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 —

$$W_t \mid W_s, W_u \;\sim\; \mathcal{N}\!\left(W_s + \frac{t-s}{u-s}(W_u - W_s),\; \frac{(t-s)(u-t)}{u-s}\right), \qquad s < t < u.$$

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.