De-Arbitraging with a Weak Smile: Application to Skew Risk

Abstract

The aim of this article is to address the methodology behind de-arbitraging a realistic volatility surface and stressing it without adding arbitrages. We derive from basic principles the constraints which the changes on the strike and the tenor axis must satisfy in order to make a volatility surface arbitrage-free. The two most influential parameterized versions of the volatility surface will then be discussed, along with their origin and their limitations. Furthermore, this review will address the issues of finding the closest arbitrage-free volatility surface through the gSVI method, a more realistic parameterized version of the volatility surface applicable to the FX, commodities, and equities markets. Finally, using examples, the methodology behind coherently stressing this arbitrage-free volatility surface will be looked at, in order to capture and isolate the risk associated with higher-order Greeks like the Vanna or the Vomma.

Keywords: arbitrage-free volatility surface; Dupire local volatility; Fokker-Planck equation; Kolmogorov forward equation; constraint optimization; search algorithm; butterfly spread; calendar spread; arbitrage frontier; SVI; gSVI; skew risk; Vanna