Quantitative Research

Quantitative Research

Industry Insights/White Papers | Credit | Fixed Income | Equities/FX | Hybrids | XVAs| Volatility| Interest Rate

Industry Insights/White Papers

Real-World Equity & Volatility Behavior: Implications for Economic Scenario Generation

In this research paper, written by Andrew McClelland PhD and Financial Engineering specialist at Numerix, we examine the differences between risk-neutral dynamics and real-world dynamics, and the important role of risk premia.

We explore why real-world dynamics are necessary for risk analysis and scenario generation, and also discuss the roles of the equity premium and volatility premium in stochastic volatility models of equity markets. Download paper

Integrating Risk into Pre-Trade Analysis:Practical Ways of Bringing Credit, Liquidity, Funding and Regulatory Costs into an Integrated Profitability Framework

In this research paper, written by Satyam Kancharla, Numerix Senior Vice President and Chief Strategy Officer, discusses the concept of Trade EVA (Economic Value Added) along with other industry best practices for integrating risk into pre-trade analytics and looking at derivative trade profitability more holistically. Download paper

Analytic Approximation for Prices of American Options, Time-Dependent Settings, Proportional and Discrete Dividends: The Decoupled Volatility Framework

This research paper, written by Yuriy Shkolnikov, introduces a fast analytic price approximations for American options on log-normal or close to log-normal underlying assets with discrete time-dependent parameters. Two types of dividends considered here are proportional and discrete - strike convention. Download paper

Model Validation: New Approaches in Testing Mathematical and Financial Correctness of Models

This research paper examines new approaches practitioners can utilize to improve their model validation processes. Author David Eliezer, PhD, Vice President and Head of Model Validation at Numerix, takes a deep dive into parallel implementation, convergence and stress testing; testing with mathematical identities; PDE testing; calibration round trip testing and financial correctness and hedge performance. Download paper

Model Risk: The Challenges of Legacy Code and Best Practices

This research paper, written by David Eliezer, PhD, Numerix Vice President and Head of Model Validation, explores the most common types and sources of model risk, and then outlines best practices for today’s model validation processes. Topic highlights include: sources of model risk; bug detection and prevention; challenges with poorly maintained legacy code; symptoms of unhealthy code; practices causing model risk and best practices for preventing unhealthy mathematical base code. Download paper

OIS & FVA Relationship: Evolution of OTC Derivative Funding Dynamics

This paper, written by Satyam Kancharla, Numerix Senior Vice President, explores the basics of OIS discounting and FVA for OTC derivatives—and then dives deeper into the relationship between the two concepts. The whitepaper also includes a case study that highlights the potential impact of FVA on trade profitability. The author concludes that FVA is a real charge that will affect the profitability of trades, and therefore, should impact decision-making about the trade. We come to see why it is imperative to know this cost in order to make the best business decisions. Download paper

The Numerix Approach to Funding Value Adjustment for General Financial Instruments

In this Numerix Research Brief, Dr. Ion Mihai, Quantitative Analyst, explores the quantitative point of view and how the inclusion of multiple sources of funding—such as unsecured money market funding, collateral and repo— leads to a nonlinear partial differential equation (PDE), featuring multiple discounting rates. Solving this nonlinear PDE is a difficult numerical problem, which is why disposing of a practical approximate implementation methodology is as important as working on rigorous theoretical foundations. Download paper

Mastering Model Risk: Assessment, Regulation and Best Practices

This paper explores the evolution of model risk, including current regulatory drivers and industry challenges. In this study, we also take a closer look at model risk analysis—examining model assessment, validation and review processes to manage the hidden risks within models themselves. We take a deeper dive into the sources of risk, including bad data, incorrect assumptions and methodology, and poor implementation and usage. In addition, this paper  outlines a case study of a large insurer with a diverse portfolio of assets, in which we attempt to validate the valuation and risk analytics utilized and provide an independent model assessment. In conclusion, we outline what we believe are the four best practices for model risk management. This article was written by Satyam Kancharla, Numerix Chief Strategy Officer & SVP of Client Solutions Group. Download paper

Considering Stochastic Mortality in Pricing Variable Annuities – Applications of the Lee Carter Model

This paper examines how the Lee Carter Model can be beneficial when pricing variable annuities. Mortality assumptions are crucial in many actuarial areas, such as the pricing of long-dated guarantees in life insurance, pensions and annuities. Due to its direct impact on various calculations in the insurance industry, appropriate assumptions can greatly help reduce mispricing risk. For example, in the past years, people lived longer than expected and the improvement of mortality has caused considerable losses to pension writers.

