A Monte Carlo EM algorithm for discretely observed Diffusions, Jump-diffusions and Lévy-driven Stochastic Differential Equations
Lindström, Erik (2012). A Monte Carlo EM algorithm for discretely observed Diffusions, Jump-diffusions and Lévy-driven Stochastic Differential Equations. International Journal of Mathematical Models and Methods in Applied Sciences, 6, (5), 643 - 651
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Published
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English
Department:
Mathematical Statistics
Mathematical Finance-lup-obsolete
Financial Mathematics Group
Research Group:
Mathematical Finance-lup-obsolete
Financial Mathematics Group
Abstract:
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), jump-diffusion
(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
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