Algorithms — to tackle the computational bottleneck associated with the calibration of the rough Bergomi volatility model. Results are encouraging as the accuracy of model calibration via Chebyshev Tensors is similar to that when using Deep Neural Nets, but with building efforts that range between 5 and 100 times more efficient in the experiments run. Our tests indicate that when using Chebyshev Tensors, the calibration of the rough Bergomi volatility model is around 40,000 times more efficient than if calibrated via “brute-force” (using the pricing function).

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Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real-time. Simultaneously we observe an increase in model sophistication on the one hand and growing demands on the quality of risk management on the other. To address the resulting computational challenges, it is natural to exploit the recurrent nature of these tasks. We concentrate on Parametric Option Pricing (POP) and show that polynomial interpolation in the parameter space promises to reduce run-times while maintaining accuracy. The attractive properties of Chebyshev interpolation and its tensorized extension enable us to identify criteria for (sub)exponential convergence and explicit error bounds. We show that these results apply to a variety of European (basket) options and affine asset models. Numerical experiments confirm our findings. Exploring the potential of the method further, we empirically investigate the efficiency of the Chebyshev method for multivariate and path-dependent options.

By Maximilian Gaß, Kathrin Glau, Mirco Mahlstedt, Maximilian Mair

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Treating high dimensionality is one of the main challenges in the development of computational methods for solving problems arising in finance, where tasks such as pricing, calibration, and risk assessment need to be performed accurately and in real-time. Among the growing literature addressing this problem, Gass et al. [14] propose a complexity reduction technique for parametric option pricing based on Chebyshev interpolation. As the number of parameters increases, however, this method is affected by the curse of dimensionality. In this article, we extend this approach to treat high-dimensional problems: Additionally exploiting low-rank structures allows us to consider parameter spaces of high dimensions. The core of our method is to express the tensorized interpolation in tensor train (TT) format and to develop an efficient way, based on tensor completion, to approximate the interpolation coefficients. We apply the new method to two model problems: American option pricing in the Heston model and European basket option pricing in the multi-dimensional Black-Scholes model. In these examples we treat parameter spaces of dimensions up to 25. The numerical results confirm the low-rank structure of these problems and the effectiveness of our method compared to advanced techniques.

By Kathrin Glau, Daniel Kressner, Francesco Statti

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The implied volatility is a crucial element of any financial toolbox, since it is used for quoting and the hedging of options as well as for model calibration. In contrast to the Black-Scholes formula its inverse, the implied volatility, is not explicitly available and numerical approximation is required. We propose a bivariate interpolation of the implied volatility surface based on Chebyshev polynomials. This yields a closed-form approximation of the implied volatility, which is easy to implement and to maintain. We prove a subexponential error decay. This allows us to obtain an accuracy close to machine precision with polynomials of a low degree. We compare the performance of the method in terms of runtime and accuracy to the most common reference methods. In contrast to existing interpolation methods, the proposed method is able to compute the implied volatility for all relevant option data. In this context, numerical experiments confirm a considerable increase in efficiency, especially for large data sets.

By Kathrin Glau, Paul Herold, Dilip B. Madan, Christian Pötz

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We present a neural network based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models -including the rough volatility family- and a range of derivative contracts. The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several numerical pricers and model families (such as rough volatility models) within the scope of applicability in industry practice. The form in which information from available data is extracted and stored influences network performance: This approach is inspired by representing the implied volatility and option prices as a collection of pixels. In a number of applications we demonstrate the prowess of this modelling approach regarding accuracy, speed, robustness and generality and also its potentials towards model recognition.

By Blanka Horvath, Aitor Muguruza, Mehdi Tomas

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Differential machine learning combines automatic adjoint differentiation (AAD) with modern machine learning (ML) in the context of risk management of financial Derivatives. We introduce novel algorithms for training fast, accurate pricing and risk approximations, online, in real-time, with convergence guarantees. Our machinery is applicable to arbitrary Derivatives instruments or trading books, under arbitrary stochastic models of the underlying market variables. It effectively resolves computational bottlenecks of Derivatives risk reports and capital calculations.

Differential ML is a general extension of supervised learning, where ML models are trained on examples of not only inputs and labels but also differentials of labels wrt inputs. It is also applicable in many situations outside finance, where high quality first-order derivatives wrt training inputs are available. Applications in Physics, for example, may leverage differentials known from first principles to learn function approximations more effectively.

In finance, AAD computes pathwise differentials with remarkable efficacy so differential ML algorithms provide extremely effective pricing and risk approximations. We can produce fast analytics in models too complex for closed form solutions, extract the risk factors of complex transactions and trading books, and effectively compute risk management metrics like reports across a large number of scenarios, backtesting and simulation of hedge strategies, or regulations like XVA, CCR, FRTB or SIMM-MVA.

