Semistatic and sparse variance‐optimal hedging

We consider hedging of a contingent claim by a 'semi-static' strategy composed of a dynamic position in one asset and static (buy-and-hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance-optimality and provide tractable formulas using Fourier-integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semi-static hedging strategy, i.e. a strategy which only uses a small subset of available hedging assets. The developed methods are illustrated in an extended numerical example where we compute a sparse semi-static hedge for a variance swap using European options as static hedging assets.

[1]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[2]  R. Tibshirani.  Regression Shrinkage and Selection via the Lasso , 1996 .

[3]  Sean Ryan Gillooly.  The CBOE Volatility Index (VIX) as a Predictor of S&P 500 Volatility and Returns , 2013 .

[4]  E. Lukács.  CERTAIN ENTIRE CHARACTERISTIC FUNCTIONS , 2005 .

[5]  Christophe Stricker,et al.  Décomposition de Kunita-Watanabe , 1993 .

[6]  Arnold Neumaier,et al.  Solving Ill-Conditioned and Singular Linear Systems: A Tutorial on Regularization , 1998, SIAM Rev..

[7]  Alan L. Lewis.  Option Valuation under Stochastic Volatility , 2000 .

[8]  Paolo Di Tella,et al.  Semi-static variance-optimal hedging in stochastic volatility models with Fourier representation , 2017, J. Appl. Probab..

[9]  김동일,et al.  LARS(Least Angle Regression)와 유전알고리즘을 결합한 변수 선택 알고리즘 , 2009 .

[10]  Affine diffusion processes: theory and applications , 2009 .

[11]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[12]  S. Crawford,et al.  Volume 1 , 2012, Journal of Diabetes Investigation.

[13]  George M. Furnival,et al.  Regressions by leaps and bounds , 2000 .

[14]  D. Steinberg,et al.  Technometrics , 2008 .

[15]  B. Eckmann,et al.  Séminaire de Probabilités V Université de Strasbourg , 1971 .

[16]  Akihiko Takahashi,et al.  A Hybrid Asymptotic Expansion Scheme: An Application to Long-Term Currency Options , 2007 .

[17]  A. Neuberger,et al.  The Log Contract , 1994 .

[18]  P. Carr,et al.  A new approach for option pricing under stochastic volatility , 2007 .

[19]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[20]  P. Friz,et al.  Moment Explosions in Stochastic Volatility Models , 2010 .

[21]  Hans Föllmer,et al.  Hedging by Sequential Regression: An Introduction to the Mathematics of Option Trading , 1988 .

[22]  Sara van de Geer,et al.  Statistics for High-Dimensional Data , 2011 .

[23]  Huyên Pham,et al.  On quadratic hedging in continuous time , 2000, Math. Methods Oper. Res..

[24]  秀俊 松井,et al.  Statistics for High-Dimensional Data: Methods, Theory and Applications , 2014 .

[25]  Aurélien Alfonsi,et al.  Affine Diffusions and Related Processes: Simulation, Theory and Applications , 2015 .

[26]  Friedrich Hubalek,et al.  Variance-Optimal Hedging for Processes with Stationary Independent Increments , 2006, math/0607112.

[27]  S. Raible,et al.  Lévy Processes in Finance: Theory, Numerics, and Empirical Facts , 2000 .

[28]  Peter Carr.  SEMI-STATIC HEDGING OF BARRIER OPTIONS UNDER POISSON JUMPS , 2011 .

[29]  M. Rosenbaum,et al.  The characteristic function of rough Heston models , 2016, 1609.02108.

[30]  Vladimir V. Piterbarg,et al.  Moment explosions in stochastic volatility models , 2005, Finance and Stochastics.

[31]  Jan Kallsen,et al.  Variance-Optimal Hedging in General Affine Stochastic Volatility Models , 2010, Advances in Applied Probability.

[32]  Jim Gatheral.  The Volatility Surface: A Practitioner's Guide , 2006 .

[33]  Euclid,et al.  The annals of applied probability : an official journal of the Institute of Mathematical Statistics. , 1991 .

[34]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[35]  M. Beiglböck,et al.  Model-independent bounds for option prices—a mass transport approach , 2011, Finance and Stochastics.

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

[37]  H. Kunita,et al.  On Square Integrable Martingales , 1967, Nagoya Mathematical Journal.

[38]  R. Lathe.  Phd by thesis , 1988, Nature.

[39]  Mathias Beiglböck,et al.  Model-independent bounds for option prices—a mass transport approach , 2011, Finance Stochastics.

[40]  M. Schweizer.  A guided tour through quadratic hedging approaches , 1999 .