The stochastic projection of mortality improvements, per the Lee Carter model, can help actuaries better quantify the mortality/longevity risk associated with certain products. This paper was written by Chao Liang, FSA and Numerix Insurance Product Specialist. Download Paper

Risk Neutral Modeling for Economic Scenario Generation: In Theory and Practice

This paper explores the theory behind and practice surrounding Risk Neutral Modeling for Economic Scenario Generation (ESG). In addition to laying out the foundations of the Risk Neutral Measure and Fundamental Theorem, this study also sets forth several practical case study examples that demonstrate the importance of risk neutral modeling and joint calibration in market consistent valuation, regulatory reporting, risk and capital forecasting—and the overall enhancement of a firm’s risk management. The paper was written by Ghali Boukfaoui, Numerix Vice President and Insurance Product Manager. Download paper

CEB TowerGroup Research Special Report - Spotlight on CVA - Trends, Perspectives & IT Implications

In this report Dushyant Shahrawat, CFA CEB TowerGroup Special Report - Spotlight on CVASenior Research Director at CEB TowerGroup discusses the trends, perspectives and IT implications for measuring and managing Counterparty Exposure via Credit Value Adjustment (CVA). This paper captures the key business, regulatory and market factors today’s Capital markets firms are facing on their path to adopting a long-term strategic approach to CVA and Cloud-based risk analytics. Download Report

TABB Group Special Report - The Risk Analytics Library: Time for a Single Source of Truth

Credit risk is now the major focal point for risk measurement and must be treated as a core and enterprise-wide competency. With Credit Value Adjustment (CVA) being seen as a primary function of the OTC derivatives market there's an even greater challenge in accounting for this calculation as part of a holistic risk assessment – alongside a much broader spectrum of risk analytics and performance requirements for Enterprise Data Management. As financial institutions seek to achieve a unified view of risk, in this paper E. Paul Rowady Jr., Senior Capital Markets Analyst at TABB Group discusses the market drivers and technical complexities behind this shift and the role an integrated platform with a centralized multi-asset risk analytics library can play in attaining a single source of truth. Download Report

Pricing & Valuations Special Report - Inside Reference Data

In this Special Report “Pricing & Valuation” authored by Michael Shashoua, Editor of Inside Reference Data, Steven R. O’Hanlon, CEO and President of Numerix joins industry thought leaders to discuss the latest developments in pricing and valuation including market trends, quantitative development, technology innovation, and the industry’s new expectations for credibility and transparency. The virtual roundtable of industry professionals provides thought-provoking commentary on new regulatory requirements and consider what end users are looking for when choosing pricing and valuations providers. Download Report

Celent Special Report - Derivatives Pricing and Risk Analytics – Industrializing Derivatives Markets

In this special report, Cubillas Ding, Research Director at leading independent research firm Celent, focuses on the challenges ahead for many market participants in the management of their derivatives portfolios. The report also includes a derivatives pricing and risk analytics vendor review and highlights five key industry challenges, ranging from the dynamic pricing of counterparty credit risk in front office workflows and the move to “market standard” OIS/CSA discounting practices in the valuation of OTC derivatives, to the industrialization of derivative-instrument pricing infrastructures and operations. As pricing and risk analytics are the lynchpin to success in meeting these challenges, the report also highlights Numerix as the clear leader in Advanced Technology, Breadth of Functionality, Customer Base and Depth of Client Service. Download Report

