Semigroup theory for pricing currency options driven by fractional Brownian motion

Abstract We study the pricing of European currency options in a financial market where the exchange rate follows a fractional Brownian motion with Hurst parameter H ∈ (0, 1). Since fractional Brownian motion is not a semimartingale, standard arbitrage pricing techniques cannot be directly applied. To address this, we reformulate the pricing problem as an abstract Cauchy problem and analyze it using semigroup theory. We establish sufficient conditions for the existence and uniqueness of mild solutions to the resulting partial differential equations. This approach provides a mathematically rigorous framework for option valuation in markets with memory effects. The theoretical results are supported by numerical simulations.