FA-61516 / Options payoff and settlement / Open access

Cox-Ross-Rubinstein American option tree: the early-exercise test prices puts as calls · case 01

American puts lose their early exercise premium.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The immediate exercise value always uses the call payoff.

VERIFIED REPAIR

Use the option's own payoff at the node price.

Unsuccessful approach: Evaluating the payoff at the terminal node price for that index uses the wrong price level.

Case contract

Inputs kind, spot S, strike K, rate r, volatility sigma, maturity T in years and steps. dt=T/steps, u=exp(sigma*sqrt(dt)), d=1/u, p=(exp(r*dt)-d)/(u-d), disc=exp(-r*dt). Roll back from terminal payoffs; at each node take max(continuation, immediate exercise at that node's price). Return the root rounded to 6.

Why this case matters

Option expiry, exercise and settlement engines move cash and shares; a wrong branch misstates obligations.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(kind, S, K, r, sigma, T, steps):
    dt = T / steps
    u = math.exp(sigma * math.sqrt(dt))
    d = 1 / u
    p = (math.exp(r * dt) - d) / (u - d)
    disc = math.exp(-r * dt)
    def pay(s):
        return max(s - K, 0.0) if kind == 'C' else max(K - s, 0.0)
    vals = [pay(S * u ** j * d ** (steps - j)) for j in range(steps + 1)]
    for i in range(steps - 1, -1, -1):
        vals = [max(disc * (p * vals[j + 1] + (1 - p) * vals[j]), max(S * u ** j * d ** (i - j) - K, 0.0)) for j in range(i + 1)]
    return round(vals[0], 6)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression early exercise value 1', ['P', 80.0, 105.0, 0.05, 0.25, 2.0, 5], 25.210167], ['regression early exercise value 2', ['P', 100.0, 100.0, 0.0, 0.4, 2.0, 5], 23.343597], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.25, 0.5, 3], 25.390902], ['normal control 1', ['C', 110.0, 100.0, 0.08, 0.4, 1.0, 3], 27.248875], ['normal control 2', ['C', 110.0, 90.0, 0.05, 0.25, 0.25, 12], 21.337289], ['normal control 3', ['C', 110.0, 105.0, 0.08, 0.25, 0.25, 5], 9.501783], ['normal control 4', ['C', 95.0, 105.0, 0.0, 0.25, 0.25, 3], 1.45145], ['normal control 5', ['C', 100.0, 90.0, 0.01, 0.15, 0.25, 25], 10.448278]], [['regression early exercise value 1', ['P', 100.0, 105.0, 0.08, 0.4, 1.0, 12], 15.430377], ['regression early exercise value 2', ['P', 80.0, 90.0, 0.08, 0.25, 0.5, 5], 10.841628], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.15, 2.0, 3], 26.088253], ['partial repair probe 2', ['P', 80.0, 100.0, 0.0, 0.15, 2.0, 3], 21.605448], ['normal control 1', ['C', 95.0, 105.0, 