FA-61741 / Options payoff and settlement / Open access

Implied volatility by bisection with arbitrage bounds: the search bracket tops out at 100% volatility · case 01

High implied volatilities are reported as 1.0.

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

ROOT CAUSE

The upper end of the bisection bracket is 1.0.

VERIFIED REPAIR

Search up to 5.0 as the contract states.

Unsuccessful approach: Doubling the bracket to 2.0 still truncates very high volatilities.

Case contract

Inputs kind, option price, S, K, r, T (no dividends). If price <= lower bound (call max(S - K e^{-rT},0), put max(K e^{-rT} - S,0)) or price >= upper bound (call S, put K e^{-rT}) return None. Otherwise bisect sigma in [1e-6, 5] for 100 iterations on the Black-Scholes price and return the midpoint rounded to 4.

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, price, S, K, r, T):
    def N(x):
        return 0.5 * (1 + math.erf(x / math.sqrt(2)))
    def bs(v):
        d1 = (math.log(S / K) + (r + 0.5 * v * v) * T) / (v * math.sqrt(T))
        d2 = d1 - v * math.sqrt(T)
        if kind == 'C':
            return S * N(d1) - K * math.exp(-r * T) * N(d2)
        return K * math.exp(-r * T) * N(-d2) - S * N(-d1)
    disc_k = K * math.exp(-r * T)
    lower = max(S - disc_k, 0) if kind == 'C' else max(disc_k - S, 0)
    upper = S if kind == 'C' else disc_k
    if price <= lower or price >= upper:
        return None
    lo, hi = 1e-6, 1.0
    for _ in range(100):
        mid = (lo + hi) / 2
        if bs(mid) > price:
            hi = mid
        else:
            lo = mid
    return round((lo + hi) / 2, 4)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression volatility search bracket 1', ['C', 91.2319, 100.0, 120.0, 0.0, 1.0], 3.5], ['regression volatility search bracket 2', ['P', 91.5993, 110.0, 100.0, 0.0, 1.0], 3.5], ['partial repair probe 1', ['P', 94.176453, 110.0, 100.0, 0.03, 2.0], 5.0], ['partial repair probe 2', ['P', 97.044553, 100.0, 100.0, 0.03, 1.0], 5.0], ['normal control 1', ['C', 1.2175, 110.0, 120.0, 0.0, 1.0], 0.1], ['normal control 2', ['P', 3.8044, 110.0, 95.0, 0.08, 1.0], 0.3], ['normal control 3', ['P', 5.59, 100.0, 100.0, 0.03, 0.25], 0.3], ['normal control 4', ['P', 2.5, 90.0, 95.0, 0.08, 2.0], 0.1239]], [['regression volatility search bracket 1', ['C', 92.3037, 100.0, 100.0, 0.08, 1.0], 3.5], ['regression volatility search bracket 2', ['C', 56.0677, 90.0, 100.0, 0.0, 2.0], 1.3], ['partial repair probe 1', ['C', 79.4077, 100.0, 95.0, 0.0, 1.0], 2.5], ['partial