In this dissertation we develop a spatially inhomogeneous Markov process as a model for financial asset prices. This model is called the Hunt variance gamma process. We define it via its infinitesimal generator, and prove that this generator induces a unique measure on the space of cadlag functions. We next describe a procedure to do computations with this model, by finding a continuous-time Markov chain approximation. This approximation is used to calibrate the model to fit the S&P 500 futures option surface. Next we investigate specific characteristics of the process, showing how it differs from both Levy and Sato processes. We conclude by using the calibrated model to answer questions about properties of the risk-neutral distribution of future stock prices. We observe a more accurate fit to the risk-neutral term structure of volatility, skewness, and kurtosis, and the presence of mean-reversion in conditional probabilities involving large jumps.