FA-61736 / Options payoff and settlement / Open access

Implied volatility by bisection with arbitrage bounds: the lower bound uses the undiscounted intrinsic value · case 01

Prices between discounted and undiscounted intrinsic are rejected or solved inconsistently.

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

ROOT CAUSE

The lower bound uses max(S-K,0) or max(K-S,0) instead of the discounted strike.

VERIFIED REPAIR

Discount the strike in the lower bound.

Unsuccessful approach: Applying the call bound formula to puts leaves put bounds wrong.

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 - K, 0) if kind == 'C' else max(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 lower arbitrage bound 1', ['P', 6.9002, 100.0, 120.0, 0.08, 2.0], 0.1], ['regression lower arbitrage bound 2', ['C', 22.0232, 110.0, 95.0, 0.08, 2.0], None], ['partial repair probe 1', ['P', 0.01, 100.0, 95.0, 0.03, 2.0], 0.0321], ['partial repair probe 2', ['P', 0.0713, 110.0, 95.0, 0.08, 2.0], 0.1], ['normal control 1', ['C', 32.8627, 100.0, 100.0, 0.0, 2.0], 0.6], ['normal control 2', ['C', 0.0, 100.0, 100.0, 0.0, 0.25], None], ['normal control 3', ['C', 100.0, 100.0, 100.0, 0.03, 2.0], None], ['normal control 4', ['C', 47.0024, 100.0, 100.0, 0.03, 0.25], 2.5]], [['regression lower arbitrage bound 1', ['P', 6.1558, 90.0, 100.0, 0.08, 1.0], 0.1352], ['regression lower arbitrage bound 2', ['C', 12.2939, 100.0, 95.0, 0.08, 1.0], None], ['partial repair probe 1', ['P', 1.1273, 100.0, 95.0, 0.03, 1.0], 0.1], ['partial repair probe 2', ['P', 2.2473, 100.0, 120.0, 0.08, 2.0], None], ['normal control 1', ['C', 19.4236, 100.0, 95.0, 0.08, 2.0], 0.1], ['normal control 2', ['C', 32.8627, 100.0, 100.0, 0.0, 2.0], 0.6], ['normal control 3', ['P', 12.0217, 90.0, 100.0, 0.0, 0.25], 0.3], ['normal control 4', ['C', 43.9475, 110.0, 95.0, 0.03, 2.0], 0.6]], [['regression lower arbitrage bound 1', ['C', 22.0232, 110.0, 95.0, 0.08, 2.0], None], ['regression lower arbitrage bound 2', ['C', 