0.0, 0.15, 1.0, 5], 2.309033], ['normal control 2', ['C', 95.0, 105.0, 0.01, 0.25, 2.0, 3], 10.862909], ['normal control 3', ['C', 80.0, 100.0, 0.01, 0.4, 1.0, 25], 6.577995], ['normal control 4', ['C', 80.0, 90.0, 0.0, 0.25, 2.0, 25], 7.653711]], [['regression early exercise value 1', ['P', 110.0, 105.0, 0.0, 0.25, 0.25, 12], 3.309833], ['regression early exercise value 2', ['P', 95.0, 100.0, 0.01, 0.25, 1.0, 25], 11.824535], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.25, 0.5, 3], 25.390902], ['partial repair probe 2', ['P', 80.0, 100.0, 0.0, 0.25, 0.5, 3], 20.925093], ['normal control 1', ['C', 80.0, 90.0, 0.0, 0.25, 0.5, 3], 1.993475], ['normal control 2', ['C', 95.0, 100.0, 0.01, 0.15, 0.25, 5], 1.165015], ['normal control 3', ['C', 80.0, 100.0, 0.08, 0.15, 1.0, 3], 0.879347], ['normal control 4', ['C', 110.0, 105.0, 0.05, 0.15, 0.5, 5], 9.10462]], [['regression early exercise value 1', ['P', 95.0, 105.0, 0.05, 0.4, 0.25, 25], 13.262218], ['regression early exercise value 2', ['P', 95.0, 100.0, 0.08, 0.15, 0.25, 5], 5.303147], ['partial repair probe 1', ['P', 80.0, 100.0, 0.0, 0.15, 0.5, 5], 20.039153], ['normal control 1', ['C', 110.0, 90.0, 0.0, 0.4, 0.25, 3], 21.805173], ['normal control 2', ['C', 100.0, 90.0, 0.0, 0.15, 2.0, 3], 13.704549], ['normal control 3', ['C', 100.0, 100.0, 0.08, 0.4, 1.0, 25], 19.526386], ['normal control 4', ['C', 80.0, 90.0, 0.0, 0.15, 0.5, 25], 0.605969], ['normal control 5', ['C', 80.0, 105.0, 0.01, 0.4, 0.25, 12], 0.754375]], [['regression early exercise value 1', ['P', 110.0, 100.0, 0.08, 0.25, 0.5, 3], 2.270558], ['regression early exercise value 2', ['P', 110.0, 90.0, 0.08, 0.4, 2.0, 5], 8.982113], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.15, 2.0, 3], 26.088253], ['partial repair probe 2', ['P', 80.0, 105.0, 0.0, 0.15, 0.5, 3], 25.0], ['normal control 1', ['C', 80.0, 105.0, 0.05, 0.4, 0.25, 12], 0.832557], ['normal control 2', ['C', 100.0, 100.0, 0.0, 0.4, 2.0, 12], 21.811807], ['normal control 3', ['C', 95.0, 100.0, 0.08, 0.15, 0.5, 3], 3.266989], ['normal control 4', ['C', 80.0, 90.0, 0.05, 0.25, 0.5, 25], 2.803238]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression early exercise value 1	`24.567697`	`25.210167`	Failed
regression early exercise value 2	`42.117121`	`23.343597`	Failed
partial repair probe 1	`25.390902`	`25.390902`	Passed
normal control 1	`27.248875`	`27.248875`	Passed
normal control 2	`21.337289`	`21.337289`	Passed
normal control 3	`9.501783`	`9.501783`	Passed
normal control 4	`1.45145`	`1.45145`	Passed
normal control 5	`10.448278`	`10.448278`	Passed

