Symmetries and conservation laws of Asian options with transaction costs and dividends under the fractional Brownian model via nonlinear self-adjointness

Standard financial models often fail to capture the intricate, non-linear dynamics of real-world markets. We develop a symmetry-based analysis of an option-pricing model that exhibits long-memory effects, obtained by driving the underlying with a fractional Brownian motion. We focus on the pricing equation for a European-exercise, fixed-strike geometric-average (Asian) option and carry out a complete Lie point-symmetry classification of that equation in logarithmic coordinates, in which the path-dependent averaging term takes a tractable form. The classification yields a solvable six-dimensional symmetry algebra, several of whose generators are genuinely fractional, carrying weights t 2 H , t 2 H + 1 tied to the Hurst parameter, together with the usual infinite-dimensional superposition freedom of a linear equation. Because the pricing operator is parabolic and is not the Euler–Lagrange equation of a variational principle, Noether’s theorem does not apply directly; instead we construct conservation laws through Ibragimov’s nonlinear self-adjointness (equivalently, through the direct multiplier method) by introducing the adjoint pricing operator. The resulting conserved currents are parametrised by solutions of the adjoint (state-price) equation and encode the martingale/no-arbitrage structure of the model. A discount-weighted adjoint solution yields an explicit, exactly verifiable conservation law, and pairing the adjoint solutions with the symmetry generators through Ibragimov’s theorem produces a bilinear, symmetry-indexed family of such laws, three of which are genuinely fractional. Numerical experiments analyse the effect of the Hurst parameter on prices and Greeks, verify the conservation law against the discretisation error, and compare the fractional model with the standard Black–Scholes benchmark.