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Optimal Option Portfolios for Student t Returns

Kyle Sung, Traian A. Pirvu

TL;DR

This paper tackles optimal option portfolios under heavy-tailed multivariate Student-t returns, addressing tail risk and the limitations of Gaussian VaR. It combines Gosset pricing, delta-gamma approximation, and Cornish-Fisher VaR to derive explicit expressions for portfolio mean and risk, including a closed-form variance-minimizing solution and a CFVaR-minimizing solution. The main contributions are explicit expressions for the mean and variance of portfolio gains, a closed-form CFVaR-optimal portfolio, and numerical evidence showing that variance and VaR portfolios diverge under tail-heavy returns. This work provides a practical framework for tail-risk aware option allocation, enabling more realistic and robust risk management in financial settings.

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.

Optimal Option Portfolios for Student t Returns

TL;DR

This paper tackles optimal option portfolios under heavy-tailed multivariate Student-t returns, addressing tail risk and the limitations of Gaussian VaR. It combines Gosset pricing, delta-gamma approximation, and Cornish-Fisher VaR to derive explicit expressions for portfolio mean and risk, including a closed-form variance-minimizing solution and a CFVaR-minimizing solution. The main contributions are explicit expressions for the mean and variance of portfolio gains, a closed-form CFVaR-optimal portfolio, and numerical evidence showing that variance and VaR portfolios diverge under tail-heavy returns. This work provides a practical framework for tail-risk aware option allocation, enabling more realistic and robust risk management in financial settings.

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.
Paper Structure (15 sections, 1 theorem, 34 equations, 2 figures, 1 table)

This paper contains 15 sections, 1 theorem, 34 equations, 2 figures, 1 table.

Table of Contents

  1. Keywords
  2. Introduction
  3. Outline
  4. The Model
  5. Gosset Formula
  6. The Delta-Gamma Approximation
  7. Variance and VaR Risk Measures
  8. Portfolio Optimization
  9. Minimizing Variance
  10. Minimizing Value at Risk
  11. Numerical Experiments
  12. Conclusion and Future Work
  13. Acknowledgments
  14. Proof of Theorem \ref{['theorem:cfvar2_optim_t_dist_returns_nonzero_drift']}
  15. Additional Experiments

Key Result

Theorem 3.1

The ${\operatorname{CFVaR}}_2^\alpha$ minimization problem eqn:valueatrisk:minimization is solved with the optimal portfolio number of shares: where ${\boldsymbol{\zeta}}$ was defined below Equation eqn:expectation_variance, and $\varepsilon^\star$ is a positive constant defined in sec:proofs.

Figures (2)

  • Figure 1: Optimal call and put option portfolios, under both variance and Cornish-Fisher VaR minimization, of five at-the-money European options with expiry in one year, written on the five stocks in the dataset by Hu01012010. Here, $\alpha = 0.01$, $\nu = 5.87$, and $\Delta t = 1/252$.
  • Figure 2: Optimal call and put option portfolios, under both variance and Cornish-Fisher VaR minimization, of five at-the-money European call options with expiry in one year, written on the five stocks in the dataset by Hu01012010. Here, $\nu = 5.87$, and $\Delta t = 1/252$.

Theorems & Definitions (2)

  • Theorem 3.1: Optimal ${\operatorname{CFVaR}}_2^\alpha$ Portfolios
  • proof