Paper: arXiv 2503.14158

Authors: Eduardo Abi Jaber, Shaun, Li

Abstract

We introduce the two-factor Quintic Ornstein-Uhlenbeck (OU) model, where volatility is modelled as a degree-five polynomial of the sum of two Ornstein-Uhlenbeck processes driven by the same Brownian motion, each mean-reverting at a different speed. We demonstrate that the model effectively captures the volatility surfaces of SPX and VIX while aligning with the skew-stickiness ratio (SSR) across maturities ranging from a few days to over two years. Furthermore, it is consistent with key empirical stylized facts, notably reproducing the Zumbach effect.

Complexity vs Empirical Score

  • Math Complexity: 7.5/10
  • Empirical Rigor: 8.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper resolves a major outstanding challenge in quantitative finance by introducing a two-factor Quintic OU stochastic volatility model that simultaneously captures SPX-VIX surfaces, SSR term structures, and key stylized facts like the Zumbach effect. It combines elegant closed-form derivation (via Gaussianity and polynomial representations) with rigorous multi-maturity empirical calibration and open-source code implementation.

Research Flowchart

  flowchart TD
    A[Research Goal: Model SPX/VIX Vol Surfaces, SSR & Zumbach Effect] --> B[Introduce Quintic OU Model: 2-Factor Volatility via Degree-5 Polynomial]
    C[Inputs: SPX & VIX Options Data across Maturities] --> D[Model Calibration & Computational Simulation]
    B --> D
    D --> E[Outcome 1: Joint Calibration of SPX & VIX Volatility Surfaces]
    D --> F[Outcome 2: Alignment with Skew-Stickiness Ratio & Zumbach Effect]

Cited References (168)

