In this paper, we consider perpetual American options under a fractional Brownian motion and give the closed-form solution for their value function. We discuss the pricing model when the underlying asset pays dividends continuously and derive the value functions. In order to get an analytical solution, we use the quadratic approximation method. By this approximation, we have Black-Scholes ordinary differential equation. Solving this equation with the boundary conditions, we get the value function and its optimal boundary.
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