A Multi-Parameter Extension of Figlewski’s Option-Pricing Formula

Abstract

The Black–Scholes (BS) model was a major conceptual breakthrough, and it established the theoretical framework within which valuation models for all kinds of derivatives are developed. But it was quickly found to be problematic as a practical tool for traders. The ubiquitous and anomalous “volatility smile” means that a set of options written on a single underlying asset are routinely priced in the market as if it had multiple different volatilities. In practice, traders replace the theoretical model with “practitioner Black–Scholes,” which is the BS equation but with a different volatility input for each option. That is, the market uses the BS equation, not the BS model. This inconsistency led Figlewski to ask: If the BS equation is used as practitioner Black–Scholes, in a way that is inconsistent with its theoretical foundation, how much better is it, if at all, than some other equation that satisfies the basic conditions necessary to avoid profitable static arbitrage (e.g., convex shape, call value increases with asset price, etc.) and has one free parameter (like implied volatility) that can be calibrated to the market, but has no pretense of economic content beyond that? For the S&P 500, it turned out that such a model-free model performed just about as well as the BS equation.

Figlewski’s equation applied only to a single maturity. In this article, Orosi extends the equation to cover the full term structure of options, with an adjustment to prevent arbitrage, and compares it against several alternatives, including a variant of practitioner Black–Scholes. In sample and out of sample, the model-free model performs comparably to the best of the BS-based alternatives.