Theta bleed to expiry

mediumTradingacceptance —~20 min

Implement theta_bleed(S, K, r, sigma, days, day_count) — how much a European call loses to time over one day: price it with Black-Scholes at T = days / day_count and again at T = (days - 1) / day_count, and return [value_now, value_tomorrow, bleed] where bleed = value_now - value_tomorrow. Use N from statistics.NormalDist.

Theta is the rent an option buyer pays every night, and it is quoted per calendar day or per trading day depending on who you ask — the day_count parameter is that choice. Return the three values rounded to 4 decimal places.

example 1

input

S=100.0, K=100.0, r=0.05, sigma=0.2, days=30, day_count=365

output

[2.4934, 2.4481, 0.0453]

example 2

input

S=100.0, K=100.0, r=0.05, sigma=0.2, days=30, day_count=252

output

[3.0512, 2.995, 0.0562]

note

Same option, quoted on a 252-day year: the bleed per day is larger.

constraints

0 < S, K ≤ 10_000
2 ≤ days ≤ 365
day_count ∈ {252, 365}

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):
    d1 = (log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
    d2 = d1 - sigma * sqrt(T)
    return S * N(d1) - K * exp(-r * T) * N(d2)


def theta_bleed(S: float, K: float, r: float, sigma: float, days: int, day_count: int) -> list:
    """[value now, value tomorrow, one-day bleed], rounded to 4 dp."""
    # TODO: a day is 1 / day_count years — 365 is hard-coded here
    now = _call(S, K, days / 365, r, sigma)
    tomorrow = _call(S, K, (days - 1) / 365, r, sigma)
    return [round(now, 4), round(tomorrow, 4), round(now - tomorrow, 4)]

ln 1, col 1tab indent · esc leave editor

python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to theta_bleed()

S=100.0, K=100.0, r=0.05, sigma=0.2, days=30, day_count=365

expected

[2.4934, 2.4481, 0.0453]

the judge calls theta_bleed() 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