Techniques
While the conditional Monte Carlo is based on the fact that
the Stratified sampling method is based on the fact that
Actually, the law of total variance says
The Conditional Monte Carlo requires the knowledge of the conditional expectation of given while the Stratified Sampling requires the knowledge of the conditional distribution of given .
The stratified sampling method is close to the composition method of simulation.
Suppose we have a random variable and we know how to compute analytically for any function . The idea of stratified sampling is
Traditionally, stratified sampling is applied with a discrete . If
then, with
where are i.i.d. random variables from the conditional distribution given .
The variance is given by
Taking ,
The implementation of stratified sampling for estimating is as follows:
For discrete random variable with
and for any , the stratified sampling estimator is
where are i.i.d. observations from given . The is estimating .
Since are independent,
thus
In general, stratified sampling estimator of by stratifying is obtained by discretizing and sampling from the conditional distribution of given .
If has the CDF , then for , define
Define the discrete random variable as
Then
and
Examples
Example (1) Arithmetic Asian Option
The discounted payoff of an Asian call option is given by
where . The price of the option is given by .
Recall that may be expressed as
where are i.i.d. . Therefore we can write where are i.i.d. .
Although we can stratify along , is not that highly correlated with , thus will not yield great variance reduction.
Instead, we can stratify along . To use this, we need to be able to compute for any and also be also to generate observations from the conditional distribution of given .
Because are i.i.d. , and hence
To generate observations from the conditional distribution of given
- generate from the conditional distribution of given
- generate given
Summarizing, the stratified sampling estimator for pricing an arithmetic Asian option goes as follows:
- Define if for some
- For each , generate observations from the conditional distribution of given . Call these samples .
- Return the estimator
Example (2) Barrier Option
the variance can be further reduced by applying the stratified sampling method by stratifying .
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