Avoiding Collateral Surprises: Managing Multi-Currency CSAs

This article explores multi-currency credit support annexes (CSAs) in the derivatives area and their potential impact on pricing. Related challenges that practitioners face in terms of collateral management and optimization are also explored. The case studies therein highlight why efficient collateral management is fundamental in the new post-crisis world, and identify why cheapest-to-deliver (CTD) collateral can be necessary for profit, effectiveness and liquidity. Recently published by gt news, this article was written by Anna Barbashov, business analyst in the Client Solutions group at Numerix. Download Article

Collateral Discounting: Rethinking the Interest Rate Pricing Framework from its Basic Concepts

gt news recently published an article written by Numerix's Anna Barbashova of the Client Solutions Group, entitled, “Collateral Discounting: Rethinking the Interest Rate Pricing Framework from its Basic Concepts.” The article is a case study representing how collateral discounting and the impact of standardisation in the market is adding a whole new level of complexity when it comes to derivative pricing and risk management. Market participants are seeking a deeper understanding when it comes to the potential consequences of moving to collateral discounting. The case studies in the article clearly demonstrate the substantial divergence in single- and multi-curve pricing and risk calculation outcomes. We come to see that the entire interest rate pricing framework needs to be rethought and carefully reviewed from its basic concepts, from curve stripping and volatility surfaces to modeling and pricing, to calibration and risk management valuations. Download Article

Hedging FX Exposures: Which Strategy is Right for Your Business?

gt news recently published an article written by Numerix's Udi Sela, Vice President, outlining several interesting hedging strategies to consider using FX options, entitled, "Hedging FX Exposures: Which Strategy is Right for Your Business?" The article addresses foreign exchange risk, examines a large Swiss multinational company and the impact on its financial statements (second half of 2011), and suggests various FX option hedging strategies. The article also explores what's been going on in the FX markets since the sub-prime crisis, including the unprecedented levels of volatility that the markets have witnessed. The impact of unpredicted volatility has been significant for the core businesses of corporations across the globe. In response, various hedging strategies have been examined in this article. Download Article

Performance Benchmark Results | Windows HPC Server 2008: Numerix CrossAsset XL and Numerix Portfolio

Numerix and Microsoft have published two joint whitepapers that detail how a high-performance computing (HPC) solution enables financial services professionals to more efficiently manage their portfolios and assess risk on an interactive and day-to-day basis. Specifically, the whitepaper presents benchmark and performance test results for typical derivative portfolio use cases. Test results showed that portfolio calculation speed increased almost linearly as more compute nodes were added to a HPC cluster.
Download Numerix CrossAsset XL and Numerix Portfolio and Windows HPC Benchmark Results Paper

Thinking Forward About Variable Annuity Pricing and Hedging

We discuss recent developments in modeling and product design of variable annuities that address key challenges in the current market, including factoring correlation into models, the impact of model selection on fair rider premiums, handling large computations, an efficient method for computing Monte Carlo VaR for GMXB portfolios, and rapid product prototyping. Download Paper

A Framework for Real-time Risk Aggregation

Many of the most serious challenges facing banks boil down to this: how to get the information required to understand risks and opportunities in a format and timeframe that enables effective decision making. Many financial institutions have no consistent view of aggregated risk across asset classes, no single view of market risk, no consolidated view of exposures by counterparty and no consolidated view of market and credit risk. At the same time, it takes too long to bring new products and businesses to market, to integrate them into the bank's risk and systems architecture and no single way to define complete trades. That's where a framework for real-time risk aggregation comes in. Download Paper

Portfolio Valuations: The Changing Landscape

Though the credit crisis appears to still be in its infancy, a clear change in the way institutions value and manage complex derivatives and structured products is currently underway. Not only has the credit crisis brought market inefficacies to light, it also has forced institutions to evaluate, from an enterprise perspective, how they participate in complex markets—in terms of valuation; model validation; risk management and overall internal controls; in addition to enterprise-wide pricing policies. Download paper