SHA-256 / cc02ac68543d3167a2f477f30f584e1f1eff78eb7af00542514b575883d8355d

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(kind, S, K, r, sigma, T, steps):
    dt = T / steps
    u = math.exp(sigma * math.sqrt(dt))
    d = 1 / u
    p = (math.exp(r * dt) - d) / (u - d)
    disc = math.exp(-r * dt)
    def pay(s):
        return max(s - K, 0.0) if kind == 'C' else max(K - s, 0.0)
    vals = [pay(S * u ** j * d ** (steps - j)) for j in range(steps + 1)]
    for i in range(steps - 1, -1, -1):
        vals = [max(disc * (p * vals[j + 1] + (1 - p) * vals[j]), pay(S * u ** j * d ** (steps - j))) for j in range(i + 1)]
    return round(vals[0], 6)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression early exercise value 1', ['P', 80.0, 105.0, 0.05, 0.25, 2.0, 5], 25.210167], ['regression early exercise value 2', ['P', 100.0, 100.0, 0.0, 0.4, 2.0, 5], 23.343597], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.25, 0.5, 3], 25.390902], ['normal control 1', ['C', 110.0, 100.0, 0.08, 0.4, 1.0, 3], 27.248875], ['normal control 2', ['C', 110.0, 90.0, 0.05, 0.25, 0.25, 12], 21.337289], ['normal control 3', ['C', 110.0, 105.0, 0.08, 0.25, 0.25, 5], 9.501783], ['normal control 4', ['C', 95.0, 105.0, 0.0, 0.25, 0.25, 3], 1.45145], ['normal control 5', ['C', 100.0, 90.0, 0.01, 0.15, 0.25, 25], 10.448278]], [['regression early exercise value 1', ['P', 100.0, 105.0, 0.08, 0.4, 1.0, 12], 15.430377], ['regression early exercise value 2', ['P', 80.0, 90.0, 0.08, 0.25, 0.5, 5], 10.841628], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.15, 2.0, 3], 26.088253], ['partial repair probe 2', ['P', 80.0, 100.0, 0.0, 0.15, 2.0, 3], 21.605448], ['normal control 1', ['C', 95.0, 105.0, 0.0, 0.15, 1.0, 5], 2.309033], ['normal control 2', ['C', 95.0, 105.0, 0.01, 0.25, 2.0, 3], 10.862909], ['normal control 3', ['C', 80.0, 100.0, 0.01, 0.4, 1.0, 25], 6.577995], ['normal control 4', ['C', 80.0, 90.0, 0.0, 0.25, 2.0, 25], 7.653711]], [['regression early exercise value 1', ['P', 110.0, 105.0, 0.0, 0.25, 0.25, 12], 3.309833], ['regression early exercise value 2', ['P', 95.0, 100.0, 0.01, 0.25, 1.0, 25], 11.824535], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.25, 0.5, 3], 25.390902], ['partial repair probe 2', ['P', 80.0, 100.0, 0.0, 0.25, 0.5, 3], 20.925093], ['normal control 1', ['C', 80.0, 90.0, 0.0, 0.25, 0.5, 3], 1.993475], ['normal control 2', ['C', 95.0, 100.0, 0.01, 0.15, 0.25, 5], 1.165015], ['normal control 3', ['C', 80.0, 100.0, 0.08, 0.15, 1.0, 3], 0.879347], ['normal control 4', ['C', 110.0, 105.0, 0.05, 0.15, 0.5, 5], 9.10462]], [['regression early exercise value 1', ['P', 95.0, 105.0, 0.05, 0.4, 0.25, 25], 13.262218], ['regression early exercise value 2', ['P', 95.0, 100.0, 0.08, 0.15, 0.25, 5], 5.303147], ['partial repair probe 1', ['P', 80.0, 100.0, 0.0, 0.15, 0.5, 5], 20.039153], ['normal control 1', ['C', 110.0, 90.0, 0.0, 0.4, 0.25, 3], 21.805173], ['normal control 2', ['C', 100.0, 90.0, 0.0, 0.15, 2.0, 3], 13.704549], ['normal control 3', ['C', 100.0, 100.0, 0.08, 0.4, 1.0, 25], 19.526386], ['normal control 4', ['C', 80.0, 90.0, 0.0, 0.15, 0.5, 25], 0.605969], ['normal control 5', ['C', 80.0, 105.0, 0.01, 0.4, 0.25, 12], 0.754375]], [['regression early exercise value 1', ['P', 110.0, 100.0, 0.08, 0.25, 0.5, 3], 2.270558], ['regression early exercise value 2', ['P', 110.0, 90.0, 0.08, 0.4, 2.0, 5], 8.982113], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.15, 2.0, 3], 26.088253], ['partial repair probe 2', ['P', 80.0, 105.0, 0.0, 0.15, 0.5, 3], 25.0], ['normal control 1', ['C', 80.0, 105.0, 0.05, 0.4, 0.25, 12], 0.832557], ['normal control 2', ['C', 100.0, 100.0, 0.0, 0.4, 2.0, 12], 21.811807], ['normal control 3', ['C', 95.0, 100.0, 0.08, 0.15, 0.5, 3], 3.266989], ['normal control 4', ['C', 80.0, 90.0, 0.05, 0.25, 0.5, 25], 2.803238]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression early exercise value 1	`68.713085`	`25.210167`	Failed
regression early exercise value 2	`71.77356`	`23.343597`	Failed
partial repair probe 1	`46.10004`	`25.390902`	Failed
normal control 1	`27.248875`	`27.248875`	Passed
normal control 2	`21.337289`	`21.337289`	Passed
normal control 3	`9.501783`	`9.501783`	Passed
normal control 4	`1.45145`	`1.45145`	Passed
normal control 5	`10.448278`	`10.448278`	Passed