By Brian Huge, Antoine Savine

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Backtesting Counterparty Credit Risk models is anything but simple. Such backtesting is becoming increasingly important in the financial industry since both the CCR capital charge and CVA management have become even more central to banks. In spite of this, there are no clear guidelines by regulators as to how to perform this backtesting. This is in contrast to Market Risk models, where the Basel Committee set a strict set of rules in 1996 which are widely followed. In this paper, the author explains a quantitative methodology to backtest counterparty risk models. He expands the three-color Basel Committee scoring scheme from the Market Risk to the Counterparty Credit Risk framework. With this methodology, each model can be assigned a color score for each chosen time horizon. Financial institutions can then use this framework to assess the need for model enhancements and to manage model risk. The author has implemented this framework in Tier-1 a financial institution; the model report it generated was sent to the regulators for IMM model approval. The model was approved a few months later.

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Published in Journal of Credit Risk, March 2014.

In this paper, the authors…

• First explain the underlying source of this risk and how it applies to CVA as well as other credit metrics, together with a review of the available methodologies.
• Further to it, they provide a critique of the different models and their view as to which is the optimal framework, and why. This is done from the standpoint of a practitioner, with special consideration of practical implementation and utilisation issues.
• After that, they extend the current state-of-the-art research in the chosen methodology with a comprehensive empirical analysis of the market-credit dependency structure. They utilise 150 case studies, providing evidence of what is the real market-credit dependency structure, and giving calibrated model parameters as of January 2013.
• Next, using these realistic calibrations, they carry out an impact study of right-way and wrong-way risk in real trades, in all relevant asset classes (equity, FX and commodities) and trade types (swaps, options and futures). This is accomplished by calculating the change in all major credit risk metrics that banks use (CVA, initial margin, exposure measurement and capital) when this risk is taken into account.
• All this is done both for collateralised and uncollateralised trades.
• Finally, based on this impact study, the authors explain why a good right and wrong way risk model (as opposed to “any” model that gives a result) is cen- tral to financial institutions, furthermore describing the consequences of not having one.

The results show how these credit metrics can vary quite significantly, both in the “right” and the “wrong” ways. This analysis also illustrates the effect of collateral; for example, how a trade can have wrong-way risk when uncollateralised, but right-way risk when collaterallised.

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Published in the Financial Analyst Journal, Mar/Apr 2015

The Basel consultation proposes to remove risk sensitive models, or put floors to them, for Counterparty Credit Risk capital calculations of OTC derivatives.

I believe that, while the Committee’s objectives are understandable, the changes proposed would have very damaging consequences indeed. For that reason I submitted my view to the committee as an independent expert in the field.

In this paper, I explain that, should the new proposals be implemented,

1. Systemic risk is created, as when simple models fail then they will fail in all banks in the world at the same time. This is precisely one of the key risks the regulatory landscape is mandated to remove from the trading system.
2. Market dislocations are created, because trading activity will be steered to those areas with low capital requirements but high relative risk as this is where return on capital will be maximised.
3. The distribution of risk is inhibited, as a consequence of  non-economic-based excessive capital flat rules compared to advanced economic-based modelled capital. This would clearly have an (unnecessary) negative economic impact.
4. Banks are un-incentivised to invest in sound risk management practices, as it sends messages to the industry that a bank does not need to be concerned about adopting its own internal risk controls.
5. Innovation is inhibited. Innovation is at the core of any healthy industry, economy and society. Why would a bank invest in research and development of better risk management processes, methodologies and systems when the potential benefits that might accrue have been removed by regulation?

No doubt there is some benefit in simplifying rules for both regulator and for some players in the industry, but those benefits need to be balanced out with the reduction in the efficacy of those rules. One of the fundamental goals of those rules is to provide a stable financial system on which the real economy can be based and grow. My intention with this document is to bring to the Committee’s attention the fact that the new proposals materially reduce the efficacy of the capital framework; i.e., that they increase the instability of the financial system and constrain real economic growth.

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Pricing a book of derivatives has become quite a complicated task, even when those derivatives are simple in nature. This is the effect of the new trading environment, highly dominated by credit, funding and capital costs. In this paper the author formally sets up a global valuation framework that accounts for market risk (risk neutral price), credit risk (bilateral CVA), funding risk (FVA) of self-default potential hedging (LVA), collateral (CollVA) and market hedging positions (HVA), as well as tail risk (KVA). These pricing metrics create a framework in which we can comprehensively value trading activity. An immediate consequence of this is the emergence of a potential difference between fair value accounting and internal accounting. This piece of work also explains the difference between both of them, and how to perform calculations in both worlds in a realistic and coherent manner, demonstrating via arbitrage-impossibility arguments that an XVA frameworks should be used in both cases.

