ABSTRACT
In the study of stochastic processes and their applications to finance, geometric Brownian motion (GBM) is the simplest model used for continuous-time asset pricing. The volatility in the GBM model is constant, i.e., the diffusion coefficient is a linear function of the underlying asset price, as is the drift coefficient. This chapter focuses on a selection of alternative models of asset price dynamics, namely, the local volatility model, the constant elasticity of variance diffusion model, the Heston stochastic volatility model, diffusions with Poisson-type jumps, and the variance gamma pure jump model. While in local volatility models a more realistic behaviour of asset prices is obtained by introducing a nonlinear volatility function, in the stochastic volatility framework, volatility changes over time according to the random process correlated with the asset price process. The variance gamma process is obtained by evaluating the Brownian motion with drift at a random time given by a gamma process.
