Summary: BLACK-SCHOLES GOES HYPERGEOMETRIC
CLAUDIO ALBANESE, GIUSEPPE CAMPOLIETI, PETER CARR, AND ALEXANDER LIPTON
ABSTRACT. We introduce a general pricing formula that extends Black-Scholes' and contains as particular
cases most analytically solveable models in the literature, including the quadratic and the constant-elasticity-
of-variance (CEV) models for European and barrier options. In addition, large families of new solutions can
be found, containing as many as seven free parameters.
It has been known since the seventies that Black-Scholes pricing formulas are a special case of more
general families of pricing formulas with more than just the volatility as an adjustable parameter. The
list of the classical extensions includes affine, quadratic and the constant-elasticity-of-variance models.
These models admit up to three adjustable parameters and have found a variety of applications to solving
pricing problems for equity, foreign exchange, interest rate and credit derivatives. In a series of working
papers, the authors have recently developed new mathematical techniques that allow to go much further
and to build several families of pricing formulas with up to seven adjustable parameters in the stationary,
driftless case, and additional flexibility in the general time dependent case. The formulas extend to
barrier options and have a similar structure as the Black-Scholes formulas, the most notable difference
being that error functions (or cumulative normal distributions) are replaced by (confluent) hypergeometric
functions, i.e. the special transcendental functions of applied mathematics and mathematical physics.
Let F denote a generic financial observable which we know is driftless. Examples could be the
forward price of a stock or foreign currency under the forward measure, a LIBOR forward rate or a swap