[Paper Review] Robust calibration and arbitrage-free interpolation of SSVI slices

Pierre Cohort, Jacopo Corbetta+2 more|arXiv (Cornell University)|Apr 13, 2018

Stochastic processes and financial applications7 references3 citations

TL;DR

This paper presents a robust, arbitrage-free calibration method for the extended SSVI (eSSVI) volatility surface model by anchoring each maturity slice to the at-the-money forward point and using a reparameterization that ensures no butterfly or calendar spread arbitrage. The key contribution is that simple piecewise linear interpolation/extrapolation of the calibrated parameters preserves no-arbitrage conditions, enabling a fast, parsimonious, and reliable volatility surface construction.

ABSTRACT

We describe a robust calibration algorithm of a set of SSVI slices (i.e. a set of 3 SSVI parameters $θ, ρ, φ$ attached to each option maturity available on the market), which grants that these slices are free of Butterfly and Calendar-Spread arbitrage. Given such a set of consistent SSVI parameters, we show that the most natural interpolation/extrapolation of the parameters provides a full continuous volatility surface free of arbitrage. The numerical implementation is straightforward, robust and quick, yielding an effective, parsimonious solution to the smile problem, which has the potential to become a benchmark one.

Motivation & Objective

• To develop a robust calibration algorithm for eSSVI parameters that guarantees absence of butterfly and calendar spread arbitrage across maturities.
• To ensure that the resulting volatility surface is continuous and free of arbitrage after interpolation/extrapolation of slice parameters.
• To provide a computationally efficient, model-free method for constructing a full arbitrage-free volatility surface from market-observed option prices.
• To enable parsimonious storage and re-use of calibrated parameters for risk management and historical surface reconstruction.

Proposed method

• Reparameterize each SSVI slice using the pair $(\rho, \psi)$, where $\psi = \theta \varphi$, and anchor the slice to the market data point closest to the forward moneyness ($k^*$, $\theta^*$) to improve calibration stability.
• Apply a one-dimensional Brent algorithm for optimization, avoiding complex black-box solvers and ensuring robustness in calibration.
• Enforce no-arbitrage conditions via explicit constraints: $\theta_t \varphi(\theta_t) \leq \frac{4}{1+|\rho|}$ and $\theta_t \varphi(\theta_t)^2 \leq \frac{4}{1+|\rho|}$, which guarantee absence of butterfly arbitrage.
• Use piecewise linear interpolation of $\theta_t$, $\rho_t$, and $\psi_t$ across maturities, which is proven to preserve no-arbitrage conditions.
• Implement short-term extrapolation via $\theta_t = \lambda \theta_1$, $\psi_t = \lambda \psi_1$, $\rho_t = \rho_1$ with $\lambda = t/T_1$, ensuring no-arbitrage in the short end.
• Apply long-term extrapolation using a continuous increasing function $u(t)$ with $u(T_N) = 0$, setting $\theta_t = \theta_N + u(t)$, $\psi_t = \psi_N$, $\rho_t = \rho_N$, preserving no-arbitrage in the long end.

Experimental results

Research questions

• RQ1 Can a robust, one-dimensional optimization-based calibration be designed for eSSVI slices that guarantees absence of butterfly and calendar spread arbitrage?
• RQ2 Does naive piecewise linear interpolation of eSSVI parameters across maturities preserve the absence of arbitrage in the resulting full volatility surface?
• RQ3 Can short- and long-term extrapolation schemes be constructed that maintain no-arbitrage conditions while ensuring continuity of the option price function at maturity zero?
• RQ4 Is the resulting eSSVI surface construction both computationally efficient and suitable for practical use in market-making and risk management?

Key findings

• The anchored reparameterization using $\rho$ and $\psi = \theta \varphi$ significantly improves calibration robustness by grounding each slice in reliable market data near the forward moneyness.
• The use of a one-dimensional Brent algorithm for calibration ensures computational efficiency and avoids convergence issues common in higher-dimensional optimization.
• Piecewise linear interpolation of $\theta_t$, $\rho_t$, and $\psi_t$ across maturities preserves no-arbitrage conditions, a non-trivial and remarkable result for a simple interpolation scheme.
• The short-term extrapolation scheme $\theta_t = \lambda \theta_1$, $\psi_t = \lambda \psi_1$, $\rho_t = \rho_1$ with $\lambda = t/T_1$ ensures no-arbitrage and continuity as $t \to 0^+$.
• The long-term extrapolation using $\theta_t = \theta_N + u(t)$, $\psi_t = \psi_N$, $\rho_t = \rho_N$ with $u(t)$ increasing and $u(T_N) = 0$ maintains no-calendar-spread arbitrage across all maturities.
• The entire framework enables a parsimonious, fast, and reliable construction of a full arbitrage-free volatility surface, suitable for storage and re-use in risk and historical analysis.