10.9901, 110.0, 100.0, 0.08, 0.25], None], ['partial repair probe 1', ['P', 2.2473, 100.0, 120.0, 0.08, 2.0], None], ['partial repair probe 2', ['P', 5.01, 90.0, 95.0, 0.08, 2.0], 0.1832], ['normal control 1', ['C', 91.9882, 100.0, 100.0, 0.0, 1.0], 3.5], ['normal control 2', ['C', 68.1279, 90.0, 120.0, 0.0, 1.0], 2.5], ['normal control 3', ['P', 104.819, 100.0, 120.0, 0.03, 2.0], 2.5], ['normal control 4', ['P', 120.0, 100.0, 120.0, 0.0, 2.0], None]], [['regression lower arbitrage bound 1', ['C', 0.01, 100.0, 100.0, 0.08, 2.0], None], ['regression lower arbitrage bound 2', ['C', 9.0363, 90.0, 95.0, 0.08, 2.0], None], ['partial repair probe 1', ['P', 0.01, 110.0, 100.0, 0.08, 0.25], 0.0927], ['partial repair probe 2', ['P', 0.2861, 100.0, 95.0, 0.03, 0.25], 0.1], ['normal control 1', ['C', 89.2978, 110.0, 95.0, 0.08, 1.0], 2.5], ['normal control 2', ['C', 22.3436, 110.0, 95.0, 0.08, 1.0], 0.1], ['normal control 3', ['C', 28.1884, 100.0, 95.0, 0.08, 0.25], 1.3], ['normal control 4', ['C', 100.0, 100.0, 95.0, 0.08, 1.0], None]], [['regression lower arbitrage bound 1', ['P', 19.1037, 100.0, 120.0, 0.03, 0.25], 0.1002], ['regression lower arbitrage bound 2', ['P', 2.8891, 110.0, 120.0, 0.08, 2.0], 0.1], ['partial repair probe 1', ['P', 7.4876, 100.0, 95.0, 0.08, 2.0], 0.3], ['partial repair probe 2', ['P', 9.99, 90.0, 100.0, 0.0, 1.0], None], ['normal control 1', ['P', 92.5421, 110.0, 120.0, 0.03, 1.0], 2.5], ['normal control 2', ['P', 100.0, 90.0, 100.0, 0.0, 2.0], None], ['normal control 3', ['P', 0.0, 100.0, 95.0, 0.03, 2.0], None], ['normal control 4', ['C', 0, 90.0, 100.0, 0.03, 0.25], 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 lower arbitrage bound 1	`None`	`0.1`	Failed
regression lower arbitrage bound 2	`0.0`	`None`	Failed
partial repair probe 1	`0.0321`	`0.0321`	Passed
partial repair probe 2	`0.1`	`0.1`	Passed
normal control 1	`0.6`	`0.6`	Passed
normal control 2	`None`	`None`	Passed
normal control 3	`None`	`None`	Passed
normal control 4	`2.5`	`2.5`	Passed

