Paper: arXiv 2601.07991

Authors: Kyle Sung, Traian A. Pirvu

Abstract

We provide an explicit solution for optimal option portfolios under variance and Value at Risk (VaR) minimization when the underlying returns follow a Student t-distribution. The novelty of our paper is the departure from the traditional normal returns setting. Our main contribution is the methodology for obtaining optimal portfolios. Numerical experiments reveal that, as expected, the optimal variance and VaR portfolio compositions differ by a significant amount, suggesting that more realistic tail risk settings can lead to potentially more realistic portfolio allocations.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper contains dense, advanced mathematics including multivariate Student t-distributions, delta-gamma approximations, moment generating functions, and explicit closed-form solutions for portfolio optimization. While it presents numerical experiments with real stock data, the summary and excerpt focus heavily on theoretical derivations and lack detailed implementation steps, code, or comprehensive backtesting results that would indicate high empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Optimal Option Portfolios for Student t Returns"] --> B["Key Methodology:<br>Solve for Min Variance & Min VaR"]
  B --> C["Data/Inputs:<br>Student t-distribution Parameters"]
  C --> D["Computational Process:<br>Explicit Analytical Solution"]
  D --> E{"Numerical Experiments"}
  E --> F["Key Finding 1:<br>Min Variance & Min VaR<br>Compositions Differ Significantly"]
  E --> G["Key Finding 2:<br>Realistic Tail Risk Yields<br>Realistic Portfolio Allocations"]

Cited References (75)

