Brownian Bridge And Stratified Sampling

Ask someone to sketch a stock's path over a year and they usually draw it left to right, day by day. But if all you actually need is where the path ends, which is all a European option cares about, drawing it day by day is backwards. It is like painting a portrait by filling in one square millimetre at a time in reading order: technically it works, but the shape of the face isn't visible until the very last stroke. A better painter blocks in the eyes and the jawline first, then fills in the detail. Simulating a Monte Carlo path works the same way, and the standard approach, draw one random step, add it, draw the next, add it, repeat hundreds of times, is the millimetre-by-millimetre painter. Every one of those steps consumes a fresh random number, and by the time you finally know where the path ends, hundreds of draws have already been spent.

Paint the big strokes first

A Brownian bridge is exactly that reordering. Instead of simulating a path step by step from today to the end, it simulates the endpoint first, then the midpoint conditional on the endpoint, then the quarter-points conditional on their neighbours, and so on, refining the path from coarse structure down to fine jitter. The mathematics behind each step is simple: given the value at two known times, the value at a time in between is normally distributed around the straight line connecting them, with a variance that depends only on how far that midpoint sits from each end,

In plain English: the bridge's value at time $t$ is a weighted average of its two known neighbours (closer to whichever one is nearer), plus a random wiggle whose size is largest exactly in the middle and shrinks to zero at the two ends, since the endpoints are already fixed.

Why this matters for quasi-Monte Carlo

A quasi-Monte Carlo path with $250$ time steps uses $250$ random numbers, one per dimension of a low-discrepancy sequence. The Sobol sequence's low-discrepancy property is strongest in its first handful of dimensions and degrades in later ones, a Sobol sequence's 200th coordinate is barely better spread than an ordinary random number. A step-by-step path spreads the payoff's dependence evenly across all $250$ dimensions, so Sobol gets no advantage there. A Brownian-bridge path concentrates the large moves, the endpoint, then the midpoint, into the first two or three dimensions, and dumps the fine, low-impact jitter into the later, weaker dimensions. The payoff ends up depending mostly on a handful of well-covered dimensions, which is what makes quasi-Monte Carlo actually pay off on multi-step, path-dependent options.

Worked example 1: building one bridge path with 4 points

Simulate a Brownian path with $W_0 = 0$, over $[0,1]$, at times $0, 0.25, 0.5, 0.75, 1$, using standard normal draws $z_1, z_2, z_3 = 1.0, 0.4, -0.6$ in bridge order.

1. Endpoint ($t=1$): $W_1 = z_1 \sqrt{1} = 1.00$.
2. Midpoint ($t=0.5$, between $W_0=0$ and $W_1=1.00$): mean $= 0.5(1.00) = 0.50$, variance $= (0.5)(0.5)/1 = 0.25$, so std $=0.5$. $W_{0.5} = 0.50 + 0.5(0.4) = 0.70$.
3. Quarter point ($t=0.25$, between $W_0=0$ and $W_{0.5}=0.70$): mean $=0.5(0.70)=0.35$, variance $=(0.25)(0.25)/0.5=0.125$, std $=0.354$. $W_{0.25} = 0.35 + 0.354(-0.6) = 0.138$.

Notice only three random numbers built four path values, and the first one already pins down where the path ends up, that is the whole point. Compare this with ordinary step-by-step simulation, which would draw four separate increments, $\Delta W_1, \Delta W_2, \Delta W_3, \Delta W_4$, one per quarter-step, and only learn the endpoint after adding up all four. Both approaches produce a path with exactly the same statistical law, the bridge is not a different model, only a different order of construction, but the bridge tells you the single most decision-relevant number, the endpoint, using only its first random draw.

Worked example 2: stratified sampling on the same endpoint

Stratified sampling is the cousin idea applied to the draw itself: instead of drawing the endpoint's normal variate freely, split the $[0,1]$ probability range into $N$ equal strata and force exactly one draw into each, so the tails are never accidentally under-sampled. With $N=4$ strata for the endpoint $W_1$: draw uniforms from $[0, 0.25), [0.25, 0.5), [0.5, 0.75), [0.75, 1)$, say $0.10, 0.40, 0.60, 0.90$, and invert each through the normal CDF to get $z = -1.28, -0.25, 0.25, 1.28$. Compare a plain random draw of 4 uniforms, which might by chance land at $0.48, 0.51, 0.55, 0.60$ (all clustered near the median, missing both tails entirely, giving $z \approx -0.05, 0.03, 0.13, 0.25$). The stratified draw guarantees representation of extreme outcomes, the very paths that determine an out-of-the-money option's value, while the naive draw can miss them by pure chance.

The paths below are built the ordinary step-by-step way, not as a bridge, but they show the raw material a bridge reorders: watch how much the path wanders between any two points, and imagine simulating the right-hand endpoint of each path first, then filling in the wandering detail afterward.

What this means in practice

Any desk pricing path-dependent products, barriers, Asians, cliquets, by Monte Carlo builds paths with a Brownian bridge whenever it also wants low-discrepancy sequences, and combines it with other variance reduction like control variates. The gain compounds: bridge construction concentrates the payoff's sensitivity into a few dimensions, and Sobol covers those few dimensions extremely well. The same reordering trick also helps plain Monte Carlo run diagnostics: because the endpoint is known after the very first random draw, a desk can price a rough, first-pass estimate of a book using far fewer total draws than a full step-by-step simulation would need, refining the path detail only for the positions that turn out to matter.