Analytical Techniques for Synthetic CDOs and Credit Default Risk Measures

Pricing and risk management of synthetic CDOs and risk management of credit portfolios are closely related problems as both require modeling of the same distribution of portfolio loss. The valuation of a single tranche CDO is equivalent in complexity to the calculation of credit default VaR for a portfolio of single name entities, while the valuation of CDO-squared is a task closely related to the calculation of credit default VaR for a portfolio of single tranche CDOs. We examine the analytical techniques developed for credit portfolio problems with a view to CDO applications and find that the saddlepoint method works better than the alternatives, leading to a new, fast technique for CDO-squared pricing and hedging. Download paper

Dynamic Model for Pricing and Hedging Heterogenous CDOs

We present a simple bottom-up dynamic credit model that can be calibrated simultaneously to the market quotes on CDO tranches and individual CDSs constituting the credit portfolio. The model is most suitable for the purpose of evaluating the hedge ratios of CDO tranches with respect to the underlying credit names. Default intensities of individual assets are modeled as deterministic functions of time and the total number of defaults accumulated in the portfolio. To overcome numerical difficulties, we suggest a semi-analytic approximation that is justified by the large number of portfolio members. We calibrate the model to the recent market quotes on CDO tranches and individual CDSs and find the hedge ratios of tranches. Results are compared with those obtained within the static Gaussian Copula model. Download paper

Overlapping Credit Portfolios

We present an accurate analytical approximation for a joint distribution function of loss of two overlapping credit portfolios using the multidimensional saddlepoint method. The same method is applied to the tail probability of loss from two tranches and to CDO-squared. Numerical examples show that the default correlations effectively destroy non-normal tails, making the conditional normal approximation viable in many practical cases. Download paper

Two-Dimensional Markovian Model for Dynamics of Aggregate Credit Loss

We propose a new model for the dynamics of the aggregate credit portfolio loss. We develop a computationally efficient method for model calibration to the market of synthetic single-tranche CDOs. The method is based on the Markovian projection technique which reduces the full model to a one-step Markov chain having the same marginal distributions of loss. We show that once the intensity function of the effective Markov chain consistent with the loss distribution implied by the tranches is found, the function rho can be recovered with a very moderate computational effort. Because our model is Markovian and has low dimensionality, it offers a convenient framework for the pricing of dynamic credit instruments, such as options on indices and tranches, by backward induction. We calibrate the model to a set of recent market quotes on CDX tranches and apply it to the pricing of trance options. Download paper

Fixed Income

Analytical Formulas for Pricing CMS Products in the LMM with Stochastic Volatility

In this paper, we develop a series of approximations for a fast analytical pricing of European constant maturity swap (CMS) products, such as CMS swaps, CMS caps/floors, and CMS spread options, for the LIBOR Market Model (LMM) with stochastic volatility. The derived formulas can also be used for model calibration to the market, including European swaptions and CMS products. The first technical achievement of this work is related to the optimal calculation of the measure change. For single-rate CMS products, we have used the standard linear regression of the measure change, with optimally calculated coefficients. For the CMS spread options, where the linear procedure does not work, we propose a new effective non-linear measure change technique. The fit quality of the new results is con¯rmed numerically using Monte Carlo simulations. The second technical advance of the article is a theoretical derivation of the generalized spread option price via two-dimensional Laplace transform presented in a closed form in terms of the complex Gamma-functions. Download paper

Markovian Projection onto a Displaced Diffusion: Generic Formulas with Applications

We develop a systematic approach to Markovian projection onto an effective displaced dif- fusion, and work out a set of computationally efficient formulas valid for a large class of non-Markovian underlying processes. The generic derivation is followed by applications, including the calculation of FX options in cross-currency models and swaption pricing in LIBOR Market Models, where we are able to recover in an unambiguous way many known analytical approximations and derive several new ones. Download paper


Generalized Vanna-Volga Method and Its Applications

We give a general treatment of the Vanna-Volga mark-to-market volatility smile correction in application to pricing of contracts with European exercise on a single underlying. The method remains applicable in cases of delayed or misaligned expiries and absolute dividends. It is also applied to cases of time-dependent instantaneous volatility, multiple underlying assets and random interest rates. We also o er computation of the underlying volatility from market data and most valuable correction using more than three traded options. Download paper

