Proof of non-convergence of the short-maturity expansion for the SABR model. Issue 9 (2nd September 2022)

• Record Type: Journal Article
• Authors: Lewis, Alan L.; Pirjol, Dan
• Is Part Of: Quantitative finance. Volume 22:Issue 9(2022)
• Page Start: 1747
• Page End: 1757
• Publication Date: 2022-09-02
• Subjects: Stochastic volatility -- Asymptotic expansions -- Singularity analysis -- Saddle point method; Finance -- Periodicals; Business mathematics -- Periodicals; Finance -- Mathematical models -- Periodicals; Investments -- Mathematics -- Periodicals; Economics -- Periodicals; Finances -- Modèles mathématiques -- Périodiques; 332.015118
• DOI: 10.1080/14697688.2022.2071759
• Languages: English
• ISSNs: 1469-7688

Abstract

Abstract : We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ( β = 1 ) SABR model. In this model, the option time-value can be represented as an integral of the form V ( T ) = ∫ 0 ∞ e − u 2 2 T g ( u ) d u with g ( u ) a 'payoff function' which is given by an integral over the McKean kernel G ( t, s ) . We study the analyticity properties of the function g ( u ) in the complex u -plane and show that it is holomorphic in the strip | ℑ ( u ) | < π . Using this result, we show that the T -series expansion of V ( T ) and implied volatility are asymptotic (non-convergent for any T >0). In a certain limit which can be defined either as the large volatility limit σ 0 → ∞ at fixed ω = 1, or the small vol-of-vol limit ω → 0 limit at fixed ω σ 0, the short maturity T -expansion for the implied volatility has a finite convergence radius T c = 1.32 ω σ 0 .