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# 9 Monte Carlo Simulation

The objective of this chapter is to convey an understanding of the basic principles of Monte Carlo methods, with a particular focus on integration1 problems. Monte Carlo methods are a class of computational algorithms that rely on stochastic sampling of a (usually highdimensional) parameter space to achieve an approximation of the desired result. In finance, these methods are used, for instance, in valuating and analyzing instruments, portfolios or investments. The various sources of uncertainty (e.g., the interest rate of a floating rate bond) that affect the result are simulated, a value for each of the simulation paths2 is computed (e.g., the value of the floating rate bond in each interest rate scenario), and the final value is determined by averaging over the range of outcomes. From a more mathematical viewpoint, each source of uncertainty can be interpreted as a random variable. In probability theory, the expectation (expected value, first moment) of such a random variable is the weighted average of all possible values that this random variable can assume.3

## 9.1 THE PRINCIPLES OF MONTE CARLO INTEGRATION

The integral of a function f can be expressed by a mean value,

where a and b are the integration limits and M[f ] is the mean value of the function over this interval. The basic idea of the Monte Carlo method is to use the sample mean for M [f ],

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