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When modelling credit spreads, there is some controversy in the market as to whether they are mean-reverting or not. This is particularly important in the context of counterparty risk, at least for risk management and capital calculations, as those models need to backtest correctly and, hence, they need to follow the “real” measure, as opposed to the “risk-neutral” one. This paper shows evidence that the credit spreads of individual corporate names, by themselves, are not mean-reverting. Our results also suggest that a mean-reversion feature should be implemented in the context of joint spread-default modelling, but not in a spread-only model.

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Published in Intelligent Risk, October 2012.

Exposure models in the context of counterparty risk have become central to financial institutions. They are a main driver of CVA pricing, capital calculation and risk management. It is general practice in the industry to use constant-volatility normal or log-normal models for it. Ignacio Ruiz and Ricardo Pachón explain some of the strong limitations of those models and show how stochastic volatility can improve the situation substantially. This is shown with illustrative examples that tackle day-to-day problems that practitioners face. Using a coupled Black-Karasinski model for the volatilty and a GBM model for the spot as an example, it is shown how stochastic volatility models can provide tangible benefits by improving netting effects, CVA pricing accuracy, regulatory capital calculation, initial margin calculations and quality of exposure management.

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Wrong Way Risk can be of crucial importance when computing counterparty risk measurements like EPE or PFE profiles. It appears when the default probability a given counterparty is not independent of its portfolio value. There are a number of approaches in the literature but, to the author’s knowledge, they all fail to provide either a computationally efficient approach or an intuitive methodology, or both. This technical note tackles this problem and describes an intuitive and fairly easy method to account for Wrong Way Risk with minimal added computational effort.

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CVA can be priced using market implied ‘risk-neutral’ or historical ‘real-world’ parameters. There is no consensus in the market as to which approach to use. This paper illustrates in simple terms the fundamental differences between the two approaches and the consequences of using each of them.

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Credit Value Adjustment (CVA) has been one of the “hot topics” in the financial industry since 2009. There have been several papers on the subject and the topic has been widely discussed in all banking conferences around the world. However, often, the fundamentals of this topic have been misunderstood or misinterpreted. Arguably, this is the result of a typical problem within quants (I am a quant): we have a tendency to explain things in unnecessarily complicated ways. This paper aims to bridge that gap. It will explain using simple examples how CVA has been priced into banking products for centuries and how it is, in fact, the root of the banking industry since its origin. It will show how CVA is nothing more than the application to modern financial derivatives of very simple and old concepts, and how its misinterpretation can lead to accounting mismanagement. The mathematical formulation for it will not be included in this text as it can be found in several publications.

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CVA,  DVA, FVA and their interaction, Part I.

This is the first part of a dual paper on FVA. This one offers a review of the concept behind FVA, and how it interacts with CVA and DVA.

There is currently a rather heated discussion about the role of Funding Value Adjustment (FVA) when pricing OTC derivative contracts. On one hand, theorists claim that value should not be accounted for as it leads to arbitrage opportunities. On the other hand, practitioners say they need to account for it, as otherwise their cost base is not reflected in the price of a contract. On the surface these claims seem to be contradictory, but upon closer examination in fact they are not. In this paper, we define FVA, explain its role, how it interacts with CVA, and how it should (and should not) be used in an organisation. We shall see, however surprising it may sound, that this debate can be seen as a semantic misunderstanding. The two sides of the argument are using the same word `price’ for two very different things: `fair price’ and `value to me’. Noting this, the paradox disappears, and we may properly understand the role of CVA, DVA and FVA in an organisation.

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CVA,  DVA, FVA and their interaction, Part II.

This is the second part of a dual paper on FVA. It explains how to calculate FVA and how to manage funding risk.

The calculation and management of funding risk for a portfolio of OTC derivatives is anything but trivial. In the first part of this paper (FVA Demystified), we discussed the ideas underlying FVA. We saw that it is an adjustment made to the price of a portfolio of derivatives to account for the future funding cost an institution might face. We also saw that it is very important to differentiate between the Price of a derivative (the amount of money we would get if we sell the derivative) and the Value to Me (the Price minus my cost of manufacturing the derivative). In this paper, we are going to investigate the practicalities of FVA. We will see how FVA can be calculated and managed. If we have a proper CVA system, calculating FVA is not too difficult, subject to a few reasonable assumptions. This can be achieved because a good CVA Monte Carlo simulation already calculates many of the inputs needed to compute FVA. Also, we will explain the role of FVA desks in current large organisations, as well as how FVA can be risk-managed and hedged. Finally, we propose a management set up for CVA and FVA, and understand why a number of institutions have decided to join both desks.

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