  1. The quintic O rnstein- U hlenbeck volatility model that jointly calibrates \text SPX & VIX smiles (June 2023)
  2. Joint SPX & VIX calibration with Gaussian polynomial volatility models: Deep pricing with quantization hints (2025)
  3. Volatility models in practice: Rough, P ath-dependent or M arkovian? (2024)
  4. Fourier- L aplace transforms in polynomial Ornstein-Uhlenbeck volatility models — → 2405.02170
  5. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  6. Consistent modeling of SPX and VIX options (2008)
  7. The rough H awkes H eston stochastic volatility model (2024)
  8. Heston stochastic vol-of-vol model for joint calibration of VIX and S&P 500 options (2018)
and 160 more
  1. A consistent pricing model for index options and volatility derivatives (2013)
  2. Decoupling the Short- and Long-Term Behavior of Stochastic Volatility (2021)
  3. Multifactor approximation of rough volatility models (2019)
  4. Lifting the H eston model (2019)
  5. Markovian structure of the V olterra H eston model (2019)
  6. Approximation of S tochastic V olterra E quations with kernels of completely monotone type (2024)
  7. Markovian approximations of stochastic V olterra equations with the fractional kernel (2023)
  8. Linear-Quadratic control for a class of stochastic V olterra equations: solvability and approximation (2021)
  9. Markovian approximation of the rough Bergomi model for Monte Carlo option pricing (2021)
  10. Fractional B rownian motion and the M arkov property (1998)
  11. The characteristic function of rough H eston models (2019)
  12. Generalized Feller processes and Markovian lifts of stochastic Volterra processes: the affine case (2020)
  13. Affine representations of fractional processes with applications in mathematical finance (2019)
  14. Log-modulated rough stochastic volatility models (2021)
  15. Strong convergence rates for Markovian representations of fractional Brownian motion (2019) — link
  16. Volatility is rough (2018)
  17. Joint pricing of VIX and SPX options with stochastic volatility and jump models (2015)
  18. Smiling twice: the H eston++ model (2018)
  19. Consistent modelling of VIX and equity derivatives using a 3/2 plus jumps model (2014)
  20. A regime-switching H eston model for VIX and S&P 500 implied volatilities (2014)
  21. Joint Modeling and Calibration of SPX and VIX by O ptimal T ransport (2022)
  22. Empirical analysis of rough and classical stochastic volatility models to the SPX and VIX markets (2022)
  23. Regime-switching stochastic volatility model: estimation and calibration to VIX options (2017)
  24. A comparison theorem for solutions of stochastic differential equations and its applications (1984)
  25. The quadratic rough H eston model and the joint S&P 500/ VIX smile calibration problem (2020)
  26. Volatility is (mostly) path-dependent (2023)
  27. NEURAL JOINT S&P 500/ VIX SMILE CALIBRATION (2023)
  28. Deep calibration of the quadratic rough Heston model (2022)
  29. Volatility Index Methodology: Cboe Volatility Index — link
  30. Affine V olterra processes (2019)
  31. Pricing under rough volatility (2016)
  32. Quadratic optimal functional quantization of stochastic processes and numerical applications (2008)
  33. Functional quantization-based stratified sampling methods (2015)
  34. Functional quantization of rough volatility and applications to the VIX (2023)
  35. High resolution quantization and entropy coding for fractional Brownian motion (2006)
  36. A regularity structure for rough volatility (2020)
  37. On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables (1918)
  38. Vector quantization and signal compression (2012)
  39. Foundations of quantization for probability distributions (2000)
  40. Optimal quadratic quantization for numerics: the Gaussian case (2003)
  41. Functional quantization for numerics with an application to option pricing (2005)
  42. A theory of regularity structures (2014)
  43. Multiscale stochastic volatility asymptotics (2003)
  44. Derivatives in financial markets with stochastic volatility (2000)
  45. Fast-reversion limit of the Heston model (2015)
  46. Optimal quantization methods and applications to numerical problems in finance (2004)
  47. On the convergence of ordinary integrals to stochastic integrals (1965)
  48. Algebraic renormalisation of regularity structures (2019)
  49. Stochastic volatility’s orderly smiles (2012)
  50. Exponential rate of convergence for L loyd’s method I (1982)
  51. http://www.quantize.maths-fi.com, 2005
  52. On Randomization of Affine Diffusion Processes with Application to Pricing of Options on VIX and S&P 500 (2022) — link
  53. Convergence of multi-dimensional quantized SDE’s (2011)
  54. Towards a theory of volatility trading (2001)
  55. Stochastic volatility modeling (2015)
  56. Turbocharging Monte Carlo pricing for the rough Bergomi model (2018)
  57. The Rough Bergomi model as H→ 0–skew flattening/blow up and non-Gaussian rough volatility (2020)
  58. Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models (2021)
  59. The joint S&P 500/ VIX smile calibration puzzle solved (2020)
  60. Dispersion-Constrained M artingale S chr "o dinger Bridges: Joint Entropic Calibration of Stochastic Volatility Models to S&P 500 and VIX Smiles (2022)
  61. Beta stochastic volatility model (2012)
  62. On the M artingale property of certain local M artingales (2012)
  63. Foundations of a pathwise volatility framework with explicit fast reversion limits (2019) — link
  64. Reconciling rough volatility with jumps — → 2303.07222
  65. Deeply learning derivatives (2018) — link
  66. A fast mean-reverting correction to Heston’s stochastic volatility model (2011)
  67. Small-time asymptotics for fast mean-reverting stochastic volatility models (2012)
  68. Short-maturity asymptotics for a fast mean-reverting Heston stochastic volatility model (2010)
  69. Does the Term-Structure of Equity At-the-Money Skew Really Follow a Power Law? (2023)
  70. Yet Another Analysis of the SP500 At-The-Money Skew: Crossover of Different Power-Law Behaviours (2023)
  71. Joint calibration to SPX and VIX options with signature-based models — → 2301.13235
  72. Smile Dynamics II (2005)
  73. Smile Dynamics III (2008)
  74. Smile Dynamics IV (2009)
  75. Long memory in continuous-time stochastic volatility models (1998)
