Robust methods for the SABR model and related processes: analysis, asymptotics and numerics

This thesis is dedicated to the study of the stochastic alpha beta rho (SABR) model and related stochastic processes both from a theoretical and a practical perspective. During its 14 years of existence, the SABR model has become industry standard and is now ubiquitous in interest rate modelling. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods, which strengthened the connection between geometry and finance. In recent years markets have moved to historically low rates for which this expansion is prone to yield inconsistent prices, thereby inducing distortions and arbitrage into modelling. As SABR is deeply embedded in the markets, there is undisputed need for uniform pricing methods—suitable for the SABR model—which eliminate the observed irregularities. Since the emergence of the aforementioned problems, the so-called arbitrage issue has been addressed in numerous approaches. Despite several excellent contributions to this matter in the past years, to date there does not seem to be a consensus about a method for adjusting the SABR formula, or about the reasons for the potential appearance of inconsistent prices under this model using different pricing tools. The aim of this thesis is to investigate the properties of the model from this perspective and to propose some effective solutions to the arbitrage problem. An analysis of the SABR model for near-zero (positive or negative) rates calls for a more general functional analytic framework than the one that Riemannian geometry and heat kernel expansions can provide. This in turn can be held accountable for the breakdown of the asymptotic formula in this region. While several available methods aim for exact approximation of the absolutely continuous part of the SABR distribution, we confirm that the asymptotics of the implied volatility for extreme strikes (for which the inconsistencies typically appear) can be fully characterized by the mass at zero, regardless of the absolutely continuous part of the distribution. Accordingly, we also propose a finite element method—tailored to the specific degeneracy of the model at the origin—in order to evaluate option prices under the SABR model. Finally we prove convergence and derive error estimates for the proposed numerical scheme.

[1]  Pascal J. Maenhout.  Robust portfolio rules and detection-error probabilities for a mean-reverting risk premium , 2006, J. Econ. Theory.

[2]  James Cunningham.  Time for Change , 2011, SIGGRAPH '11.

[3]  Christoph Schwab,et al.  Fast deterministic pricing of options on Lévy driven assets , 2002 .

[4]  Othmane Islah.  Solving SABR in Exact Form and Unifying it with LIBOR Market Model , 2009 .

[5]  Rama Cont,et al.  A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models , 2005, SIAM J. Numer. Anal..

[6]  David Jerison,et al.  Subelliptic, second order differential operators , 1987 .

[7]  D. Hernández-Hernández,et al.  Robust utility maximization in a stochastic factor model , 2006 .

[8]  Qi Wu.  Analytical Solutions of the SABR Stochastic Volatility Model , 2012 .

[9]  M. Yor,et al.  Mathematical Methods for Financial Markets , 2009 .

[10]  Marcel Nutz,et al.  A Quasi-Sure Approach to the Control of Non-Markovian Stochastic Differential Equations , 2011, ArXiv.

[11]  Jan Obloj,et al.  Fine-tune your smile: Correction to Hagan et al , 2007, 0708.0998.

[12]  Christoph Schwab,et al.  Wavelet Discretizations of Parabolic Integrodifferential Equations , 2003, SIAM J. Numer. Anal..

[13]  J. V. Casteren,et al.  Stochastic Spectral Theory for Selfadjoint Feller Operators , 2000 .

[14]  Jim Gatheral,et al.  Fast Ninomiya-Victoir Calibration of the Double-Mean-Reverting Model , 2013 .

[15]  The Heston Riemannian distance function , 2013, 1302.2337.

[16]  H. Berestycki,et al.  Computing the implied volatility in stochastic volatility models , 2004 .

[17]  L. Szpruch,et al.  An Euler-type method for the strong approximation of the Cox–Ingersoll–Ross process , 2012, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[18]  P. Lions,et al.  Correlations and bounds for stochastic volatility models , 2007 .

[19]  Pascal J. Maenhout.  Robust Portfolio Rules and Asset Pricing , 2004 .

[20]  Gérard Ben Arous,et al.  Methods de laplace et de la phase stationnaire sur l'espace de wiener , 1988 .

[21]  Stefan Gerhold,et al.  The Hartman-Watson Distribution revisited: Asymptotics for Pricing Asian Options , 2010, 1011.4830.

[22]  Patrick S. Hagan,et al.  Arbitrage-Free SABR , 2014 .

[23]  Peter K. Friz,et al.  Varadhan's formula, conditioned diffusions, and local volatilities , 2013, 1311.1545.

[24]  Zhi-Ming Ma,et al.  Introduction to the theory of (non-symmetric) Dirichlet forms , 1992 .

[25]  Elton P. Hsu,et al.  ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS , 2009 .