repair probe 2', ['C', 70.1631, 110.0, 100.0, 0.03, 0.25], 3.5], ['normal control 1', ['P', 9.2391, 100.0, 100.0, 0.08, 2.0], 0.3], ['normal control 2', ['P', 3.6659, 100.0, 95.0, 0.0, 0.25], 0.3], ['normal control 3', ['P', 92.311635, 100.0, 100.0, 0.08, 1.0], None], ['normal control 4', ['P', 92.192326, 100.0, 95.0, 0.03, 1.0], None]], [['regression volatility search bracket 1', ['C', 87.8484, 110.0, 100.0, 0.0, 1.0], 2.5], ['regression volatility search bracket 2', ['C', 56.8158, 110.0, 100.0, 0.03, 1.0], 1.3], ['partial repair probe 1', ['C', 101.9338, 110.0, 100.0, 0.08, 1.0], 3.5], ['partial repair probe 2', ['P', 116.453464, 110.0, 120.0, 0.03, 1.0], 5.0], ['normal control 1', ['C', 0.4365, 90.0, 95.0, 0.03, 0.25], 0.1], ['normal control 2', ['P', 0.0, 110.0, 95.0, 0.0, 2.0], None], ['normal control 3', ['P', 10.0, 90.0, 100.0, 0.0, 0.25], None], ['normal control 4', ['C', 8.8425, 100.0, 100.0, 0.08, 1.0], 0.1]], [['regression volatility search bracket 1', ['P', 77.5605, 100.0, 120.0, 0.03, 0.25], 3.5], ['regression volatility search bracket 2', ['C', 58.7023, 100.0, 120.0, 0.08, 0.25], 3.5], ['partial repair probe 1', ['C', 102.1427, 110.0, 95.0, 0.08, 1.0], 3.5], ['partial repair probe 2', ['P', 92.29, 100.0, 100.0, 0.0, 2.0], 2.5], ['normal control 1', ['C', 0.01, 100.0, 100.0, 0.08, 2.0], None], ['normal control 2', ['P', 36.4553, 90.0, 120.0, 0.0, 2.0], 0.3], ['normal control 3', ['C', 0.01, 90.0, 120.0, 0.08, 0.25], 0.1967], ['normal control 4', ['C', 0.01, 90.0, 120.0, 0.03, 1.0], 0.0951]], [['regression volatility search bracket 1', ['C', 55.7612, 110.0, 95.0, 0.0, 0.25], 2.5], ['regression volatility search bracket 2', ['C', 70.09, 110.0, 120.0, 0.03, 2.0], 1.3], ['partial repair probe 1', ['C', 54.2681, 110.0, 100.0, 0.0, 0.25], 2.5], ['partial repair probe 2', ['P', 88.7705, 110.0, 100.0, 0.03, 1.0], 3.5], ['normal control 1', ['C', 12.9118, 110.0, 100.0, 0.03, 2.0], None], ['normal control 2', ['C', 110.0, 110.0, 95.0, 0.08, 0.25], None], ['normal control 3', ['P', 87.696053, 100.0, 95.0, 0.08, 1.0], None], ['normal control 4', ['P', 19.99, 100.0, 120.0, 0.0, 2.0], None]]]
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 volatility search bracket 1	`1.0`	`3.5`	Failed
regression volatility search bracket 2	`1.0`	`3.5`	Failed
partial repair probe 1	`1.0`	`5.0`	Failed
partial repair probe 2	`1.0`	`5.0`	Failed
normal control 1	`0.1`	`0.1`	Passed
normal control 2	`0.3`	`0.3`	Passed
normal control 3	`0.3`	`0.3`	Passed
normal control 4	`0.1239`	`0.1239`	Passed

