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Ultra-short-term volatility surfaces

Federico M. Bandi, Nicola Fusari, Guido Gazzani, Roberto Renò

Abstract

Options with maturities below one week, hereafter "ultra-short-term" options, have seen a sharp increase in trading activity in recent years. Yet, these instruments are difficult to price jointly using classical pricing models due to the pronounced oscillations observed in the at-the-money implied-volatility term structure across ultra-short-term tenors. We propose Edgeworth++, a parsimonious jump-diffusion model featuring a nonparametric stochastic volatility component, which provides flexibility in capturing implied-volatility smiles for each tenor, combined with a deterministic shift extension, which allows the model to fit rich at-the-money implied-volatility shapes across tenors. We derive a local (in tenor) expansion of the process characteristic function suited to value ultra-short-term options. The expansion leads to fast and accurate option pricing in closed form via standard Fourier inversion. We discuss the benefits of the proposed approach relative to benchmarks.

Ultra-short-term volatility surfaces

Abstract

Options with maturities below one week, hereafter "ultra-short-term" options, have seen a sharp increase in trading activity in recent years. Yet, these instruments are difficult to price jointly using classical pricing models due to the pronounced oscillations observed in the at-the-money implied-volatility term structure across ultra-short-term tenors. We propose Edgeworth++, a parsimonious jump-diffusion model featuring a nonparametric stochastic volatility component, which provides flexibility in capturing implied-volatility smiles for each tenor, combined with a deterministic shift extension, which allows the model to fit rich at-the-money implied-volatility shapes across tenors. We derive a local (in tenor) expansion of the process characteristic function suited to value ultra-short-term options. The expansion leads to fast and accurate option pricing in closed form via standard Fourier inversion. We discuss the benefits of the proposed approach relative to benchmarks.

Paper Structure

This paper contains 23 sections, 5 theorems, 98 equations, 12 figures, 6 tables.

Table of Contents

  1. Introduction
  2. The pricing model
  3. Characteristic function expansion with displacement
  4. Piecewise deterministic displacement
  5. Edgeworth++
  6. Data
  7. Benchmarks
  8. Rough Heston++
  9. 2F Heston Merton
  10. Estimation
  11. Performance
  12. Pricing speed
  13. 0DTEs
  14. Ultra-short-term surfaces
  15. Discussion
  16. ...and 8 more sections

Key Result

Theorem 1

Let $Z_{\tau}^c$ be defined as in Eq. (zeta). Assume $X^c$, defined in Eq. dyn, to be $D_W^{(5)}$.This is a technical, but innocuous, assumption of differentiability discussed in the Appendix. Then, for $\tau>0,$ we have where $\psi(u)$ is an integrable function over $\mathbb{R}$ of order $u^{-3}$, as $|u|\rightarrow\infty$, and with $\tilde{\phi}(s)=1+\phi(s)/\sigma_0,$ for all $s\in{\mathbb R}

Figures (12)

  • Figure 1: The figure reports volume (i.e., number of contracts traded) in SPX options as a fraction of total volume associated with various tenors: between 0 and 7 days (1w), between 8 and 30 days (1m), between 31 and 60 days (2m), between 61 and 120 days (6m) and between 121 and 365 days (12m). The data covers the period from 2014 to 2023. Data source: OptionMetrics.
  • Figure 2: ATM implied-volatility term structure on a typical day: Thursday, August 18, 2022. Data source: CBOE.
  • Figure 3: We report the histograms of two summary statistics for the ATM implied-volatility term structures associated with options with expiries between 0 and 7 calendar days (ultra-short-term options) and expiries between 8 and 30 calendar days. Data are from May 6, 2022 to May 11, 2023. Data source: CBOE. Panel A: Average absolute value of the first derivative of the ATM implied-volatility term structure, defined as $\frac{1}{n-1} \sum_{i=1}^{n-1} \left| \frac{\sigma_{i+1} - \sigma_i}{\tau_{i+1} - \tau_i} \right|$, where $\tau_1, \tau_2, \ldots, \tau_n$ denote the expiries and $\sigma_i = \sigma(\tau_i)$ denote the corresponding implied volatilities. Panel B: Average absolute value of the second derivative of the ATM implied-volatility term structure defined using a discrete approximation based on triplets of consecutive implied volatilities.
  • Figure 4: Number of options in each daily surface for each of the $6$ maturities. The data covers the period May 6, 2022 to May 11, 2023. Data source: CBOE.
  • Figure 5: Histograms of the daily RMSEs of estimated models in our samples on 0DTEs, from May 6, 2022 to May 11, 2023. Data source: CBOE.
  • ...and 7 more figures

Theorems & Definitions (13)

  • Theorem 1
  • proof
  • Corollary 1: to Theorem \ref{['th:expansion']}
  • proof
  • Corollary 2: to Theorem \ref{['th:expansion']}
  • proof
  • Remark 1: Displaced Black and Scholes model
  • Remark 2: Interpreting the displacement
  • Theorem 2
  • proof
  • ...and 3 more