Pricing vanilla, barrier, and lookback options under two-scale stochastic volatility driven by two approximate fractional Brownian motions

Min-Ku Lee; Jeong-Hoon Kim; Min-Ku Lee; Jeong-Hoon Kim

doi:10.3934/math.20241248

AIMS Mathematics

2024, Volume 9, Issue 9: 25545-25576. doi: 10.3934/math.20241248

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Pricing vanilla, barrier, and lookback options under two-scale stochastic volatility driven by two approximate fractional Brownian motions

Min-Ku Lee ^{1
,},
Jeong-Hoon Kim ^{2
,
,}

1.
Department of Mathematics, Kunsan National University, Kunsan 54150, Republic of Korea; Email: mgcorea@kunsan.ac.kr
2.
Department of Mathematics, Yonsei University, Seoul 03722, Republic of Korea; Email: jhkim96@yonsei.ac.kr

Received: 29 June 2024 Revised: 13 August 2024 Accepted: 26 August 2024 Published: 02 September 2024
MSC : 91G20, 35Q91, 60J70

In this paper, we proposed a stochastic volatility model in which the volatility was given by stochastic processes representing two characteristic time scales of variation driven by approximate fractional Brownian motions with two Hurst exponents. We obtained an approximate closed-form formula for a European vanilla option price and the corresponding implied volatility formula based on singular and regular perturbations and a Mellin transform. The explicit formula for the implied volatility allowed us to find the slope of the implied volatility skew with respect to the Hurst exponent and time-to-maturity. The proposed model allows the market volatility behavior to be captured uniformly in time-to-maturity. We conducted an empirical analysis to find the validity of the proposed model by comparing it with other models and Monte Carlo simulation. Further, we extended the pricing result for the vanilla option to two path-dependent exotic (barrier and lookback) options and obtained the corresponding price formulas explicitly.
- approximate fractional Brownian motion,
- Hurst exponent,
- two-scale,
- stochastic volatility,
- option pricing
Citation: Min-Ku Lee, Jeong-Hoon Kim. Pricing vanilla, barrier, and lookback options under two-scale stochastic volatility driven by two approximate fractional Brownian motions[J]. AIMS Mathematics, 2024, 9(9): 25545-25576. doi: 10.3934/math.20241248

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Abstract

In this paper, we proposed a stochastic volatility model in which the volatility was given by stochastic processes representing two characteristic time scales of variation driven by approximate fractional Brownian motions with two Hurst exponents. We obtained an approximate closed-form formula for a European vanilla option price and the corresponding implied volatility formula based on singular and regular perturbations and a Mellin transform. The explicit formula for the implied volatility allowed us to find the slope of the implied volatility skew with respect to the Hurst exponent and time-to-maturity. The proposed model allows the market volatility behavior to be captured uniformly in time-to-maturity. We conducted an empirical analysis to find the validity of the proposed model by comparing it with other models and Monte Carlo simulation. Further, we extended the pricing result for the vanilla option to two path-dependent exotic (barrier and lookback) options and obtained the corresponding price formulas explicitly.

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