Higher order approximation of option prices in Barndorff-Nielsen and Shephard models

TY - JOUR

T1 - Higher order approximation of option prices in Barndorff-Nielsen and Shephard models

AU - Roux, Alet

AU - Guinea Julia, Álvaro

N1 - © 2024 Informa UK Limited, trading as Taylor & Francis Group. This is an author-produced version of the published paper. Uploaded in accordance with the University’s Research Publications and Open Access policy.

PY - 2024/9/10

Y1 - 2024/9/10

N2 - We present an approximation method based on the mixing formula (Hull and White 1987, Romano and Touzi 1997) for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the ase where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené (2022). We obtain asymptotic results for the error of the N th order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process.

AB - We present an approximation method based on the mixing formula (Hull and White 1987, Romano and Touzi 1997) for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the ase where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das and Langrené (2022). We obtain asymptotic results for the error of the N th order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein–Uhlenbeck process or a gamma Ornstein–Uhlenbeck process.

KW - Barndorff-Nielsen and Shephard models

KW - Stochastic volatility

KW - Option pricing

KW - Closed-form approximation

KW - Taylor expansion

U2 - 10.1080/14697688.2024.2394220

DO - 10.1080/14697688.2024.2394220

M3 - Article

SN - 1469-7688

JO - Quantitative Finance

JF - Quantitative Finance

ER -