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Heun Solutions to the SABR Model

After investigation of the relative volatility of the spot price in the SABR model, we find that it is related to the general Fuschian differential equation with four singularities known as the Heun equation. The boundary behavior at zero and at infinity of the spot price in the SABR model, are identified for the first time through a classical Feller test. Expressions for the moment generating function of the log price are found analytically through solving an nonhomogeneous Heun equation. For the case of zero correlation between spot and vol, the differential equation to solve simplifies to an equation with three singularities for which the solutions can be expressed in terms of Hypergeometric functions.