Efficient Calibration to FX Options by Markovian Projection in Cross-Currency LIBOR Market Models

We revisit the cross-currency LIBOR Market Model armed with the technique of Markovian projection. We derive an efficient approximation for FX options and show how the FX skew can be modeled consistently with the interest rate skew in a common multifactor model.. Download paper

Hybrid Models

Markovian Projection to a Displaced Volatility Heston Model

Markovian Projection is an optimal approximation of a complex underlying process with a simpler one, keeping essential properties of the initial process. The Heston process, as the Markovian Projection target, is an example. In this article, we generalize the results of Markovian Projection onto a Heston model to a wider class of approximating models, a Heston model with displaced volatility. As an important application, we derive an effective approximation for FX/EQ options for the Heston model, coupled with correlated Gaussian interest rates. The main technical result is an option evaluation for correlated Heston/Lognormal processes. Unlike the case of exactly solvable (affine) zero correlation or its uncorrelated displacement generalization, considered by Andreasen, non-trivial correlations destroy affine structure and exact solvability. Using the powerful technique of Markovian Projection onto a Heston model with displaced volatility, we produce an effective approximation and present its numerical confirmation. Download paper


Funding Value Adjustment for General Financial Instruments: Theory and Practice

This research paper by the Numerix Quantitative Development Team, details a new methodology for calculating Funding Value Adjustment (FVA) for vanilla and exotic deals at both the trade, and portfolio level. The proposed methodology for FVA expands upon Vladimir Piterbarg's framework for valuation in the presence of real collateral. This Numerix approach is one of the first that allows the computation of the FVA calculation for arbitrary instrument types; even the most complex and bespoke instruments with optionality, callability and triggers. It also allows for an efficient deal-by-deal computation, which is attractive for parallelization. Download paper

Exposure & CVA for Large Portfolios of Vanilla Swaps: The Thin-Out Optimization

In this article we present an efficient optimization for calculating the exposure and CVA for large portfolios of vanilla swaps. The optimization is based on a "thin-out" procedure applied to fixed payment streams, which reduces a very frequent stream of payments to a much less frequent one. We compute the exposure and CVA for a large portfolio of fixed-for-floating swaps, and find that our approximation reduces the computation time for the portfolio to that of a single swap, with a roughly annual schedule. The approximation is highly accurate, entirely model independent and can be applied to various instruments such as FX-forwards, cross-currency swaps etc. Download paper

Risk and CVA for Exotic Derivatives: The Universal Modeling

At the 9th annual Quant Congress USA, Dr. Alexander Antonov, Senior Vice President, Quantitative Research at Numerix, presented “Risk and CVA for Exotic Derivatives: the Universal Modeling.” The presentation addressed key issues, such as: Calculation of the portfolio exposure in a self-consistent way using arbitrage-free model calibrated to both implied market and real-world projections; A new automatic method of exposure calculations (at the same time as pricing) especially attractive for exotic portfolios avoiding cumbersome exercise aggregation; And the efficient CVA calculation using the simulated information. Download Presentation

Algorithmic Exposure and CVA for Exotic Derivatives

In this article, we develop the algorithmic approach for Counterparty exposure calculation and automate its application to arbitrary complicated instruments. Assuming that the portfolio is priced by the backward (American) Monte-Carlo method, our approach allows calculating the credit exposure as a pricing by-product, essentially without modifications in the usual pricing procedure. In particular, for the exposure calculation of callable instruments, we manage to avoid a cumbersome aggregation of exercise indicators, applying them sequentially in parallel with the main pricing.

