Valuation of options subject to default risk under a mixed fractional and multiscale stochastic volatility model

• Published in: Advances in continuous and discrete models Vol. 2025; no. 1; p. 93
• Main Authors: Lee, Min-Ku, Kim, Jeong-Hoon
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 12.05.2025; Springer Nature B.V
• Subjects: Accuracy; Analysis; Approximation; Arbitrage; Brownian motion; Default; Difference and Functional Equations; Functional Analysis; Mathematics; Mathematics and Statistics; Monte Carlo simulation; Ordinary Differential Equations; Parameter sensitivity; Partial Differential Equations; Securities prices; Stochastic models; Volatility; Stochastic volatility; Option pricing; Fractional Brownian motion; Multiscale; Default risk
• ISSN: 2731-4235; 1687-1839; 2731-4235; 1687-1847
• DOI: 10.1186/s13662-025-03953-5

Based on the recent trend in developing stochastic volatility models in terms of a fractional Brownian motion, we introduce a volatility model for option’s underlying asset and option issuer’s total assets by incorporating the short and long memory properties into the fast and slow scale variations of volatility, respectively. We derive a closed-form formula explicitly for the approximate price of a European option with the option issuer’s default risk. We study the accuracy of the approximation and the sensitivity of the price formula to the fractional volatility parameters and provide a comparative analysis of the Black–Scholes (constant volatility) and the multiscale stochastic volatility models with or without the memory property in view of model fitting performance to Monte Carlo simulation data.