Techniques
Basic idea:
Hence,
has less variance than the standard Monte Carlo estimator .
In other words, part of the expectation calculation can help reduce the variance of the simulation.
Remarks:
- For any choice of the random variable , the conditional Monte Carlo method yields a variance reduction. But a good choice of yields a high correlation between and
- We can find precisely in an analytical form only when we assume a “nice” model for the underlying stochastic process
- Under several examples of SDE, such conditional expectations are readily computable
Examples
Example (1): Rare event probability
Suppose we want to estimate for some . This arises in the case of a digital option. For example, for some random variables . Hence
Denote and , we can then introduce the conditioning on the maximum,
Therefore
Denote , which is the complementary of the CDF , then
Example (2): Barrier Options
Suppose we want to find the price of a European option that has a payoff at expiration given by
The price of this option is given by
If we know the transition density of given , then we can use the conditional Monte Carlo method:
where and can be calculated with BS European call formula (with , maturity time , interest rate , and strike price , ).
Example (3): Knock-in Option
Consider the digital knock-in option with a payoff
With , we get
where is the first such that
To find , we can use the importance sampling method, which requires choices of two changes of measure. Alternatively, we have
Loading Comments...