[26]  Aurélien Alfonsi,et al.  High order discretization schemes for the CIR process: Application to affine term structure and Heston models , 2010, Math. Comput..

[27]  D. Hernández-Hernández,et al.  A control approach to robust utility maximization with logarithmic utility and time-consistent penalties , 2007 .

[28]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[29]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[30]  J. Lions,et al.  Problèmes aux limites non homogènes et applications , 1968 .

[31]  Aurélien Alfonsi,et al.  On the discretization schemes for the CIR (and Bessel squared) processes , 2005, Monte Carlo Methods Appl..

[32]  Carlos E. Kenig,et al.  The local regularity of solutions of degenerate elliptic equations , 1982 .

[33]  J. García-cuerva,et al.  Weighted norm inequalities and related topics , 1985 .

[34]  G. Burton.  Sobolev Spaces , 2013 .

[35]  Richard J. Martin,et al.  Mean Reversion Pays, but Costs , 2011, 1103.4934.

[36]  Jesper Andreasen,et al.  ZABR -- Expansions for the Masses , 2011 .

[37]  T. Sargent,et al.  Robust Control and Model Uncertainty , 2001 .

[38]  M. Rockner,et al.  Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations , 2005, math/0511708.

[39]  Albo Carlos Cavalheiro,et al.  Weighted Sobolev Spaces and Degenerate Elliptic Equations , 2008 .

[40]  Oleg Reichmann,et al.  Numerical option pricing beyond Lévy , 2012 .

[41]  Philipp Doersek,et al.  A Semigroup Point Of View On Splitting Schemes For Stochastic (Partial) Differential Equations , 2010, 1011.2651.

[42]  M. Hutzenthaler,et al.  Strong convergence rates and temporal regularity for Cox-Ingersoll-Ross processes and Bessel processes with accessible boundaries , 2014, 1403.6385.

[43]  Jim Gatheral,et al.  Arbitrage-free SVI volatility surfaces , 2012, 1204.0646.

[44]  Mark D. Schroder,et al.  Optimal lifetime consumption-portfolio strategies under trading constraints and generalized recursiv , 2003 .

[45]  Alexandre Antonov,et al.  Advanced Analytics for the SABR Model , 2012 .

[46]  Garleanu Nicolae,et al.  Dynamic Trading with Predictable Returns and Transaction Costs , 2009 .

[47]  Leif Andersen.  Simple and efficient simulation of the Heston stochastic volatility model , 2008 .

[48]  Antoine Jacquier,et al.  Marginal Density Expansions for Diffusions and Stochastic Volatility II: Applications , 2014 .

[49]  Marc Yor,et al.  Exponential functionals and principal values related to brownian motion : a collection of research papers , 1997 .

[50]  Denis Talay,et al.  Worst case model risk management , 2002, Finance Stochastics.

[51]  Christoph Schwab,et al.  Wavelet solution of degenerate Kolmogoroff forward equations for exotic contracts in finance , 2013 .

[52]  Fabien Le Floch,et al.  Finite Difference Techniques for Arbitrage Free SABR , 2014 .

[53]  Nadia Sidorova,et al.  Construction of surface measures for Brownian motion , 2009 .

[54]  P. Henry-Labordère.  Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing , 2008 .

[55]  B. Muckenhoupt,et al.  Weighted norm inequalities for the Hardy maximal function , 1972 .

[56]  A. Lindsay,et al.  Results on the CEV Process, Past and Present , 2010 .

[57]  M. Quenez.  Optimal Portfolio in a Multiple-Priors Model , 2004 .

[58]  S. Iyengar.  Hitting Lines with Two-Dimensional Brownian Motion , 1985 .

[59]  Bin Chen,et al.  A LOW-BIAS SIMULATION SCHEME FOR THE SABR STOCHASTIC VOLATILITY MODEL , 2012 .

[60]  G. Ben Arous,et al.  Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus , 1988 .

[61]  M. Hutzenthaler,et al.  Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations , 2013, 1309.5595.

[62]  Pierre Collin-Dufresne,et al.  Strategic Asset Allocation with Predictable Returns and Transaction Costs , 2012 .

[63]  Nicolas Bouleau,et al.  Dirichlet Forms and Analysis on Wiener Space , 1991 .

[64]  P. Friz,et al.  REGULAR VARIATION AND SMILE ASYMPTOTICS , 2006, math/0603146.

[65]  E. Benhamou,et al.  Local Time for the SABR Model: Connection with the 'Complex' Black Scholes and Application to CMS and Spread Options , 2007 .

