Discovering parametrizations of implied volatility with symbolic regression

Abstract

We investigate the data-driven discovery of parametric representations for implied volatility slices. Using symbolic regression, we search for simple analytic formulas that approximate the total implied variance as a function of log-moneyness and maturity. Our approach generates candidate parametrizations directly from market data without imposing a predefined functional form. We compare the resulting formulas with the widely used SVI parametrization in terms of accuracy and simplicity. Numerical experiments indicate that symbolic regression can identify compact parametrizations with competitive fitting performance.

Figures (6)

• Figure 1: Expression tree of the SVI parametrization \ref{['eq:SVI']}.
• Figure 2: Complexity vs. log-loss of symbolic regression results and comparison to SVI. The six expressions $f_1, \dots f_6$ on the efficient frontier with smaller loss than SVI are marked.
• Figure 3: Log-Loss distribution of expression $f_4$ vs. SVI
• Figure 4: Three fits of the $f_4$-parametrization, representing the $10\%$-, the $50\%$- (median) and $90\%$-quantile in terms of fitting error. The SVI-fit for each slice is also shown.
• Figure 5: Fit of the discovered parametrizations $\hat{w}_1^\text{C6}$ and $\hat{w}_2^\text{C6}$ to two C6-curves from the data set D2. Note that curves and data points are virtually indistinguishable in the left plot.