The Unspanned Stochastic-Local Volatility Model (USLV) Combines Benefits of Stochastic and Local Volatility Modeling for Derivatives Pricing & Hedging

New York, NY – April 2, 2013 – Numerix (, the leading provider of cross-asset analytics for derivatives valuations and risk management, today announced the results of its latest quantitative research paper “USLV: Unspanned Stochastic Local Volatility Model”[1] authored by Dr. Igor Halperin, Executive Director of Model Risk and Development at JPMorgan and Dr. Andrey Itkin, Senior Vice President, Head of Quantitative Development at Numerix, both Adjunct Professors at the Polytechnic Institute of New York University.

The paper establishes a new framework for modeling stochastic-local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity and FX, as well as hybrid derivatives. The USLV model extends the Unspanned Stochastic Volatility model proposed by Dr. Peter Carr and Profs. Xavier Gabaix and Liuren Wu in 2011[2] by adding a local volatility layer to the model. The latter added stochastic volatility into the class of Linearity-Generating processes (LGP), originally discovered by X. Gabaix in 2007[3].  

This approach is an attractive alternative, both theoretically and practically to the affine models, as well as to Heath-Jarrow-Morton (HJM)-like models. This is because the new framework is more economic than HJM-like models in terms of the number of stochastic Markov processes that need to be modelled. For example, the number of Markov drivers for a four-factor specification of the LGP model is four for both interest rate and equity derivatives, while a four-factor HJM model would require seven Markov drivers.

“The USLV model resulted from our attempt to construct a generic multi-factor stochastic-local volatility model that would be able to provide unified modeling methodology and shared implementation for derivatives pricing for various asset classes,” said Dr. Andrey Itkin, Senior Vice President, Head of Quantitative Development at Numerix. “Though adding a local volatility layer is a seemingly simple change, it requires a totally different numerical approach, and this is exactly where our experience with numerical methods helped.”

Dr. Igor Halperin, Executive Director of Model Risk and Development at JPMorgan, adds: “Our framework for local volatility enables perfect matching of an arbitrary number of European vanilla options quotes with different strikes and maturities. This allows us to price both vanilla and exotic derivatives, especially if vanilla options are being used to hedge the exotics. By merging such flexibility with a multi-factor LGP framework, we hope to produce an accurate and efficient stochastic local volatility model that could work across different asset classes.”

The USLV model outlines an efficient numerical scheme for pricing derivatives for a 4-factor specification, namely two curve factors plus two volatility factors enabling a unified implementation of a multi-factor stochastic volatility model that would work for both interest rate and equity derivatives. Based on the Markov chain approximation, this approach was inspired by a similar construction developed by Dr. Igor Halperin for credit portfolio derivatives. Known as the Markovian Bivariate Spread-Loss model (BSLP model)[4], it has been adopted by a number of Wall Street firms, and is available through the Numerix model library.

“The joint research of Drs. Itkin and Halperin represents the best of what the quantitative finance industry has to offer in terms of expertise, innovation and leadership,” said Steven R. O’Hanlon, Chief Executive Officer & President of Numerix. “While the implementation and empirical validation of the new framework is still underway, this important research is sure to bring further benefits and efficiencies to the complex world of OTC derivatives pricing.”

1 USLV: Unspanned Stochastic Local Volatility Model

2 Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing

3 Linearity-Generating Processes: A modeling Tool Yielding Closed Forms for Asset Prices

4 M. Arnsdorf and I. Halperin, “BSLP: Markovian Bivariate Spread-Loss Model for Portfolio Credit Derivatives,” Journal of Computational Finance, 12 (2007), 77-107 (

About Numerix
Numerix is the award winning, leading independent analytics institution providing cross-asset solutions for structuring, pre-trade price discovery, trade capture, valuation and portfolio management of derivatives and structured products. Since its inception in 1996, over 700 clients and 75 partners across more than 25 countries have come to rely on Numerix analytics for speed and accuracy in valuing and managing the most sophisticated financial instruments. With offices in New York, London, Paris, Frankfurt, Milan, Stockholm, Tokyo, Hong Kong, Singapore, Dubai, South Korea, India and Australia, Numerix brings together unparalleled expertise across all asset classes and engineering disciplines. For more information please visit

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