TY - JOUR
T1 - Proof of non-convergence of the short-maturity expansion for the SABR model
AU - Lewis, Alan L.
AU - Pirjol, Dan
N1 - Publisher Copyright:
© 2022 Informa UK Limited, trading as Taylor & Francis Group.
PY - 2022
Y1 - 2022
N2 - We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ((Formula presented.)) SABR model. In this model, the option time-value can be represented as an integral of the form (Formula presented.) with (Formula presented.) a ‘payoff function’ which is given by an integral over the McKean kernel (Formula presented.). We study the analyticity properties of the function (Formula presented.) in the complex u-plane and show that it is holomorphic in the strip (Formula presented.). Using this result, we show that the T-series expansion of (Formula presented.) 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 (Formula presented.) at fixed (Formula presented.), or the small vol-of-vol limit (Formula presented.) limit at fixed (Formula presented.), the short maturity T-expansion for the implied volatility has a finite convergence radius (Formula presented.).
AB - We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ((Formula presented.)) SABR model. In this model, the option time-value can be represented as an integral of the form (Formula presented.) with (Formula presented.) a ‘payoff function’ which is given by an integral over the McKean kernel (Formula presented.). We study the analyticity properties of the function (Formula presented.) in the complex u-plane and show that it is holomorphic in the strip (Formula presented.). Using this result, we show that the T-series expansion of (Formula presented.) 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 (Formula presented.) at fixed (Formula presented.), or the small vol-of-vol limit (Formula presented.) limit at fixed (Formula presented.), the short maturity T-expansion for the implied volatility has a finite convergence radius (Formula presented.).
KW - Asymptotic expansions
KW - Saddle point method
KW - Singularity analysis
KW - Stochastic volatility
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U2 - 10.1080/14697688.2022.2071759
DO - 10.1080/14697688.2022.2071759
M3 - Article
AN - SCOPUS:85130292170
SN - 1469-7688
VL - 22
SP - 1747
EP - 1757
JO - Quantitative Finance
JF - Quantitative Finance
IS - 9
ER -