SHA-256 / 3d9e2d424dec85492539075f11f37422b058e03e095ab9b08d0c27940a86c606

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)
    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 lower arbitrage bound 1', ['P', 6.9002, 100.0, 120.0, 0.08, 2.0], 0.1], ['regression lower arbitrage bound 2', ['C', 22.0232, 110.0, 95.0, 0.08, 2.0], None], ['partial repair probe 1', ['P', 0.01, 100.0, 95.0, 0.03, 2.0], 0.0321], ['partial repair probe 2', ['P', 0.0713, 110.0, 95.0, 0.08, 2.0], 0.1], ['normal control 1', ['C', 32.8627, 100.0, 100.0, 0.0, 2.0], 0.6], ['normal control 2', ['C', 0.0, 100.0, 100.0, 0.0, 0.25], None], ['normal control 3', ['C', 100.0, 100.0, 100.0, 0.03, 2.0], None], ['normal control 4', ['C', 47.0024, 100.0, 100.0, 0.03, 0.25], 2.5]], [['regression lower arbitrage bound 1', ['P', 6.1558, 90.0, 100.0, 0.08, 1.0], 0.1352], ['regression lower arbitrage bound 2', ['C', 12.2939, 100.0, 95.0, 0.08, 1.0], None], ['partial repair probe 1', ['P', 1.1273, 100.0, 95.0, 0.03, 1.0], 0.1], ['partial repair probe 2', ['P', 2.2473, 100.0, 120.0, 0.08, 2.0], None], ['normal control 1', ['C', 19.4236, 100.0, 95.0, 0.08, 2.0], 0.1], ['normal control 2', ['C', 32.8627, 100.0, 100.0, 0.0, 2.0], 0.6], ['normal control 3', ['P', 12.0217, 90.0, 100.0, 0.0, 0.25], 0.3], ['normal control 4', ['C', 43.9475, 110.0, 95.0, 0.03, 2.0], 0.6]], [['regression lower arbitrage bound 1', ['C', 22.0232, 110.0, 95.0, 0.08, 2.0], None], ['regression lower arbitrage bound 2', ['C', 10.9901, 110.0, 100.0, 0.08, 0.25], None], ['partial repair probe 1', ['P', 2.2473, 100.0, 120.0, 0.08, 2.0], None], ['partial repair probe 2', ['P', 5.01, 90.0, 95.0, 0.08, 2.0], 0.1832], ['normal control 1', ['C', 91.9882, 100.0, 100.0, 0.0, 1.0], 3.5], ['normal control 2', ['C', 68.1279, 90.0, 120.0, 0.0, 1.0], 2.5], ['normal control 3', ['P', 104.819, 100.0, 120.0, 0.03, 2.0], 2.5], ['normal control 4', ['P', 120.0, 100.0, 120.0, 0.0, 2.0], None]], [['regression lower arbitrage bound 1', ['C', 0.01, 100.0, 100.0, 0.08, 2.0], None], ['regression lower arbitrage bound 2', ['C', 9.0363, 90.0, 95.0, 0.08, 2.0], None], ['partial repair probe 1', ['P', 0.01, 110.0, 100.0, 0.08, 0.25], 0.0927], ['partial repair probe 2', ['P', 0.2861, 100.0, 95.0, 0.03, 0.25], 0.1], ['normal control 1', ['C', 89.2978, 110.0, 95.0, 0.08, 1.0], 2.5], ['normal control 2', ['C', 22.3436, 110.0, 95.0, 0.08, 1.0], 0.1], ['normal control 3', ['C', 28.1884, 100.0, 95.0, 0.08, 0.25], 1.3], ['normal control 4', ['C', 100.0, 100.0, 95.0, 0.08, 1.0], None]], [['regression lower arbitrage bound 1', ['P', 19.1037, 100.0, 120.0, 0.03, 0.25], 0.1002], ['regression lower arbitrage bound 2', ['P', 2.8891, 110.0, 120.0, 0.08, 2.0], 0.1], ['partial repair probe 1', ['P', 7.4876, 100.0, 95.0, 0.08, 2.0], 0.3], ['partial repair probe 2', ['P', 9.99, 90.0, 100.0, 0.0, 1.0], None], ['normal control 1', ['P', 92.5421, 110.0, 120.0, 0.03, 1.0], 2.5], ['normal control 2', ['P', 100.0, 90.0, 100.0, 0.0, 2.0], None], ['normal control 3', ['P', 0.0, 100.0, 95.0, 0.03, 2.0], None], ['normal control 4', ['C', 0, 90.0, 100.0, 0.03, 0.25], 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 lower arbitrage bound 1	`0.1`	`0.1`	Passed
regression lower arbitrage bound 2	`None`	`None`	Passed
partial repair probe 1	`None`	`0.0321`	Failed
partial repair probe 2	`None`	`0.1`	Failed
normal control 1	`0.6`	`0.6`	Passed
normal control 2	`None`	`None`	Passed
normal control 3	`None`	`None`	Passed
normal control 4	`2.5`	`2.5`	Passed

