Heston Monte Carlo pricer

hardQuantacceptance —~45 min

Implement heston_call(S, K, T, r, v0, kappa, theta, xi, rho, steps, n, seed) — a European call under Heston stochastic volatility by Euler simulation with full truncation. Call random.seed(seed); for each path and step draw z1 = random.gauss(0, 1) then w = random.gauss(0, 1), set z2 = rho·z1 + sqrt(1 - rho²)·w, update v = max(v + kappa·(theta - v)·dt + xi·sqrt(max(v, 0))·sqrt(dt)·z2, 0) and S = S·exp((r - v/2)·dt + sqrt(v)·sqrt(dt)·z1) using the variance from the START of the step. Discount the mean payoff.

Five parameters and a correlation buy you a volatility smile that Black-Scholes cannot draw; the workbench's Heston surrogate is trained on this exact simulation. Return the price rounded to 4 decimal places.

example 1

input

S=100.0, K=100.0, T=1.0, r=0.02, v0=0.04, kappa=1.5, theta=0.04, xi=0.3, rho=0.0, steps=20, n=400, seed=7

output

9.2308

example 2

input

S=100.0, K=100.0, T=1.0, r=0.02, v0=0.04, kappa=1.5, theta=0.04, xi=0.3, rho=-0.7, steps=20, n=400, seed=7

output

9.1822

note

Same parameters with rho = −0.7: the skew appears.

constraints

-1 < rho < 1
kappa, theta, xi > 0
steps ≤ 40, n ≤ 1_000 (it has to run in your head, not on a cluster)

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, sqrt


def heston_call(S, K, T, r, v0, kappa, theta, xi, rho, steps, n, seed):
    """Heston call by Euler Monte Carlo, rounded to 4 dp."""
    random.seed(seed)
    dt = T / steps
    total = 0.0
    for _ in range(n):
        s, v = S, v0
        for _ in range(steps):
            z1 = random.gauss(0, 1)
            w = random.gauss(0, 1)
            # TODO: the variance shock is correlated with the price shock through rho
            z2 = w
            v_pos = max(v, 0.0)
            s = s * exp((r - v_pos / 2) * dt + sqrt(v_pos) * sqrt(dt) * z1)
            v = max(v + kappa * (theta - v) * dt + xi * sqrt(v_pos) * sqrt(dt) * z2, 0.0)
        total += max(s - K, 0.0)
    return round(exp(-r * T) * total / n, 4)

ln 1, col 1tab indent · esc leave editor

python 3.13 · pyodide

+ 6 hidden on submit

input · keyword arguments to heston_call()

S=100.0, K=100.0, T=1.0, r=0.02, v0=0.04, kappa=1.5, theta=0.04, xi=0.3, rho=0.0, steps=20, n=400, seed=7

expected

9.2308

the judge calls heston_call() 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