Efficient estimation for the Greeks of Asian options under mixed fractional Brownian motion

Greeks are widely utilized to quantify the impact of diverse market factors on option prices, and the rapid, accurate estimation of these sensitivities is paramount for effective risk management. Among the available approaches, the finite difference method stands as the most prevalent, yet it suffers from a critical limitation: poor numerical stability. To address these inherent drawbacks, this paper uses a novel methodology called the Malliavin calculus approach. Specifically, we derive explicit expressions for the Greeks of continuous Asian options by leveraging Malliavin calculus, and transform the computation of Greeks into the calculation of Malliavin weight. Our analysis is built upon an underlying asset price model driven by a mixed fractional Brownian motion, which effectively captures key market characteristics such as long memory and self-similarity. This methodology circumvents the need for direct differentiation of the option price, thus offering enhanced generality and applicability. By applying this method, we successfully derive the Greeks for both continuous geometric and arithmetic Asian options, respectively. Some numerical results demonstrate that the Malliavin calculus approach outperforms the finite difference method in terms of computational efficiency.