A Monte Carlo EM algorithm for discretely observed Diffusions, Jump-diffusions and Lévy-driven Stochastic Differential Equations

Stochastic differential equations driven by standard Brownian motion(s) or Lévy processes are by far the most popular models in mathematical finance, but are also frequently used in engineering and science. A key feature of the class of models is that the parameters are easy to interpret for anyone working with ordinary differential equations, making connections between statistics and other scientific fields far smoother. We present an algorithm for computing the (historical probability measure) maximum likelihood estimate for parameters in diffusions, jump-diffusions and Lévy processes. This is done by introducing a simple, yet computationally efficient, Monte Carlo Expectation Maximization algorithm. The smoothing distribution is computed using resampling, making the framework very general. The algorithm is evaluated on diffusions (CIR, Heston), jumpdiffusion (Bates) and Lévy processes (NIG, NIG-CIR) on simulated data and market data from S & P 500 and VIX, all with satisfactory results. Keywords— Bates model, Heston model, Jump-Diffusion, Lévy process, parameter estimation, Monte Carlo Expectation Maximization, NIG, Stochastic differential equation.