Multivariable integrals by Monte Carlo

Integrals by sampling

A multiple integral adds up a function over a region, $\iint_R f\,dA$. Few of them can be solved by hand, yet finance is full of them: the expected payoff of a derivative, the probability of a loss, the price of a basket option are all integrals over many dimensions.

Monte Carlo estimates the integral by sampling. Draw $N$ random points $\mathbf{x}_i$ uniformly in a region of known volume $V$ that contains the domain, average $f$ over them and multiply by $V$. The estimate is itself random, with a standard error $\sigma/\sqrt{N}$: to cut the error in half you need four times as many points.

Its great advantage is dimension. A grid with 10 points per axis needs $10^d$ points in $d$ dimensions, which is $10^{10}$ for $d=10$. Monte Carlo's error still shrinks like $1/\sqrt{N}$ whatever $d$ is. For areas and volumes it reduces to hit-or-miss: the fraction of random points that land inside the shape, times the volume of the box.

• Estimate: $V$ times the average of $f$ over uniform random points.
• Standard error: $\sigma/\sqrt{N}$; four times the points halve it.
• Hit-or-miss: area or volume is the fraction of points inside times the box volume.

Lab · what's graded

• An exact double integral15 pts
• The standard error15 pts
• How many more points15 pts
• A volume by hit-or-miss15 pts
• Why Monte Carlo in many dimensions10 pts
• The Monte Carlo recipe10 pts
• Why sampling works20 pts