Abstract
The common practice of using different volatilities for options of different strikes in the Black‐Scholes (1973) model imposes inconsistent assumptions on underlying securities. The phenomenon is referred to as the volatility smile. This paper addresses this problem by replacing the Brownian motion or, alternatively, the Geometric Brownian motion in the Black‐Scholes model with a two‐piece quadratic or linear function of the Brownian motion. By selecting appropriate parameters of this function we obtain a wide range of shapes of implied volatility curves with respect to option strikes. The model has closed‐form solutions for European options, which enables fast calibration of the model to market option prices. The model can also be efficiently implemented in discrete time for pricing complex options.
G1