Analytical solutions to the generalized Black-Scholes PDE with the help of an adiabatic approximation to the Schrödinger PDE

For particular forms of a general volatility function, analytical solutions of the Black-Scholes PDE can be found. However, it tends to be the case that the more 'realistic' the volatility function is, for instance with a volatility smile, analytical solutions become difficult to obtain. In this paper we first convert the Black-Scholes PDE into a time dependent Schrödinger PDE. Then by using the well known separation of variables technique we obtain the time independent version of this PDE. In this format we can exploit a useful technique, introduced by Jeffreys (Proc. London Math. Soc. (1923)) and Rayleigh (Proc. Roy. Soc. (1912)), to solving this time independent Schrödinger PDE. Since we obtain thus an adiabatic approximation of an initial value problem we are particularly careful in respecting the Merton conditions when finding expressions for the coefficients of the approximation. We show that for some particular forms of the volatility function those coefficients can not be found without violating some of the Merton conditions. Fortunately enough for other volatility functions, such as the volatility smile, this violation does not occur. Finally, to find the analytical solution to the Black-Scholes PDE for allowable volatility functions we need to convert back the Schrödinger PDE into the Black-Scholes PDE. For the cases treated in the paper, the solution is not a linear combination of cumulative distribution functions