Binomial-tree American put

hardFinanceacceptance —~40 min

Implement american_put(S, K, T, r, sigma, steps) — the price of an American put on a Cox-Ross-Rubinstein binomial tree with steps steps: u = exp(sigma·sqrt(dt)), d = 1/u, risk-neutral p = (exp(r·dt) - d) / (u - d). Roll back from expiry, and at every node take the larger of continuation value and immediate exercise K - S_node.

The put is the option where early exercise actually matters, and the tree is the honest way to see it. The workbench's American pricer bot is a neural surrogate trained on exactly this tree. Return the price rounded to 4 decimal places.

example 1

input

S=100.0, K=100.0, T=1.0, r=0.0, sigma=0.2, steps=50

output

7.9258

example 2

input

S=100.0, K=110.0, T=1.0, r=0.06, sigma=0.2, steps=50

output

11.6613

note

With r = 6% and the put deep in the money, exercising early beats waiting.

constraints

0 < S, K ≤ 10_000
0 < T ≤ 5
0 ≤ r ≤ 0.2
10 ≤ steps ≤ 500

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

from math import exp, sqrt


def american_put(S: float, K: float, T: float, r: float, sigma: float, steps: int) -> float:
    """CRR tree price of an American put, rounded to 4 dp."""
    dt = T / steps
    u = exp(sigma * sqrt(dt))
    d = 1 / u
    p = (exp(r * dt) - d) / (u - d)
    disc = exp(-r * dt)
    values = [max(K - S * u ** j * d ** (steps - j), 0.0) for j in range(steps + 1)]
    for i in range(steps - 1, -1, -1):
        nxt = []
        for j in range(i + 1):
            cont = disc * (p * values[j + 1] + (1 - p) * values[j])
            # TODO: an American holder may exercise here — compare with K - S at this node
            nxt.append(cont)
        values = nxt
    return round(values[0], 4)

ln 1, col 1tab indent · esc leave editor

python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to american_put()

S=100.0, K=100.0, T=1.0, r=0.0, sigma=0.2, steps=50

expected

7.9258

the judge calls american_put() 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