Variance Swaps on Defaultable Assets and Market Implied Time-Changes

We compute the value of a variance swap when the underlying is modeled as a Markov process time changed by a L\'{e}vy subordinator. In this framework, the underlying may exhibit jumps with a state-dependent L\'{e}vy measure, local stochastic volatility and have a local stochastic default intensity. Moreover, the L\'{e}vy subordinator that drives the underlying can be obtained directly by observing European call/put prices. To illustrate our general framework, we provide an explicit formula for the value of a variance swap when the underlying is modeled as (i) a L\'evy subordinated geometric Brownian motion with default and (ii) a L\'evy subordinated Jump-to-default CEV process (see \citet{carr-linetsky-1}). {In the latter example, we extend} the results of \cite{mendoza-carr-linetsky-1}, by allowing for joint valuation of credit and equity derivatives as well as variance swaps.

[1]  Pascal J. Maenhout,et al.  Individual Stock-Option Prices and Credit Spreads , 2004 .

[2]  Vadim Linetsky,et al.  A jump to default extended CEV model: an application of Bessel processes , 2006, Finance Stochastics.

[3]  佐藤 健一.  Lévy processes and infinitely divisible distributions , 2013 .

[4]  Roger Lee,et al.  Variance swaps on time-changed Lévy processes , 2012, Finance Stochastics.

[5]  S. Bochner.  Diffusion Equation and Stochastic Processes. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[6]  F. Neubrander,et al.  LAPLACE TRANSFORM METHODS FOR EVOLUTION EQUATIONS , 2000 .

[7]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[8]  Douglas T. Breeden,et al.  Prices of State-Contingent Claims Implicit in Option Prices , 1978 .

[9]  René L. Schilling,et al.  Bernstein Functions: Theory and Applications , 2010 .

[10]  V. Linetsky,et al.  Time-Changed CIR Default Intensities with Two-Sided Mean-Reverting Jumps , 2014, 1403.5402.

[11]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[12]  P. Carr,et al.  Option Pricing, Interest Rates and Risk Management: Towards a Theory of Volatility Trading , 2001 .

[13]  V. Linetsky.  PRICING EQUITY DERIVATIVES SUBJECT TO BANKRUPTCY , 2006 .

[14]  T. Bielecki,et al.  Credit Risk: Modeling, Valuation And Hedging , 2004 .

[15]  J. Bertoin.  Subordinators: Examples and Applications , 1999 .

[16]  Peter Carr,et al.  A Simple Robust Link between American Puts and Credit Protection , 2008 .

[17]  R. Phillips,et al.  On the generation of semigroups of linear operators. , 1952 .

[18]  P. Carr,et al.  Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation , 2006 .

[19]  P. Carr,et al.  Hedging Under the Heston Model with Jump-to-Default , 2007 .

[20]  Rama Cont,et al.  A CONSISTENT PRICING MODEL FOR INDEX OPTIONS AND VOLATILITY DERIVATIVES , 2009 .

[21]  Jow-Ran Chang,et al.  Cross-Market Hedging Strategies for Credit Default Swaps under a Markov Regime-Switching Framework , 2012, The Journal of Fixed Income.

[22]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[23]  H. McKean.  Elementary solutions for certain parabolic partial differential equations , 1956 .

[24]  Oleg Bondarenko.  Estimation of Risk-Neutral Densities Using Positive Convolution Approximation , 2002 .

[25]  B. Davies,et al.  Numerical Inversion of the Laplace Transform: A Survey and Comparison of Methods , 1979 .

[26]  E. C. Titchmarsh,et al.  The Laplace Transform , 1991, Heat Transfer 1.

[27]  V. B. Uvarov,et al.  Special Functions of Mathematical Physics: A Unified Introduction with Applications , 1988 .