SHA-256 / 03321b612329d2787280bffebc29fe4dae2780a6d4603fad59a5623c57f5df30

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(kind, S, K, r, sigma, T, steps):
    dt = T / steps
    u = math.exp(sigma * math.sqrt(dt))
    d = 1 / u
    p = (math.exp(r * dt) - d) / (u - d)
    disc = math.exp(-r * dt)
    def pay(s):
        return max(s - K, 0.0) if kind == 'C' else max(K - s, 0.0)
    vals = [pay(S * u ** j * d ** (steps - j)) for j in range(steps + 1)]
    for i in range(steps - 1, -1, -1):
        vals = [max(disc * (p * vals[j + 1] + (1 - p) * vals[j]), pay(S * u ** j * d ** (i - j))) for j in range(i + 1)]
    return round(vals[0], 6)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression early exercise value 1', ['P', 80.0, 105.0, 0.05, 0.25, 2.0, 5], 25.210167], ['regression early exercise value 2', ['P', 100.0, 100.0, 0.0, 0.4, 2.0, 5], 23.343597], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.25, 0.5, 3], 25.390902], ['normal control 1', ['C', 110.0, 100.0, 0.08, 0.4, 1.0, 3], 27.248875], ['normal control 2', ['C', 110.0, 90.0, 0.05, 0.25, 0.25, 12], 21.337289], ['normal control 3', ['C', 110.0, 105.0, 0.08, 0.25, 0.25, 5], 9.501783], ['normal control 4', ['C', 95.0, 105.0, 0.0, 0.25, 0.25, 3], 1.45145], ['normal control 5', ['C', 100.0, 90.0, 0.01, 0.15, 0.25, 25], 10.448278]], [['regression early exercise value 1', ['P', 100.0, 105.0, 0.08, 0.4, 1.0, 12], 15.430377], ['regression early exercise value 2', ['P', 80.0, 90.0, 0.08, 0.25, 0.5, 5], 10.841628], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.15, 2.0, 3], 26.088253], ['partial repair probe 2', ['P', 80.0, 100.0, 0.0, 0.15, 2.0, 3], 21.605448], ['normal control 1', ['C', 95.0, 105.0, 0.0, 0.15, 1.0, 5], 2.309033], ['normal control 2', ['C', 95.0, 105.0, 0.01, 0.25, 2.0, 3], 10.862909], ['normal control 3', ['C', 80.0, 100.0, 0.01, 0.4, 1.0, 25], 6.577995], ['normal control 4', ['C', 80.0, 90.0, 0.0, 0.25, 2.0, 25], 7.653711]], [['regression early exercise value 1', ['P', 110.0, 105.0, 0.0, 0.25, 0.25, 12], 3.309833], ['regression early exercise value 2', ['P', 95.0, 100.0, 0.01, 0.25, 1.0, 25], 11.824535], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.25, 0.5, 3], 25.390902], ['partial repair probe 2', ['P', 80.0, 100.0, 0.0, 0.25, 0.5, 3], 20.925093], ['normal control 1', ['C', 80.0, 90.0, 0.0, 0.25, 0.5, 3], 1.993475], ['normal control 2', ['C', 95.0, 100.0, 0.01, 0.15, 0.25, 5], 1.165015], ['normal control 3', ['C', 80.0, 100.0, 0.08, 0.15, 1.0, 3], 0.879347], ['normal control 4', ['C', 110.0, 105.0, 0.05, 0.15, 0.5, 5], 9.10462]], [['regression early exercise value 1', ['P', 95.0, 105.0, 0.05, 0.4, 0.25, 25], 13.262218], ['regression early exercise value 2', ['P', 95.0, 100.0, 0.08, 0.15, 0.25, 5], 5.303147], ['partial repair probe 1', ['P', 80.0, 100.0, 0.0, 0.15, 0.5, 5], 20.039153], ['normal control 1', ['C', 110.0, 90.0, 0.0, 0.4, 0.25, 3], 21.805173], ['normal control 2', ['C', 100.0, 90.0, 0.0, 0.15, 2.0, 3], 13.704549], ['normal control 3', ['C', 100.0, 100.0, 0.08, 0.4, 1.0, 25], 19.526386], ['normal control 4', ['C', 80.0, 90.0, 0.0, 0.15, 0.5, 25], 0.605969], ['normal control 5', ['C', 80.0, 105.0, 0.01, 0.4, 0.25, 12], 0.754375]], [['regression early exercise value 1', ['P', 110.0, 100.0, 0.08, 0.25, 0.5, 3], 2.270558], ['regression early exercise value 2', ['P', 110.0, 90.0, 0.08, 0.4, 2.0, 5], 8.982113], ['partial repair probe 1', ['P', 80.0, 105.0, 0.0, 0.15, 2.0, 3], 26.088253], ['partial repair probe 2', ['P', 80.0, 105.0, 0.0, 0.15, 0.5, 3], 25.0], ['normal control 1', ['C', 80.0, 105.0, 0.05, 0.4, 0.25, 12], 0.832557], ['normal control 2', ['C', 100.0, 100.0, 0.0, 0.4, 2.0, 12], 21.811807], ['normal control 3', ['C', 95.0, 100.0, 0.08, 0.15, 0.5, 3], 3.266989], ['normal control 4', ['C', 80.0, 90.0, 0.05, 0.25, 0.5, 25], 2.803238]]]
for label, args, expected in fixtures[N-1]:
    check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
regression early exercise value 1	`25.210167`	`25.210167`	Passed
regression early exercise value 2	`23.343597`	`23.343597`	Passed
partial repair probe 1	`25.390902`	`25.390902`	Passed
normal control 1	`27.248875`	`27.248875`	Passed
normal control 2	`21.337289`	`21.337289`	Passed
normal control 3	`9.501783`	`9.501783`	Passed
normal control 4	`1.45145`	`1.45145`	Passed
normal control 5	`10.448278`	`10.448278`	Passed

SHA-256 / 642c67746189590d088903b9c6fb99f21e5c47cc01269f9e16b27384977d7ab3

Verification & scope

A deterministic toy contract stated explicitly in the contract field; no claim of conformance to any exchange or clearing rulebook. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:46:56.047783+00:00.

Case digest / aa05efacce6ce895b8caa7e5dbf51c09449fcb6f36931e3ca7ef99d100537ab6