We explain how the obtained exposure can be integrated into the Credit Valuation Adjustment (CVA), based on the extension of the pricing model with a Counterparty credit component. The presented approach to the exposure computation is formulated in an arbitrary probability measure. To perform the measure change we use the cross-currency model semantics and calibrate the model to the real-world measure using indexes projections. Download paper


Decoupled American Option Pricing Method: Computation of Implied Volatilities and Further Applications

In this paper we introduce a method for volatility computation from listed prices of American options on a un-derlying close to log-normal. From prices of American calls and puts, traded at an exchange at multiple strikes we compute the underlying volatility and implied volatility of an untraded European contract at each strike. Download paper

Efficient Analytic Price Approximation for American Options

Dr. Yuriy Shkolnikov, Director of Quantitative Research at Numerix, presented “Efficient Analytic Price Approximation for American Options. Discrete Time-Dependent Parameters” at Modeling High Frequency Data in Finance 3 at Stevens Institute of Technology. The presentation addressed key issues such as: An efficient analytic approximation for American options on a log-normal underlying with time-dependent parameters, proportional or discrete dividends – strike convention; The “Decoupled Volatility Method” framework, designed to price American options on a general underlying with proportional or discrete dividends efficiently and timely; The comparison of results and computational times for the presented approximation and trinomial tree for a log-normal underlying, proportional or discrete dividends; The inverse problem of extraction of the time-dependent Implied Volatility curve and Underlying Volatility surface for a general underlying with proportional or discrete dividends. Download Presentation

Interest Rate

Advanced Analytics for the SABR Model

This paper presents advanced analytical formulas for SABR model option pricing. The first technical result consists of a new exact formula for the zero correlation case. The second result is an effective approximation of the general correlation case. These formulas are important in volatility surface construction and CMS product replication because they provide correct behavior for far strikes and reduced approximation error. The latter is also helpful for dynamic SABR models. Download paper

Projection on a Quadratic Model by Asymptotic Expansion with an Application to LMM Swaption

This paper develops a technique of parameter averaging and Markovian projection on a quadratic volatility model based on a term-by-term matching of the asymptotic expansions of option prices in volatilities. In doing so, the paper revisits the procedure of asymptotic expansion and show that the use of the product formula for iterated Itô integrals leads to a considerable simplification in comparison with the approach currently prevalent in the literature. Results are applied to the classic problem of LIBOR Market Model (LMM) swaption pricing. We confirm numerically that the retention of the quadratic term gives a marked improvement over the standard approximation based on the projection on a displaced diffusion. Download paper

Markovian Projection Onto a Heston Model

This paper develops a systematic approach to the reduction of dimensionality of smile-enabled models by projecting them onto a displaced version of the two-dimensional Heston process. The projection is the key for deriving efficient, analytical approximations to European option prices in such models. This is a further development of the method of Markovian projection previously used for projecting on the displaced-diffusion process (with skew but without smile). The method is derived in a generic form and has a wide range of suitable applications. Examples for spread and basket options are given. Download paper

Interest Rate Modelling Framework in Discrete Rolling Spot Measure

This paper presents advanced analytical formulas for SABR model option pricing. The first technical result consists of a new exact formula for the zero correlation case. The second result is an effective approximation of the general correlation case. These formulas are important in volatility surface construction and CMS product replication because they provide correct behavior for far strikes and reduced approximation error. The latter is also helpful for dynamic SABR models. Download paper

Analytical Approximations for Short Rate Models

In this article, we present the analytical approximation of zero-coupon bonds and swaption prices for general short rate models. The approximation is based on regular and singular expansions with respect to the small volatility and contains a low-dimensional integration. The model in hand assumes the short rate is an arbitrary function of a multi-dimensional Gaussian underlying process. The high approximation accuracy is confirmed by numerical experiments. We have treated two special classes of the model. The first one is a Generalized multi-factor Black-Karasinski (BK) model. The second one is a new Bounded short rate model where the rates evolve between certain user-defined limits. This model is particularly attractive for scenario generation and, due to the proposed swaption approximation, can be easily calibrated to the implied market. Download paper


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Numerix provided BlackRock Solutions the models it needed to expand into FX exotics

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Northern Trust leverages Numerix & Prism Valuation to expand its OTC derivative valuation platform

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