Analytic approximations for foreign equity options under a stochastic volatility with fast mean reversion

Jaegi Jeon; Geonwoo Kim; Jaegi Jeon; Geonwoo Kim

doi:10.3934/math.2025849

AIMS Mathematics

2025, Volume 10, Issue 8: 18997-19017. doi: 10.3934/math.2025849

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Research article

Analytic approximations for foreign equity options under a stochastic volatility with fast mean reversion

Jaegi Jeon ¹,
Geonwoo Kim ^{2
,
,}

1.
Graduate School of Data Science, Chonnam National University, Gwangju 61186, Republic of Korea
2.
School of Natural Sciences, Seoul National University of Science and Technology, Seoul 01811, Republic of Korea

Received: 21 May 2025 Revised: 06 August 2025 Accepted: 13 August 2025 Published: 21 August 2025
MSC : 91G20

In this paper, we investigate analytical approximations for foreign equity options under a stochastic volatility model with fast mean reversion. Foreign equity options are financial derivatives whose payoffs are determined by the price of an underlying asset and the foreign exchange rate, and they involve two currencies where the currency of the actual payoff differs from the currency of the underlying asset. We consider a stochastic volatility model to capture key stylized facts observed in financial markets, such as volatility clustering and the volatility smile, by modeling the volatilities of both the foreign asset and the exchange rate as rapidly mean-reverting processes. Specifically, we employ asymptotic expansion techniques to derive analytical pricing formulas for two types of options: Option struck in foreign currency and option struck in the domestic currency. This methodology provides accurate price approximations without complex numerical methods. Additionally, we provide some examples to show the importance of model parameters in foreign equity option pricing.
- foreign equity option,
- stochastic volatility,
- fast mean reversion,
- asymptotic expansion
Citation: Jaegi Jeon, Geonwoo Kim. Analytic approximations for foreign equity options under a stochastic volatility with fast mean reversion[J]. AIMS Mathematics, 2025, 10(8): 18997-19017. doi: 10.3934/math.2025849

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Abstract

In this paper, we investigate analytical approximations for foreign equity options under a stochastic volatility model with fast mean reversion. Foreign equity options are financial derivatives whose payoffs are determined by the price of an underlying asset and the foreign exchange rate, and they involve two currencies where the currency of the actual payoff differs from the currency of the underlying asset. We consider a stochastic volatility model to capture key stylized facts observed in financial markets, such as volatility clustering and the volatility smile, by modeling the volatilities of both the foreign asset and the exchange rate as rapidly mean-reverting processes. Specifically, we employ asymptotic expansion techniques to derive analytical pricing formulas for two types of options: Option struck in foreign currency and option struck in the domestic currency. This methodology provides accurate price approximations without complex numerical methods. Additionally, we provide some examples to show the importance of model parameters in foreign equity option pricing.

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