GBM path

easyMathsacceptance —~15 min

Implement gbm_path(S0, mu, sigma, dt, steps, seed) — a geometric Brownian motion path. Call random.seed(seed), then for each step draw z = random.gauss(0, 1) and set S = S·exp((mu - sigma²/2)·dt + sigma·sqrt(dt)·z). Return the steps simulated prices (not including S0).

It is the null model every option price and every Monte Carlo VaR in the workbench is built on. Return the prices rounded to 4 decimal places.

example 1

input

S0=100.0, mu=0.05, sigma=0.2, dt=1.0, steps=4, seed=7

output

[97.9047, 111.7518, 110.0639, 106.4896]

example 2

input

S0=100.0, mu=0.05, sigma=0.2, dt=0.25, steps=4, seed=7

output

[98.2075, 104.1389, 102.5772, 100.144]

constraints

S0 > 0
0 < dt ≤ 1
1 ≤ steps ≤ 252

teacher · 2 hints · the first is free

↗ read the theory

the chapter this problem is cut from

▶ open in the workbench

run the same math on a real tape

●solution.py· utf-8 · spaces:4

import random
from math import exp


def gbm_path(S0: float, mu: float, sigma: float, dt: float, steps: int, seed: int) -> list:
    """GBM prices after each step, rounded to 4 dp."""
    random.seed(seed)
    s = S0
    out = []
    for _ in range(steps):
        z = random.gauss(0, 1)
        # TODO: the shock scales with sqrt(dt), not dt
        s = s * exp((mu - 0.5 * sigma ** 2) * dt + sigma * dt * z)
        out.append(round(s, 4))
    return out

ln 1, col 1tab indent · esc leave editor

python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to gbm_path()

S0=100.0, mu=0.05, sigma=0.2, dt=1.0, steps=4, seed=7

expected

[97.9047, 111.7518, 110.0639, 106.4896]

the judge calls gbm_path() 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