  76. The smile of stochastic volatility: Revisiting the Bergomi-Guyon expansion (2021)
  77. Model risk adjusted hedge ratios (2009)
  78. Volatility has to be rough (2021)
  79. Things we think we know (2023)
  80. Modeling and pricing long memory in stock market volatility (1996)
  81. Multifractional stochastic volatility models (2014)
  82. On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility (2007)
  83. Asymptotic analysis for stochastic volatility: martingale expansion (2011)
  84. Time reversal invariance in finance (2009)
  85. Volatility conditional on price trends (2010)
  86. The Zumbach effect under rough Heston (2020)
  87. Rough volatility: fact or artefact? (2024)
  88. Path Shadowing Monte-Carlo — → 2308.01486
  89. Empirical properties of asset returns: stylized facts and statistical issues (2001)
  90. Asset returns and volatility clustering in financial time series (2011)
  91. Multiple time scales in volatility and leverage correlations: a stochastic volatility model (2004)
  92. A long memory property of stock market returns and a new model (1993)
  93. Correlations in economic time series (1997)
  94. Modelling fluctuations of financial time series: from cascade process to stochastic volatility model (2000)
  95. Studies of stock market volatility changes (1976)
  96. Asymmetric volatility and risk in equity markets (2000)
  97. Leverage effect in financial markets: The retarded volatility model (2001)
  98. Maturity cycles in implied volatility (2004)
  99. An intuitive introduction to fractional and rough volatilities (2021)
  100. Stochastic volatility as a simple generator of apparent financial power laws and long memory (2001)
  101. The VIX future in Bergomi models: Fast approximation formulas and joint calibration with S&P 500 skew (2022)
  102. What type of process underlies options? A simple robust test (2003)
  103. A GMM approach to estimate the roughness of stochastic volatility (2023)
  104. Long range dependence in financial markets (2005)
  105. Affine Volterra processes with jumps (2023)
  106. Rough volatility, path-dependent PDEs and weak rates of convergence — → 2304.03042
  107. Functional central limit theorems for rough volatility (2024)
  108. Large and moderate deviations for stochastic Volterra systems (2022)
  109. A weak solution theory for stochastic V olterra equations of convolution type (2021)
  110. Large deviation principle for V olterra type fractional stochastic volatility models (2018)
  111. Rough fractional diffusions as scaling limits of nearly unstable heavy tailed H awkes processes (2016)
  112. A martingale approach for fractional B rownian motions and related path dependent PDE s (2019)
  113. Arbitrage pricing with stochastic volatility (1992)
  114. Forests, cumulants, martingales (2022)
  115. Parabolic Systems (1969)
  116. Algebraic Topology (2002)
  117. Affine processes and applications in finance (2003)
  118. Analysis of F ourier transform valuation formulas and applications (2010)
  119. Contingent claims and market completeness in a stochastic volatility model (1997)
  120. Signature SDE s from an affine and polynomial perspective — → 2302.01362
  121. Why the rotation count algorithm works (2006)
  122. The characteristic function of G aussian stochastic volatility models: an analytic expression (2022)
  123. Stock price distributions with stochastic volatility: an analytic approach (1991)
  124. Stochastic volatility with an O rnstein– U hlenbeck process: an extension (1999)
  125. A novel pricing method for European options based on F ourier-cosine series expansions (2009)
  126. Laplace transforms and suprema of stochastic processes (2002)
  127. Signature volatility models: pricing and hedging with Fourier (2024)
  128. Option valuation using the fast F ourier transform (1999)
  129. A simple option formula for general jump-diffusion and other exponential L 'e vy processes (2001)
  130. Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing (2000)
  131. Mathematical methods for foreign exchange: A financial engineer’s approach (2001)
  132. Second order multiscale stochastic volatility asymptotics: stochastic terminal layer analysis and calibration (2016)
  133. A Fourier transform method for spread option pricing (2010)
  134. Affine forward variance models (2019)
  135. Transform analysis and asset pricing for affine jump-diffusions (2000)
  136. Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics (2001)
  137. Pricing with a smile (1994)
  138. Managing smile risk (2002)
  139. The vol smile problem (2002)
  140. Under lineare methoden in der wahr scheinlichkeitsrechnung (1947)
  141. Probability Theory II (1978)
  142. https://ir.cboe.com/news/news-details/2024/Cboe-GLOBAL-MARKETS-REPORTS-TRADING-VOLUME-FOR-DECEMBER-AND-FULL-YEAR-2023/default.aspx
  143. https://www.cboe.com/insights/posts/volatility-insights-evaluating-the-market-impact-of-spx-0-dte-options/
  144. On V IX futures in the rough B ergomi model (2018)
  145. Mimicking the one-dimensional marginal distributions of processes having an It ^o differential (1986)
  146. Prices of state-contingent claims implicit in option prices (1978)
  147. Computing the SSR — → 2406.16131
  148. Computing skew-stickiness (2023)
  149. The fine-structure of volatility feedback I : M ulti-scale self-reflexivity (2014)
  150. Heterogeneous volatility cascade in financial markets (2001)
  151. Revisiting stylized facts for modern stock markets (2023)
  152. Capital accumulation and economic growth (1961)
  153. Volatility clustering in financial markets: empirical facts and agent-based models (2007)
  154. Volatility processes and volatility forecast with longmemory (2003)
  155. Smile Dynamics and Rough Volatility (2024)
  156. Contribution `a l’ 'e tude de la repr 'e sentation d’une fonction arbitraire par des int 'e grales d 'e finies (1910)
  157. Fast exact joint S&P 500/VIX smile calibration in discrete and continuous time (2024)
  158. Option valuation with conditional heteroskedasticity and nonnormality (2010)
  159. The SSR under Quadratic Rough Heston (2025)
  160. Capturing Smile Dynamics with the Quintic Volatility Model: SPX, Skew-Stickiness Ratio and VIX — → 2503.14158

8 of 168 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.