By Maurice Holt
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Extra resources for 2nd Int'l Conference on Numerical Methods in Fluid Dynamics
In fact many of the financial applications that use Monte Carlo simulation involve the evaluation of various stochastic integrals which are related to the probabilities of particular events occurring. In many cases, however, the assumptions of constant volatility and a lognormal distribution for ST are quite restrictive. Real financial applications may require a variety of extensions to the standard Black–Scholes model. Common requirements are for: nonlognormal distributions, time-varying volatilities, caps, floors, barriers, etc.
4 Monte Carlo integration using random numbers. It can be seen that the pseudo-random sequence gives the worst performance. But as the number of points increases, its approximation to the integral improves. Of the quasi-random sequences, it can be seen that the Faure sequence has the worst performance, while both the Sobol and Neiderreiter sequences give rapid convergence to the solution. Finance literature contains many references to the benefits of using quasirandom numbers for computing important financial integrals.
The corresponding cumulative probability distribution functions F (θ ) and F (r) can be found by evaluating the following integrals: 2 F (θ ) = θ 1 2π dθ = 0 θ 2π and r F (r) = re−r 2 /2 dr = −e−r 0 2 /2 r 0 = 1 − e−r 2 /2 We now want to draw variates rˆ and θˆ from the probability distributions f (r) and f (θ ) respectively. To do this we will use the result (see for example Evans, Hastings, and Peacock (2000)), that a uniform variate u, ¯ from the distribution U(0, 1) can be transformed into a variate v¯ from the distribution f (v) by using ¯ or equivalently F (v) ¯ = u.
2nd Int'l Conference on Numerical Methods in Fluid Dynamics by Maurice Holt