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Unformatted text preview: Lecture 21 Monte Carlo Methods in Finance 21.1 What Monte Carlo for? Monte Carlo simulation, considered as mathematical experimentation, can help in statistical mod- eling, optimization for a function of many variables, numerical integration over a high-dimensional space, and a lot more ... Example 21.1: Statistical modeling J J ' ' H H J J P P diagnostics statistical inference (bottom-up) descriptive statistics fitted model (top-down) Monte Carlo simulation proposed model synthetic data real data See the diagram of reconstruction cycle. Consider the usual procedure: Given a data set, summarize it, propose a model, fit the model (statistical inference), generate synthetic data from the fitted model (this is the simulation step!), compare the simulated data with the real one. May need to start over again, etc. Such cases are plenty: regression, time series, spatial stat, with many applications. Why need a model? For prediction, better understanding, generalization, etc. 1 Example 21.2: Optimization Maybe the last resort when calculus, discrete search, etc. cannot do it. Optimization via simulation is usually very time-consuming. Consider the 2D Ising model in statistical physics. Minimizing the energy function U ( x ) is equivalent to finding the ground state x , i.e. the mode of the Gibbs distribution. This can be done by simulated annealing (SA). See Lius book, sec. 1.3 for the Ising model, sec. 10.2 for SA. Geman and Geman (1984) applies SA to image analysis. Example 21.3: Numerical integration Consider the integral I = Z D g ( x ) ( x ) dx (21.1) where the function g and probability density are defined over a domain D IR d . How to compute I numerically?...
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This note was uploaded on 11/17/2011 for the course STOR 890 taught by Professor Staff during the Spring '08 term at UNC.
- Spring '08