SHA-256 / b31159e23f35103743cec8ad0585ca85ea8a21617f73f3e3a1791e02f99efd96

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(kind, price, S, K, r, T):
    def N(x):
        return 0.5 * (1 + math.erf(x / math.sqrt(2)))
    def bs(v):
        d1 = (math.log(S / K) + (r + 0.5 * v * v) * T) / (v * math.sqrt(T))
        d2 = d1 - v * math.sqrt(T)
        if kind == 'C':
            return S * N(d1) - K * math.exp(-r * T) * N(d2)
        return K * math.exp(-r * T) * N(-d2) - S * N(-d1)
    disc_k = K * math.exp(-r * T)
    lower = max(S - disc_k, 0) if kind == 'C' else max(disc_k - S, 0)
    upper = S if kind == 'C' else disc_k
    if price <= lower or price >= upper:
        return None
    lo, hi = 1e-6, 2.0
    for _ in range(100):
        mid = (lo + hi) / 2
        if bs(mid) > price:
            hi = mid
        else:
            lo = mid
    return round((lo + hi) / 2, 4)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression volatility search bracket 1', ['C', 91.2319, 100.0, 120.0, 0.0, 1.0], 3.5], ['regression volatility search bracket 2', ['P', 91.5993, 110.0, 100.0, 0.0, 1.0], 3.5], ['partial repair probe 1', ['P', 94.176453, 110.0, 100.0, 0.03, 2.0], 5.0], ['partial repair probe 2', ['P', 97.044553, 100.0, 100.0, 0.03, 1.0], 5.0], ['normal control 1', ['C', 1.2175, 110.0, 120.0, 0.0, 1.0], 0.1], ['normal control 2', ['P', 3.8044, 110.0, 95.0, 0.08, 1.0], 0.3], ['normal control 3', ['P', 5.59, 100.0, 100.0, 0.03, 0.25], 0.3], ['normal control 4', ['P', 2.5, 90.0, 95.0, 0.08, 2.0], 0.1239]], [['regression volatility search bracket 1', ['C', 92.3037, 100.0, 100.0, 0.08, 1.0], 3.5], ['regression volatility search bracket 2', ['C', 56.0677, 90.0, 100.0, 0.0, 2.0], 1.3], ['partial repair probe 1', ['C', 79.4077, 100.0, 95.0, 0.0, 1.0], 2.5], ['partial repair probe 2', ['C', 70.1631, 110.0, 100.0, 0.03, 0.25], 3.5], ['normal control 1', ['P', 9.2391, 100.0, 100.0, 0.08, 2.0], 0.3], ['normal control 2', ['P', 3.6659, 100.0, 95.0, 0.0, 0.25], 0.3], ['normal control 3', ['P', 92.311635, 100.0, 100.0, 0.08, 1.0], None], ['normal control 4', ['P', 92.192326, 100.0, 95.0, 0.03, 1.0], None]], [['regression volatility search bracket 1', ['C', 87.8484, 110.0, 100.0, 0.0, 1.0], 2.5], ['regression volatility search bracket 2', ['C', 56.8158, 110.0, 100.0, 0.03, 1.0], 1.3], ['partial repair probe 1', ['C', 101.9338, 110.0, 100.0, 0.08, 1.0], 3.5], ['partial repair probe 2', ['P', 116.453464, 110.0, 120.0, 0.03, 1.0], 5.0], ['normal control 1', ['C', 0.4365, 90.0, 95.0, 0.03, 0.25], 0.1], ['normal control 2', ['P', 0.0, 110.0, 95.0, 0.0, 2.0], None], ['normal control 3', ['P', 10.0, 90.0, 100.0, 0.0, 0.25], None], ['normal control 4', ['C', 8.8425, 100.0, 100.0, 0.08, 1.0], 0.1]], [['regression volatility search bracket 1', ['P', 77.5605, 100.0, 120.0, 0.03, 0.25], 3.5], ['regression volatility search bracket 2', ['C', 58.7023, 100.0, 120.0, 0.08, 0.25], 3.5], ['partial repair probe 1', ['C', 102.1427, 110.0, 95.0, 0.08, 1.0], 3.5], ['partial repair probe 2', ['P', 92.29, 100.0, 100.0, 0.0, 2.0], 2.5], ['normal control 1', ['C', 0.01, 100.0, 100.0, 0.08, 2.0], None], ['normal control 2', ['P', 36.4553, 90.0, 120.0, 0.0, 2.0], 0.3], ['normal control 3', ['C', 0.01, 90.0, 120.0, 0.08, 0.25], 0.1967], ['normal control 4', ['C', 0.01, 90.0, 120.0, 0.03, 1.0], 0.0951]], [['regression volatility search bracket 1', ['C', 55.7612, 110.0, 95.0, 0.0, 0.25], 2.5], ['regression volatility search bracket 2', ['C', 70.09, 110.0, 120.0, 0.03, 2.0], 1.3], ['partial repair probe 1', ['C', 54.2681, 110.0, 100.0, 0.0, 0.25], 2.5], ['partial repair probe 2', ['P', 88.7705, 110.0, 100.0, 0.03, 1.0], 3.5], ['normal control 1', ['C', 12.9118, 110.0, 100.0, 0.03, 2.0], None], ['normal control 2', ['C', 110.0, 110.0, 95.0, 0.08, 0.25], None], ['normal control 3', ['P', 87.696053, 100.0, 95.0, 0.08, 1.0], None], ['normal control 4', ['P', 19.99, 100.0, 120.0, 0.0, 2.0], None]]]
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 volatility search bracket 1	`2.0`	`3.5`	Failed
regression volatility search bracket 2	`2.0`	`3.5`	Failed
partial repair probe 1	`2.0`	`5.0`	Failed
partial repair probe 2	`2.0`	`5.0`	Failed
normal control 1	`0.1`	`0.1`	Passed
normal control 2	`0.3`	`0.3`	Passed
normal control 3	`0.3`	`0.3`	Passed
normal control 4	`0.1239`	`0.1239`	Passed

