Greeks by finite difference

mediumFinanceacceptance —~25 min

Implement greeks_fd(S, K, T, r, sigma, kind) — delta, gamma and vega of a European option by central finite differences on the Black-Scholes price: bump S by 0.01 for delta and gamma, bump sigma by 0.0001 for vega (report vega per unit of sigma). kind is "call" or "put"; price puts with put-call parity.

When the pricer is a black box — a tree, a Monte Carlo, a neural surrogate — this is how you still get Greeks. Return [delta, gamma, vega] rounded to 4 decimal places.

example 1

input

S=100.0, K=100.0, T=1.0, r=0.05, sigma=0.2, kind="call"

output

[0.6368, 0.0188, 37.524]

example 2

input

S=100.0, K=100.0, T=1.0, r=0.05, sigma=0.2, kind="put"

output

[-0.3632, 0.0188, 37.524]

constraints

0 < S, K ≤ 10_000
0 < T ≤ 5
0.01 ≤ sigma ≤ 2
kind ∈ {call, put}

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 _price(S, K, T, r, sigma, kind):
    d1 = (log(S / K) + (r + 0.5 * sigma ** 2) * T) / (sigma * sqrt(T))
    d2 = d1 - sigma * sqrt(T)
    call = S * N(d1) - K * exp(-r * T) * N(d2)
    # TODO: kind == "put" — put-call parity, not the call price
    return call


def greeks_fd(S: float, K: float, T: float, r: float, sigma: float, kind: str) -> list:
    """[delta, gamma, vega] by central differences, each rounded to 4 dp."""
    h, hv = 0.01, 0.0001
    up = _price(S + h, K, T, r, sigma, kind)
    mid = _price(S, K, T, r, sigma, kind)
    dn = _price(S - h, K, T, r, sigma, kind)
    delta = (up - dn) / (2 * h)
    gamma = (up - 2 * mid + dn) / h ** 2
    vega = (_price(S, K, T, r, sigma + hv, kind) - _price(S, K, T, r, sigma - hv, kind)) / (2 * hv)
    return [round(delta, 4), round(gamma, 4), round(vega, 4)]

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python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to greeks_fd()

S=100.0, K=100.0, T=1.0, r=0.05, sigma=0.2, kind="call"

expected

[0.6368, 0.0188, 37.524]

the judge calls greeks_fd() 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