[66]  Antoine Jacquier,et al.  Marginal Density Expansions for Diffusions and Stochastic Volatility I: Theoretical Foundations , 2011, 1111.2462.

[67]  Antoine Jacquier,et al.  Shapes of Implied Volatility with Positive Mass at Zero , 2013, SIAM J. Financial Math..

[68]  J. Kallsen,et al.  Time Change Representation of Stochastic Integrals , 2002 .

[69]  Josef Teichmann,et al.  Functional Analytic (Ir-)Regularity Properties of SABR-type Processes , 2017 .

[70]  Andrew Lesniewski,et al.  Probability Distribution in the SABR Model of Stochastic Volatility , 2015 .

[71]  Adam Metzler,et al.  On the first passage problem for correlated Brownian motion , 2010 .

[72]  Karl-Theodor Sturm,et al.  Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties. , 1994 .

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

[74]  Pierre Gauthier,et al.  Fitting the Smile, Smart Parameters for SABR and Heston , 2009 .

[75]  Karl-Theodor Sturm,et al.  New Directions in Dirichlet Forms , 1998 .

[76]  Richard White,et al.  The SABR/LIBOR Market Model: Pricing, Calibration and Hedging for Complex Interest-Rate Derivatives , 2009 .

[77]  D. Elworthy.  Geometric aspects of diffusions on manifolds , 1988 .

[78]  Alberto Torchinsky,et al.  Real-Variable Methods in Harmonic Analysis , 1986 .

[79]  C. Schwab,et al.  Computational Methods for Quantitative Finance: Finite Element Methods for Derivative Pricing , 2013 .

[80]  Chi Seng Pun,et al.  Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities , 2014, SIAM J. Control. Optim..

[81]  Alexandre Antonov,et al.  The Free Boundary SABR: Natural Extension to Negative Rates , 2015 .

[82]  Kerstin Vogler,et al.  Table Of Integrals Series And Products , 2016 .

[83]  D. Filipović.  Term-Structure Models: A Graduate Course , 2009 .

[84]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[85]  Revaz Tevzadze,et al.  Robust utility maximization for a diffusion market model with misspecified coefficients , 2013, Finance Stochastics.

[86]  M. Yor.  On some exponential functionals of Brownian motion , 1992, Advances in Applied Probability.

[87]  P. Henry-Labordère.  Unifying the Bgm and Sabr Models: a Short Ride in Hyperbolic Geometry , 2006, physics/0602102.

[88]  Shaolin Ji,et al.  Ambiguous volatility, possibility and utility in continuous time , 2011, 1103.1652.

[89]  Jim Gatheral,et al.  Implied Volatility from Local Volatility: A Path Integral Approach , 2014 .

[90]  Second‐order operators with degenerate coefficients , 2006, math/0601307.

[91]  Hyukjae Park,et al.  Efficient Valuation Method for the SABR Model , 2013, 1308.0665.

[92]  Louis Paulot,et al.  Asymptotic Implied Volatility at the Second Order with Application to the SABR Model , 2009, 0906.0658.

[93]  Bohumír Opic,et al.  How to define reasonably weighted Sobolev spaces , 1984 .

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

[95]  Marc Yor,et al.  Exponential functionals of Brownian motion, I: Probability laws at fixed time , 2005 .

[96]  Othmane Islah.  Heun Solutions to the SABR Model , 2011 .

[97]  Jan Kallsen,et al.  Derivative pricing based on local utility maximization , 2002, Finance Stochastics.

[98]  J. Ramírez,et al.  Small-time Gaussian behavior of symmetric diffusion semi-groups , 2003 .

[99]  M. Röckner,et al.  Dirichlet forms on topological vector spaces: closability and a Cameron-Martin formula , 1990 .

[100]  Yacine Ait-Sahalia.  Closed-Form Likelihood Expansions for Multivariate Diffusions , 2002, 0804.0758.

[101]  John M. Lee.  Riemannian Manifolds: An Introduction to Curvature , 1997 .

[102]  Peter K. Friz,et al.  Semi-closed form cubature and applications to financial diffusion models , 2010 .

[103]  P. Hagan,et al.  MANAGING SMILE RISK , 2002 .

[104]  Karl-Theodor Sturm,et al.  ON THE GEOMETRY DEFINED BY DIRICHLET FORMS , 1995 .

[105]  Nicolae Gârleanu,et al.  Dynamic portfolio choice with frictions , 2016, J. Econ. Theory.

[106]  Björn Böttcher,et al.  Feller Processes: The Next Generation in Modeling. Brownian Motion, Lévy Processes and Beyond , 2010, PloS one.

[107]  Archil Gulisashvili,et al.  Analytically Tractable Stochastic Stock Price Models , 2012 .

