<p>In this work, we extend the Bismut-Elworthy-Li formula to facilitate the sensitivity analysis of market products modeled by semi-linear mean-field stochastic differential equations driven by fractional Brownian motion. We establish an expression representation for the stochastic flow of the solutions in relation to their Malliavin derivatives, after showing the existence, uniqueness, and weak differentiability of these solutions. The framework is applied to analyze the sensitivity of variance swaps to the initial conditions, calculate the vega of the derivatives price with stochastic volatility modeled by mean-field equations, and assess the sensitivity of path-dependent Asian options to the initial states.</p>