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# Conclusions and Observations

The accuracy in the use of Monte Carlo scheme to calculate risk-neutral expectations and price financial instruments depends mainly on the value of dt, the number of Monte Carlo simulations, and the method of approximation that we use to simulate the SDEs. The first two are parameters of our system that we can easily change, but the third one is an equation of our procedure. In our simulation results (figures 2.7 − 2.12), we can see that all approximations start with a different accuracy value or mean error defined in {45}, follow their own path and, eventually, as dt becomes smaller (dt < 1/15), they converge to the same value.

The Euler scheme, the most famous in the literature, is easy to implement. However, this is the only advantage, because as we can see in our results, it is the worst approximation of all. We recommend the use of it only when we price financial instruments with very small dt (dt < 1/40). Otherwise, Milstein is the better solution. The 1D Milstein scheme is easy to implement and is very famous also. The results are acceptable but never as good as its second dimension version. The accuracy of the 2D Milstein scheme depends directly on the approximation method you use to simulate its double Ito integral {24}. The conclusions of the five methods defined are:

• The subdivision method proposed by Kloeden (page 13) has an error when dt > 1 in the initial conditions {26}. They, without any doubt, have to be equal to zero. If not, all the results will be wrong or worse than the simple Euler scheme. Using its correction (page 14), this method can give excellent results and the accuracy depends on the approximation formulae {28}. The main problem with this method is, as we increase NK to obtain better results, the computation time increases exponentially to simulate it. Therefore, the implementation in practice does not work. However, this method explains to us in an easy way how the double integral works. We use this method by simulating one million of double integrals {24}, and the results help us to determine the best method.

• The Fourier Lévy formulae {33} is a bad approximation, because it gives a larger error than the simple 1D Milstein scheme. We can see in the results that it follows the same path as if we approximate the double Ito integral {24} to a simple numerical number (I(2,1) = 1/ 2 ∆W1∆W2). However, the deduction of the formulae and its structure are the key to obtain the rest of the approximations.

• The Exact Fourier Lévy formulae {37} is the second best method and is an exact copy of the Fourier Levy formulae with a simple modification. Because of the assumption {31}, this method has less accuracy than the Real Variance formulae when dt is big. Otherwise, this method gives an excellent approximation. We can see in the results that for dt ≤ 1/13, this method gives the best mean error.

• The Real Variance formulae {41} is the best method of all approximations. Its formulae is an accumulation of all the method properties we outline above. As we can see in the results, it gives the best mean and variance for the error defined in {45} for all dt > 1/13.

• The Fast Real Variance formulae {43} is a descendent of the Real Variance formulae, and we strongly recommend only the use of it if we price the option using dt = 1. Otherwise, the original one {41} is the best solution.

In summary, we can say, if we use the 2D Milstein scheme to calculate the option price, and if we use dt = 1, we recommend the use of the Fast Real Variance formulae {43}, for 1 > dt > 1/13 the Real Variance formulae {41} is the best, and for 1/13 ≥ dt > 1/20 the Exact Fourier Lévy formulae {37}, otherwise, for dt ≤ 1/20 the simple 1D Milstein scheme can give the same results.

An important point to note is that using the mean of different Monte Carlo simulations with different sets of independent random numbers gives us a big opportunity to appreciate more the accuracy of the approximation methods described in this chapter. For example, to calculate the right mean error {45} or right probability and converge the error to one value, we need for our results a combination of 1, 000, 000 simulations taking into account the number of Monte Carlo simulations times the number of independent sets or subroutines.

Prof. Klaus Schmitz

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