  1. Portfolio Risk Management (2023)
  2. Non-linear equity portfolio variance reduction under a mean–variance framework—A delta–gamma approach (2013) — link
  3. Nonlinear portfolio selection using approximate parametric Value-at-Risk (2013) — link
  4. Portfolio Selection (1952)
  5. Monte Carlo Methods in Financial Engineering (2003)
  6. Accelerating Quadratic Optimization with Reinforcement Learning (2021) — link
  7. Pricing European options with a log Student’s t-distribution: A Gosset formula (2010) — link
  8. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python (2020) — link
and 67 more
  1. The Pricing of Options and Corporate Liabilities (1973)
  2. Multivariate unified skew-t distributions and their properties (2024) — link
  3. Cumulative Prospect Theory with Generalized Hyperbolic Skewed $t$ Distribution (2018) — link
  4. Quadratic Forms in Random Variables: Theory and Applications (1992) — link
  5. On Nonlinear Fractional Programming (1967)
  6. Fractional Programming. II, on Dinkelbach’s Algorithm (1976)
  7. Toward Maximum Diversification (2008) — link
  8. Log Student’s t-distribution-based option sensitivities: Greeks for the Gosset formulae (2013) — link
  9. Option Pricing with Heavy-Tailed Distributions of Logarithmic Returns (2019) — link
  10. In-sample and out-of-sample Sharpe ratios of multi-factor asset pricing models (2024) — link
  11. Portfolio Optimization under Correlation Constraint (2020)
  12. Risk Management under Omega Measure (2017)
  13. Risk Minimization and Portfolio Diversification (2016)
  14. Numerical Analysis for Spread Option Pricing Model of Markets with Finite Liquidity: Full Feedback Model (2016)
  15. Multi-period Investment Strategies under Cumulative Prospect Theory (2019)
  16. Numerical Analysis for Spread Option Pricing Model of Markets with Finite Liquidity: First-order Feedback Model (2014)
  17. Non-Linear Equity Portfolio Variance Reduction under Delta-Gamma Approach Analysis (2013)
  18. Mean-Variance-VaR Portfolios: MIQP Formulation and Performance Analysis (2023)
  19. Hedging Crash Risk in Optimal Portfolio Selection (2020)
  20. A Novel Nonlinear Value-at-Risk Method for Modeling Risk of Option Portfolio with Multivariate Mixture of Normal Distributions (2013)
  21. Systemic Risk of Optioned Portfolio: Controllability and Optimization (2023)
  22. Value-at-risk computation by fourier inversion with explicit error bounds (2009)
  23. Matplotlib: A 2D Graphics Environment (2007) — link
  24. seaborn: statistical data visualization (2021) — link
  25. Array programming with NumPy (2020) — link
  26. Portfolio optimization under a generalized hyperbolic skewed t distribution and exponential utility (2016) — link
  27. Value-at-Risk computation by Fourier inversion with explicit error bounds (2009) — link
  28. Portfolio optimization for student t and skewed t returns (2010) — link
  29. Matrix differential calculus with applications in statistics and econometrics (2019)
  30. The Matrix Cookbook (2012)
  31. The Percentile Points of Distributions Having Known Cumulants (1960)
  32. On a Formula for the Product-Moment Coefficient of any Order of a Normal Frequency Distribution in any Number of Variables (1918)
  33. The Sharpe ratio (1998)
  34. Optimal Portfolio Choice with Parameter Uncertainty (2007)
  35. An Analytic Derivation of the Efficient Portfolio Frontier (1972)
  36. Moments of random vectors with skew t distribution and their quadratic forms (2003) — link
  37. Distributions Generated by Perturbation of Symmetry with Emphasis on a Multivariate Skew t-Distribution (2003) — link
  38. Stochastic representations and probabilistic characteristics of multivariate skew-elliptical distributions (2024) — link
  39. Statistical Applications of the Multivariate Skew Normal Distribution (1999)
  40. The t Copula and Related Copulas (2005)
  41. Portfolio optimization under the generalized hyperbolic distribution: optimal allocation, performance and tail behavior (2021) — link
  42. Generalized skew-elliptical distributions and their quadratic forms (2005) — link
  43. The Generalized Hyperbolic Skew Student’s $t$-Distribution (2006) — link
  44. Optimal Portfolios with Skewed and Heavy-Tailed Distributions (2006) — link
  45. Robust mixture modeling using the skew t distribution (2007) — link
  46. Moments of skew-normal random vectors and their quadratic forms (2001) — link
  47. A General Class of Multivariate Skew-Elliptical Distributions (2001) — link
  48. yfinance (2026) — link
  49. A Comparison of Tail Behaviour of Stock Market Returns (2014) — link
  50. Extreme Value Estimation of Boom and Crash Statistics (2006) — link
  51. Efficient and Scalable Parametric High-Order Portfolios Design via the Skew-$t$ Distribution (2023) — link
  52. On quadratic forms in multivariate generalized hyperbolic random vectors (2021) — link
  53. The skew elliptical distributions and their quadratic forms (2003) — link
  54. Minimizing CVaR and VaR for a portfolio of derivatives (2006)
  55. A new closed-form formula for pricing European options under a skew Brownian motion (2018) — link
  56. The R package \texttt sn : The skew-normal and related distributions such as the skew-$t$ and the SUN (version 2.1.3). (2026) — link
  57. Portfolio optimization under multivariate affine generalized hyperbolic distributions (2022) — link
  58. A stein type lemma for the multivariate generalized hyperbolic distribution (2017) — link
  59. Continuous time mean-variance portfolio optimization with piecewise state-dependent risk aversion (2016) — link
  60. The multivariate skew-normal distribution (1996) — link
  61. Statistical Applications of the Multivariate Skew Normal Distribution (1999) — link
  62. Multivariate skew-normal distributions with applications in insurance (2006) — link
  63. Skewed bivariate models and nonparametric estimation for the CTE risk measure (2008) — link
  64. Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models? (2012) — link
  65. Fitting asset returns to skewed distributions: Are the skew-normal and skew-student good models? (2014) — link
  66. Bayesian Conjugacy in Probit, Tobit, Multinomial Probit and Extensions: A Review and New Results (2023) — link
  67. Skew-elliptical distributions with applications in risk theory (2017) — link

0 of 75 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.