Annals of Finance

, Volume 13, Issue 4, pp 401–434 | Cite as

Analysis of variance based instruments for Ornstein–Uhlenbeck type models: swap and price index

  • Aziz Issaka
  • Indranil SenGupta
Research Article


In this paper a couple of variance dependent instruments in the financial market are studied. Firstly, a number of aspects of the variance swap in connection to the Barndorff-Nielsen and Shephard model are studied. A partial integro-differential equation that describes the dynamics of the arbitrage-free price of the variance swap is formulated. Under appropriate assumptions for the first four cumulants of the driving subordinator, a Večeř-type theorem is proved. The bounds of the arbitrage-free variance swap price are also found. Finally, a price-weighted index modulated by market variance is introduced. The large-basket limit dynamics of the price index and the “error term” are derived. Empirical data driven numerical examples are provided in support of the proposed price index.


Barndorff-Nielsen and Shephard model Variance swap Stochastic volatility Price index Weak convergence 

JEL Classification

C02 D53 G10 L16 



The authors would like to thank the anonymous reviewers for their careful reading of the manuscript and for suggesting points to improve the quality of the paper.


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsNorth Dakota State UniversityFargoUSA

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