15 - Applications of Monte Carlo methods

Summary

In this chapter we discuss some applications of Monte Carlo methods to the analysis of actuarial and financial data. We first re-visit the tests of model misspecification introduced in Chapter 13. For an asymptotic test, Monte Carlo simulation can be used to improve the performance of the test when the sample size is small, in terms of getting more accurate critical values or p-values. When the asymptotic distribution of the test is unknown, as for the case of the Kolmogorov–Smirnov test when the hypothesized distribution has some unknown parameters, Monte Carlo simulation may be the only way to estimate the critical values or p-values.

The Monte Carlo estimation of critical values is generally not viable when the null hypothesis has some nuisance parameters, i.e. parameters that are not specified and not tested under the null. For such problems, the use of bootstrap may be applied to estimate the p-values. Indeed, bootstrap is one of the most powerful and exciting techniques in statistical inference and analysis. We shall discuss the use of bootstrap in model testing, as well as the estimation of the bias and mean squared error of an estimator.

The last part of this chapter is devoted to the discussion of the simulation of asset-price processes. In particular, we consider both pure diffusion processes that generate lognormally distributed asset prices, as well as jump–diffusion processes that allow for discrete jumps as a random event with a random magnitude.