1. Lecture 4 - Monte Carlo improvements via variance reduction techniques: antithetic sampling
Antithetic variates: for any one path obtained by a gaussian variate vector draw {zi}, we generate a mirror image by changing the sign of all random numbers {zi}. Then we compute the pay-off along the two paths.
The second mirrored path has the same probability of the original one. So that we can combine the two pay-offs in a new estimator:
The Monte Carlo error for the improved estimate is:
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2. Monte Carlo improvements via variance reduction techniques: control variates
Control variates: the Monte Carlo simulation is carried out both for the original problem as well as a similar problem for which we have a closed form solution. Being
f : the option value we want to estimate and
y : the analytical exact value of the auxiliary option
an improved estimator of f is:
As a consequence of the above relations, control variates become more and more efficient as the auxiliary option is more correlated (or anti-correlated) with the original option, i.e. when the two problems are similar.
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3. Monte Carlo improvements via variance reduction techniques: control variates and p
Let us back to the problem of estimate p. A good choice for a control variable is a polygon with n edges inscribed in the circle. Indeed:
a closed formula exists for any value of n
it has an high superposition with the circle
Stochastic term
The error scales again as 1/sqrt(N) but with a smaller proportional constant :
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4. Monte Carlo improvements: low discrepancy sequences – Quasi Monte Carlo
In the standard Monte Carlo the error decreases slowly, as 1/sqrt(N), with number of samples, N, because draws do not fill in the space in a regular way. Indeed some gaps are present (clustering effects).
In low discrepancy sequences the points are chosen in order to fill in the space more regularly and uniformly, without inhomogeneities. As a result the function to be integrated converges not as one over the square root of the number of samples (N) but much more closely as one over N (Quasi Monte Carlo).
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5. Monte Carlo improvements: low discrepancy sequences – definition and results
The most famous algorithms to generate low discrepancy numbers are the Sobol and Halton sequences.
The discrepancy is a measure of how inhomogeneous a set of D-dimensional vectors of random numbers fills in a unit hypercube.
By definition, in a low discrepancy sequence (in D dimension), the discrepancy scales with the number of draws, N, as:
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6. Monte Carlo improvements: low discrepancy sequences – high dimensional behavior
Pros: For a given precision, a lower number of scenarios are needed.
Cons:
The convergence speed depends on problem dimension (making the method inefficient in very high dimensions).
Quasi Monte carlo simulation are not reliable when high dimensions are involved, with a breakdown of homogeneity along some hyper-planes.
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7. Monte Carlo pros and cons
Can be used in high dimensional problems.
Easy to implement
Easily extensible to any type of pay-off
Cons:
Heavy from a computational point of view.
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8. Conclusion
We have presented a powerful numerical technique to price exotic options: the Monte Carlo method.
Numerical methods in finance will become more and more important due to the rapid growth in financial markets of the exotic products, with a clear trend to increase the complexity embedded in the exotic options.
References: P. Jackel - Monte Carlo Methods in Finance, Wiley Finance, (2002).
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