SHA-256 / 19f9537c3523452fbf74451497be916656b2385faf1d7c0e8e9b89767a305d45

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 lower arbitrage bound 1', ['P', 6.9002, 100.0, 120.0, 0.08, 2.0], 0.1], ['regression lower arbitrage bound 2', ['C', 22.0232, 110.0, 95.0, 0.08, 2.0], None], ['partial repair probe 1', ['P', 0.01, 100.0, 95.0, 0.03, 2.0], 0.0321], ['partial repair probe 2', ['P', 0.0713, 110.0, 95.0, 0.08, 2.0], 0.1], ['normal control 1', ['C', 32.8627, 100.0, 100.0, 0.0, 2.0], 0.6], ['normal control 2', ['C', 0.0, 100.0, 100.0, 0.0, 0.25], None], ['normal control 3', ['C', 100.0, 100.0, 100.0, 0.03, 2.0], None], ['normal control 4', ['C', 47.0024, 100.0, 100.0, 0.03, 0.25], 2.5]], [['regression lower arbitrage bound 1', ['P', 6.1558, 90.0, 100.0, 0.08, 1.0], 0.1352], ['regression lower arbitrage bound 2', ['C', 12.2939, 100.0, 95.0, 0.08, 1.0], None], ['partial repair probe 1', ['P', 1.1273, 100.0, 95.0, 0.03, 1.0], 0.1], ['partial repair probe 2', ['P', 2.2473, 100.0, 120.0, 0.08, 2.0], None], ['normal control 1', ['C', 19.4236, 100.0, 95.0, 0.08, 2.0], 0.1], ['normal control 2', ['C', 32.8627, 100.0, 100.0, 0.0, 2.0], 0.6], ['normal control 3', ['P', 12.0217, 90.0, 100.0, 0.0, 0.25], 0.3], ['normal control 4', ['C', 43.9475, 110.0, 95.0, 0.03, 2.0], 0.6]], [['regression lower arbitrage bound 1', ['C', 22.0232, 110.0, 95.0, 0.08, 2.0], None], ['regression lower arbitrage bound 2', ['C', 10.9901, 110.0, 100.0, 0.08, 0.25], None], ['partial repair probe 1', ['P', 2.2473, 100.0, 120.0, 0.08, 2.0], None], ['partial repair probe 2', ['P', 5.01, 90.0, 95.0, 0.08, 2.0], 0.1832], ['normal control 1', ['C', 91.9882, 100.0, 100.0, 0.0, 1.0], 3.5], ['normal control 2', ['C', 68.1279, 90.0, 120.0, 0.0, 1.0], 2.5], ['normal control 3', ['P', 104.819, 100.0, 120.0, 0.03, 2.0], 2.5], ['normal control 4', ['P', 120.0, 100.0, 120.0, 0.0, 2.0], None]], [['regression lower arbitrage bound 1', ['C', 0.01, 100.0, 100.0, 0.08, 2.0], None], ['regression lower arbitrage bound 2', ['C', 9.0363, 90.0, 95.0, 0.08, 2.0], None], ['partial repair probe 1', ['P', 0.01, 110.0, 100.0, 0.08, 0.25], 0.0927], ['partial repair probe 2', ['P', 0.2861, 100.0, 95.0, 0.03, 0.25], 0.1], ['normal control 1', ['C', 89.2978, 110.0, 95.0, 0.08, 1.0], 2.5], ['normal control 2', ['C', 22.3436, 110.0, 95.0, 0.08, 1.0], 0.1], ['normal control 3', ['C', 28.1884, 100.0, 95.0, 0.08, 0.25], 1.3], ['normal control 4', ['C', 100.0, 100.0, 95.0, 0.08, 1.0], None]], [['regression lower arbitrage bound 1', ['P', 19.1037, 100.0, 120.0, 0.03, 0.25], 0.1002], ['regression lower arbitrage bound 2', ['P', 2.8891, 110.0, 120.0, 0.08, 2.0], 0.1], ['partial repair probe 1', ['P', 7.4876, 100.0, 95.0, 0.08, 2.0], 0.3], ['partial repair probe 2', ['P', 9.99, 90.0, 100.0, 0.0, 1.0], None], ['normal control 1', ['P', 92.5421, 110.0, 120.0, 0.03, 1.0], 2.5], ['normal control 2', ['P', 100.0, 90.0, 100.0, 0.0, 2.0], None], ['normal control 3', ['P', 0.0, 100.0, 95.0, 0.03, 2.0], None], ['normal control 4', ['C', 0, 90.0, 100.0, 0.03, 0.25], 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 lower arbitrage bound 1	`0.1`	`0.1`	Passed
regression lower arbitrage bound 2	`None`	`None`	Passed
partial repair probe 1	`0.0321`	`0.0321`	Passed
partial repair probe 2	`0.1`	`0.1`	Passed
normal control 1	`0.6`	`0.6`	Passed
normal control 2	`None`	`None`	Passed
normal control 3	`None`	`None`	Passed
normal control 4	`2.5`	`2.5`	Passed

SHA-256 / 03b929b79d1c31085957d072a23a9410b54f47c940c7b2cbaa6c539acdb4f9a0

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 2026-09-29T14:46:58.062938+00:00.

Case digest / c2a027235805a8b161446cd18937bfa927bca9dd735822b199789187afe03b47