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USLV: Unspanned Stochastic Local Volatility Model


  • Igor Halperin
  • Andrey Itkin


We propose a new framework for modeling stochastic local volatility, with potential applications to modeling derivatives on interest rates, commodities, credit, equity, FX etc., as well as hybrid derivatives. Our model extends the linearity-generating unspanned volatility term structure model by Carr et al. (2011) by adding a local volatility layer to it. We outline efficient numerical schemes for pricing derivatives in this framework for a particular four-factor specification (two "curve" factors plus two "volatility" factors). We show that the dynamics of such a system can be approximated by a Markov chain on a two-dimensional space (Z_t,Y_t), where coordinates Z_t and Y_t are given by direct (Kroneker) products of values of pairs of curve and volatility factors, respectively. The resulting Markov chain dynamics on such partly "folded" state space enables fast pricing by the standard backward induction. Using a nonparametric specification of the Markov chain generator, one can accurately match arbitrary sets of vanilla option quotes with different strikes and maturities. Furthermore, we consider an alternative formulation of the model in terms of an implied time change process. The latter is specified nonparametrically, again enabling accurate calibration to arbitrary sets of vanilla option quotes.

Suggested Citation

  • Igor Halperin & Andrey Itkin, 2013. "USLV: Unspanned Stochastic Local Volatility Model," Papers 1301.4442,, revised Mar 2013.
  • Handle: RePEc:arx:papers:1301.4442

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    References listed on IDEAS

    1. Mark Britten-Jones & Anthony Neuberger, 2000. "Option Prices, Implied Price Processes, and Stochastic Volatility," Journal of Finance, American Finance Association, vol. 55(2), pages 839-866, April.
    2. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    3. Pierre Collin-Dufresne & Robert S. Goldstein, 2002. "Do Bonds Span the Fixed Income Markets? Theory and Evidence for Unspanned Stochastic Volatility," Journal of Finance, American Finance Association, vol. 57(4), pages 1685-1730, August.
    4. Igor Halperin, 2009. "Implied Multi-Factor Model for Bespoke CDO Tranches and other Portfolio Credit Derivatives," Papers 0910.2696,
    5. Mario Cerrato & Chia Chun Lo & Konstantinos Skindilias, 2011. "Adaptive continuous time Markov chain approximation model to general jump-diffusions," Working Papers 2011_16, Business School - Economics, University of Glasgow.
    6. Xavier Gabaix, 2007. "Linearity-Generating Processes: A Modelling Tool Yielding Closed Forms for Asset Prices," NBER Working Papers 13430, National Bureau of Economic Research, Inc.
    7. repec:dau:papers:123456789/1392 is not listed on IDEAS
    8. Puzanova, Natalia, 2011. "A hierarchical model of tail dependent asset returns for assessing portfolio credit risk," Discussion Paper Series 2: Banking and Financial Studies 2011,16, Deutsche Bundesbank.
    9. Helyette Geman & P. Carr & D. Madan & M. Yor, 2003. "Stochastic Volatility for Levy Processes," Post-Print halshs-00144385, HAL.
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    Cited by:

    1. Konstantinos Skindilias & Chia Lo, 2015. "Local volatility calibration during turbulent periods," Review of Quantitative Finance and Accounting, Springer, vol. 44(3), pages 425-444, April.
    2. Andrey Itkin, 2015. "LSV models with stochastic interest rates and correlated jumps," Papers 1511.01460,, revised Nov 2016.

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