Translating eSSVI Calendar Constraints into Calibration Bounds

Summary

The document describes a calibration question for the eSSVI implied volatility surface, where parameters are fitted slice by slice across maturities. It focuses on a calendar-consistency inequality that links the current slice parameters to those of the previously calibrated slice. The proposed procedure searches candidate values for the correlation parameter and, for each candidate, minimizes the fit error over the curvature parameter within bounds derived from that inequality.

The central issue is how to translate the coupled constraint into lower and upper bounds for the curvature parameter. Algebraically, the inequality can be rearranged for a fixed correlation candidate, with care needed for the sign of the coefficient multiplying that parameter; when that coefficient is zero, the constraint instead reduces to a feasibility check. The document only presents the implementation question and does not include a solution or empirical calibration results. Its discussion is limited to this constraint-handling step, rather than a complete fitting method or validation of the resulting surface.

Key ideas

• The eSSVI calibration described fits two parameters separately for each maturity slice.
• The calendar constraint couples the new slice parameters to those of the preceding slice.
• For each fixed correlation candidate, the inequality can be rearranged to bound the curvature parameter, subject to the coefficient's sign.
• A zero coefficient requires checking feasibility directly rather than dividing to obtain bounds.
• The document raises the constraint translation problem but supplies no empirical results.

Tags

# ESSVI calibration problem in translating parameter bounds


# ESSVI calibration problem in translating parameter bounds












I am trying to implement the calibration algorithm presented in the "ESSVI Implied Volatility Surface" white paper from Factset by Akhundzadeh et al.

The eSSVI model includes 2 variables that must be calibrated to option market IV data slice by slice $(ρ_2,ψ_2)$ with a specific bound constraint to take note of:

$$ |\rho_2 \psi_2 - \rho_1 \psi_1| \leq \psi_2 - \psi_1, \qquad \text{for all } \; T_n > T_1 $$

where $\rho_1$, $\psi_1$ are the parameters obtained from calibrating the previous slice.

Since eSSVI's parameter bounds are dependent on another one of the variables you are calibrating for, you must make $n$ guesses for $ρ_2$ $[ρ_{2i}\ldots ρ_{2n}]$, find the bounds for $ψ_2$ corresponding to guess for $ρ_2$ using the above inequality $[ψ_{2il},ψ_{2iu}]\ldots$ , pass $[ρ_{2i}\ldots ρ_{2n}]$ as a parameter to your minimization function

$$f(ρ_{2i}), \: \text{minimize w.r.t } \: ψ_2,$$ for each, obtain a vector of minimization values $[f(p_{2i})\ldots f(p_{2n})]$, than use the minimum of the vector to center your next $n$ guesses for $ρ_2$.

In the above bound constraint, one must translate the inequality into a lower/upperbound for $ψ_2$ before continuing onto calibration in the next steps. However, you cannot mathematically back this out. Am I misinterpreting the implementation in general, or is there a workaround to this?

(P.S. First time posting so apologies for the non-clarity if found.)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.