Simulation / Compute function expectation
Compute function integral
Suppose we want to find the value of:
in some region with volume .
Now, we can estimate this integral by estimating the proportion of random points that fall under , then multiplied by .
Important in Exotics pricing as there’s no analytical solutions (e.g. Asian Options)
The speed Monte Carlo method converges to the correct result as we increase the sample size / number of simulations.
Monte Carlo has convergence rate .
Quasi Monte Carlo has convergence rate .
Reduce the variance of Monte Carlo simulated result with regard to the true result.
Increase number of simulations
Not very feasible, as in order to decrease variance / noise linearly, we have to increase simulations exponentially.
Quasi Monte Carlo (QMC)
We take any prime number r where r>=2. Any integer n has a unique expansion with base r. We can then generate a sequence of numbers in the interval [0, 1), which are equally spaced within the interval.
For example, r = 3 and n = 7. We can write 7 in the form of base 3 as below:
Now, if we reflect this number about its “decimal point”, we get a new number in [0, 1):
We keep on doing this for every number n, we will generate a sequence in interval [0, 1). And we observe that the newly generated number keep filling the gaps in proceeding sequence. For example, for n = 1 to 9, we have below Faure Sequence:
In a general form, for any given number n, we can represent n in base r:
Then, we can find the corresponding Faure number in interval [0, 1) as follows: