Editing Monte Carlo methods in finance (section)

==== [[Control variates|Control variate method]] ====
It is also natural to use a [[control variate]]. Let us suppose that we wish to obtain the Monte Carlo value of a derivative ''H'', but know the value analytically of a similar derivative I. Then ''H''* = (Value of ''H'' according to Monte Carlo) + B*[(Value of ''I'' analytically) &minus; (Value of ''I'' according to same Monte Carlo paths)] is a better estimate, where B is covar(H,I)/var(H).

The intuition behind that technique, when applied to derivatives, is the following: note that the source of the variance of a derivative will be directly dependent on the risks (e.g. delta, vega) of this derivative. This is because any error on, say, the estimator for the forward value of an underlier, will generate a corresponding error depending on the delta of the derivative with respect to this forward value. The simplest example to demonstrate this consists in comparing the error when pricing an at-the-money call and an at-the-money straddle (i.e. call+put), which has a much lower delta.

Therefore, a standard way of choosing the derivative ''I'' consists in choosing a [[replicating portfolio]]s of options for ''H''. In practice, one will price ''H'' without variance reduction, calculate deltas and vegas, and then use a combination of calls and puts that have the same deltas and vegas as control variate.