SHA-256 / 7f5fb5fd24e44b96fa709aae33402ec8782d4d273d73cf9de710dabb6a95e26b

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(kind, price, S, K, r, T):
    def N(x):
        return 0.5 * (1 + math.erf(x / math.sqrt(2)))
    def bs(v):
        d1 = (math.log(S / K) + (r + 0.5 * v * v) * T) / (v * math.sqrt(T))
        d2 = d1 - v * math.sqrt(T)
        if kind == 'C':
            return S * N(d1) - K * math.exp(-r * T) * N(d2)
        return K * math.exp(-r * T) * N(-d2) - S * N(-d1)
    disc_k = K * math.exp(-r * T)
    lower = max(S - disc_k, 0) if kind == 'C' else max(disc_k - S, 0)
    upper = S if kind == 'C' else disc_k
    if price <= lower or price >= upper:
        return None
    lo, hi = 1e-6, 5.0
    for _ in range(100):
        mid = (lo + hi) / 2
        if bs(mid) > price:
            hi = mid
        else:
            lo = mid
    return round((lo + hi) / 2, 4)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression volatility search bracket 1', ['C', 91.2319, 100.0, 120.0, 0.0, 1.0], 3.5], ['regression volatility search bracket 2', ['P', 91.5993, 110.0, 100.0, 0.0, 1.0], 3.5], ['partial repair probe 1', ['P', 94.176453, 110.0, 100.0, 0.03, 2.0], 5.0], ['partial repair probe 2', ['P', 97.044553, 100.0, 100.0, 0.03, 1.0], 5.0], ['normal control 1', ['C', 1.2175, 110.0, 120.0, 0.0, 1.0], 0.1], ['normal control 2', ['P', 3.8044, 110.0, 95.0, 0.08, 1.0], 0.3], ['normal control 3', ['P', 5.59, 100.0, 100.0, 0.03, 0.25], 0.3], ['normal control 4', ['P', 2.5, 90.0, 95.0, 0.08, 2.0], 0.1239]], [['regression volatility search bracket 1', ['C', 92.3037, 100.0, 100.0, 0.08, 1.0], 3.5], ['regression volatility search bracket 2', ['C', 56.0677, 90.0, 100.0, 0.0, 2.0], 1.3], ['partial repair probe 1', ['C', 79.4077, 100.0, 95.0, 0.0, 1.0], 2.5], ['partial repair probe 2', ['C', 70.1631, 110.0, 100.0, 0.03, 0.25], 3.5], ['normal control 1', ['P', 9.2391, 100.0, 100.0, 0.08, 2.0], 0.3], ['normal control 2', ['P', 3.6659, 100.0, 95.0, 0.0, 0.25], 0.3], ['normal control 3', ['P', 92.311635, 100.0, 100.0, 0.08, 1.0], None], ['normal control 4', ['P', 92.192326, 100.0, 95.0, 0.03, 1.0], None]], [['regression volatility search bracket 1', ['C', 87.8484, 110.0, 100.0, 0.0, 1.0], 2.5], ['regression volatility search bracket 2', ['C', 56.8158, 110.0, 100.0, 0.03, 1.0], 1.3], ['partial repair probe 1', ['C', 101.9338, 110.0, 100.0, 0.08, 1.0], 3.5], ['partial repair probe 2', ['P', 116.453464, 110.0, 120.0, 0.03, 1.0], 5.0], ['normal control 1', ['C', 0.4365, 90.0, 95.0, 0.03, 0.25], 0.1], ['normal control 2', ['P', 0.0, 110.0, 95.0, 0.0, 2.0], None], ['normal control 3', ['P', 10.0, 90.0, 100.0, 0.0, 0.25], None], ['normal control 4', ['C', 8.8425, 100.0, 100.0, 0.08, 1.0], 0.1]], [['regression volatility search bracket 1', ['P', 77.5605, 100.0, 120.0, 0.03, 0.25], 3.5], ['regression volatility search bracket 2', ['C', 58.7023, 100.0, 120.0, 0.08, 0.25], 3.5], ['partial repair probe 1', ['C', 102.1427, 110.0, 95.0, 0.08, 1.0], 3.5], ['partial repair probe 2', ['P', 92.29, 100.0, 100.0, 0.0, 2.0], 2.5], ['normal control 1', ['C', 0.01, 100.0, 100.0, 0.08, 2.0], None], ['normal control 2', ['P', 36.4553, 90.0, 120.0, 0.0, 2.0], 0.3], ['normal control 3', ['C', 0.01, 90.0, 120.0, 0.08, 0.25], 0.1967], ['normal control 4', ['C', 0.01, 90.0, 120.0, 0.03, 1.0], 0.0951]], [['regression volatility search bracket 1', ['C', 55.7612, 110.0, 95.0, 0.0, 0.25], 2.5], ['regression volatility search bracket 2', ['C', 70.09, 110.0, 120.0, 0.03, 2.0], 1.3], ['partial repair probe 1', ['C', 54.2681, 110.0, 100.0, 0.0, 0.25], 2.5], ['partial repair probe 2', ['P', 88.7705, 110.0, 100.0, 0.03, 1.0], 3.5], ['normal control 1', ['C', 12.9118, 110.0, 100.0, 0.03, 2.0], None], ['normal control 2', ['C', 110.0, 110.0, 95.0, 0.08, 0.25], None], ['normal control 3', ['P', 87.696053, 100.0, 95.0, 0.08, 1.0], None], ['normal control 4', ['P', 19.99, 100.0, 120.0, 0.0, 2.0], None]]]
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 volatility search bracket 1	`3.5`	`3.5`	Passed
regression volatility search bracket 2	`3.5`	`3.5`	Passed
partial repair probe 1	`5.0`	`5.0`	Passed
partial repair probe 2	`5.0`	`5.0`	Passed
normal control 1	`0.1`	`0.1`	Passed
normal control 2	`0.3`	`0.3`	Passed
normal control 3	`0.3`	`0.3`	Passed
normal control 4	`0.1239`	`0.1239`	Passed

SHA-256 / 59dfc3e9603224462fed12aa568e07d8d391753c2d9d24153567720d98806550

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 2026-09-29T14:46:58.092734+00:00.

Case digest / fd4926193f68dd378bb541ad0aa1357c4a18b00fb7191f3f9b99662e8ca5d83f