Financial Engineering with Copulas Explained (Financial Engineering Explained) by Mai Jan-Frederik & Scherer Matthias

Author:Mai, Jan-Frederik & Scherer, Matthias [Mai, Jan-Frederik]
Language: eng
Format: epub
ISBN: 9781137346322
Publisher: Palgrave Macmillan
Published: 2014-10-02T23:00:00+00:00

The justification of Algorithm 5.0.3 stems from the strong law of large numbers, which states that for a sequence of independent and identically distributed random variables Z1, Z2, . . . with finite mean it follows almost surely that

Applying this statement to the random variables verifies why Algorithm 5.0.3 makes sense.

FAQ 5.0.4 (What are the pros and cons of the Monte-Carlo method?)

On the one hand, the most obvious drawback of the Monte-Carlo method is that it requires quite a lot of runtime, because one needs to simulate a huge number n of scenarios to create a sufficiently accurate estimate for the desired expectation value. In particular, if one wishes to calibrate the model’s parameters to observed data, then one needs to re-evaluate the expected value in question multiple times for different parameter sets. Another disadvantage is that, depending on the specific problem, the variance of the random variable f (X1, . . . , Xd) in question might be huge. In order to obtain a reasonable confidence interval in such a case, we might require a very high number of samples n. For this reason, it is always important to provide a confidence interval together with a Monte-Carlo estimator, because otherwise one cannot judge its accuracy.

On the other hand, the most striking advantage of the Monte-Carlo method is its applicability to very complex models. It is often the only method feasible, in particular if the dimension d is large. Moreover, provided the variance σ2 of f(X1, . . . , Xd) is finite, the standard deviation of the Monte-Carlo estimate equals times σ, implying that the convergence rate of the Monte-Carlo estimator is known and problem-invariant. Furthermore, applying the central limit theorem, an asymptotic (1 – α)-confidence interval can be computed quite easily by adding and subtracting the value from the Monte-Carlo estimate for the mean, for example for α = 0.05 or α = 0.01. If σ is unknown, which is typically the case, it may simply be replaced with its canonical estimator

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