Quantitative Research

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Industry Insights/White Papers

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

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

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

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. Request paper

Numerix CrossAsset XL and Windows HPC Server 2008 R2: Faster Performance for Valuation and Risk Management in Complex Derivative Portfolios

Numerix and Microsoft HPC published a joint whitepaper entitled, “Numerix CrossAsset XL and Windows HPC Server 2008 R2: Faster Performance for Valuation and Risk Management in Complex Derivative Portfolios,” that details 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 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

Credit

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. Request 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. Request 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. Request 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. Request 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. Request 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. Request paper

Equities/FX

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. Request 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. Request paper