Barrier-touch probability

hardMathsacceptance —~35 min

Implement touch_probability(S, B, T, r, sigma) — the probability that a geometric Brownian motion starting at S touches an upper barrier B > S at some point before T, under risk-neutral drift. With m = ln(B / S), nu = r - sigma²/2 and s = sigma·sqrt(T), the reflection principle gives P = Φ((-m + nu·T) / s) + (B / S)^(2·nu / sigma²)·Φ((-m - nu·T) / s). Use statistics.NormalDist().cdf for Φ.

This is the probability behind every knock-in, every stop-loss and the /learn probability chapter's third way. Return P rounded to 4 decimal places.

example 1

input

S=100.0, B=110.0, T=1.0, r=0.02, sigma=0.2

output

0.6337

note

Here r = sigma²/2 exactly, so the drift term vanishes.

example 2

input

S=100.0, B=110.0, T=1.0, r=0.05, sigma=0.2

output

0.6783

constraints

0 < S < B
0 < T ≤ 5
0 ≤ r ≤ 0.2
0.01 ≤ sigma ≤ 2

teacher · 2 hints · the first is free

↗ read the theory

the chapter this problem is cut from

●solution.py· utf-8 · spaces:4

from math import log, sqrt
from statistics import NormalDist

N = NormalDist().cdf


def touch_probability(S: float, B: float, T: float, r: float, sigma: float) -> float:
    """P(GBM touches B before T), rounded to 4 dp."""
    m = log(B / S)
    s = sigma * sqrt(T)
    # TODO: this is the driftless reflection formula — add the drift term nu = r - sigma²/2
    return round(2 * N(-m / s), 4)

ln 1, col 1tab indent · esc leave editor

python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to touch_probability()

S=100.0, B=110.0, T=1.0, r=0.02, sigma=0.2

expected

0.6337

the judge calls touch_probability() with exactly these inputs · outputs are compared to 6 decimal places

● standing bypython 3.13 in your browser · pyodide 0.28.3 · judged on the server to 6 dp⌘↵ run · ⌘⇧↵ submit

paper only · educational · not advice