Implied volatility by bisection

mediumFinanceacceptance —~25 min

Implement implied_vol(price, S, K, T, r) — the volatility sigma at which the Black-Scholes call price equals the observed price. Bisect on sigma between 0.01 and 3.0 until the interval is narrower than 1e-8, pricing with N from statistics.NormalDist.

Quotes come as prices; models think in vols. Inverting the pricer is how every implied-vol surface on every screen gets built, and the workbench's IV solver bot does exactly this — thousands of times per chain.

Return sigma rounded to 4 decimal places.

example 1

input

price=10.4506, S=100.0, K=100.0, T=1.0, r=0.05

output

0.2

example 2

input

price=8.26, S=100.0, K=100.0, T=0.5, r=0.05

output

0.25

constraints

price is attainable: 0 < price < S
0 < T ≤ 5
0 ≤ r ≤ 0.2

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 log, sqrt, exp
from statistics import NormalDist

N = NormalDist().cdf


def _call(S, K, T, r, sigma):
    # TODO: the denominator is sigma * sqrt(T) — this only works when T == 1
    d1 = (log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * T)
    d2 = d1 - sigma * sqrt(T)
    return S * N(d1) - K * exp(-r * T) * N(d2)


def implied_vol(price: float, S: float, K: float, T: float, r: float) -> float:
    """Bisect sigma in [0.01, 3.0] until the bracket is under 1e-8. 4 dp."""
    lo, hi = 0.01, 3.0
    while hi - lo > 1e-8:
        mid = (lo + hi) / 2
        if _call(S, K, T, r, mid) > price:
            hi = mid
        else:
            lo = mid
    return round((lo + hi) / 2, 4)

ln 1, col 1tab indent · esc leave editor

python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to implied_vol()

price=10.4506, S=100.0, K=100.0, T=1.0, r=0.05

expected

0.2

the judge calls implied_vol() 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