[108]  David Hobson,et al.  Comparison results for stochastic volatility models via coupling , 2010, Finance Stochastics.

[109]  Antoine Jacquier,et al.  An Explicit Euler Scheme with Strong Rate of Convergence for Financial SDEs with Non-Lipschitz Coefficients , 2014, SIAM J. Financial Math..

[110]  D. Schmeidler.  Subjective Probability and Expected Utility without Additivity , 1989 .

[111]  Anis Matoussi,et al.  ROBUST UTILITY MAXIMIZATION IN NONDOMINATED MODELS WITH 2BSDE: THE UNCERTAIN VOLATILITY MODEL , 2015 .

[112]  Seng-Kee Chua,et al.  On Weighted Sobolev Spaces , 1996, Canadian Journal of Mathematics.

[113]  B. Jourdain,et al.  Loss of martingality in asset price models with lognormal stochastic volatility , 2022 .

[114]  A. Grigor’yan.  Heat Kernel and Analysis on Manifolds , 2012 .

[115]  M. Forde,et al.  The large-maturity smile for the SABR and CEV-heston models , 2013 .

[116]  Alexander Schied,et al.  Optimal Investments for Robust Utility Functionals in Complete Market Models , 2005, Math. Oper. Res..

[117]  Paul Doust.  No-arbitrage SABR , 2012 .

[118]  Small time asymptotics of diffusion processes , 2006, math/0601350.

[119]  Alexander Ostermann,et al.  Exponential splitting for unbounded operators , 2009, Math. Comput..

[120]  Nicolas Bouleau,et al.  Energy image density property and the lent particle method for Poisson measures , 2008, 0807.1963.

[121]  Option Valuation in Multivariate SMM/SABR Models (with an Application to the CMS Spread) , 2010 .

[122]  J. Graver,et al.  Graduate studies in mathematics , 1993 .

[123]  I. Chavel.  Eigenvalues in Riemannian geometry , 1984 .

[124]  Leif Andersen,et al.  Moment Explosions in Stochastic Volatility Models Moment Explosions in the Black–scholes and Exponential Lévy Model Moment Explosions in the Heston Model , 2022 .

[125]  E. Mayerhofer,et al.  The limits of leverage , 2015, 1506.02802.

[126]  Reinhold Schneider,et al.  Multiresolution weighted norm equivalences and applications , 2004, Numerische Mathematik.

[127]  M. Schroder.  On the integral of geometric Brownian motion , 2002, math/0205063.

[128]  D. Talay,et al.  Expansion of the global error for numerical schemes solving stochastic differential equations , 1990 .

[129]  Björn Böttcher,et al.  On the construction of Feller processes with unbounded coefficients , 2011 .

[130]  Martin Forde,et al.  Sharp tail estimates for the correlated SABR model , 2015 .

[131]  W. Ziemer.  Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , 1989 .

[132]  Lorenzo Garlappi,et al.  Portfolio Selection with Parameter and Model Uncertainty: A Multi-Prior Approach , 2004 .

[133]  A. Etheridge,et al.  On the moments and the interface of the symbiotic branching model , 2010, 1012.1759.

[134]  Dirichlet Forms and Degenerate Elliptic Operators , 2006, math/0601349.

[135]  I. Gilboa,et al.  Maxmin Expected Utility with Non-Unique Prior , 1989 .

[136]  Alexandre Antonov,et al.  Mixing SABR Models for Negative Rates , 2015 .

[137]  Archil Gulisashvili.  Left-wing asymptotics of the implied volatility in the presence of atoms , 2013 .

[138]  Karl-Theodor Sturm,et al.  Is a diffusion process determined by its intrinsic metric , 1997 .

[139]  P. Donnelly.  MARKOV PROCESSES Characterization and Convergence (Wiley Series in Probability and Mathematical Statistics) , 1987 .

[140]  Roger Lee.  THE MOMENT FORMULA FOR IMPLIED VOLATILITY AT EXTREME STRIKES , 2004 .

[141]  Archil Gulisashvili,et al.  Asymptotic Formulas with Error Estimates for Call Pricing Functions and the Implied Volatility at Extreme Strikes , 2009, SIAM J. Financial Math..

[142]  On additive time-changes of Feller processes , 2009, 0909.4881.

[143]  Douglas Kennedy.  Stochastic Financial Models , 2010 .

[144]  Costis Skiadas.  Robust control and recursive utility , 2003, Finance Stochastics.

[145]  The Heat-Kernel Most-Likely-Path Approximation , 2011 .

[146]  A. Jacquier,et al.  Mass at Zero and Small-Strike Implied Volatility Expansion in the SABR Model , 2015 .