FA-61546 / Options payoff and settlement / Open access

European option value with continuous dividend yield: d2 subtracts sigma*T · case 01

Values are wrong for every maturity other than one year.

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

ROOT CAUSE

d2 subtracts sigmaT instead of sigmasqrt(T).

VERIFIED REPAIR

Use d2 = d1 - sigma*sqrt(T).

Unsuccessful approach: Subtracting sigma^2*sqrt(T) mixes variance and volatility.

Case contract

Inputs kind, spot S, strike K, rate r, dividend yield q, volatility sigma and calendar days to expiry. T = days/365. At days == 0 return intrinsic value. Otherwise d1 = (ln(S/K) + (r - q + sigma^2/2)T)/(sigma sqrt T), d2 = d1 - sigma sqrt T, call = S e^{-qT} N(d1) - K e^{-rT} N(d2), put = K e^{-rT} N(-d2) - S e^{-qT} N(-d1). Round 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, q, sigma, days):
    def N(x):
        return 0.5 * (1 + math.erf(x / math.sqrt(2)))
    T = days / 365
    if days == 0:
        return round(max(S - K, 0.0) if kind == 'C' else max(K - S, 0.0), 6)
    d1 = (math.log(S / K) + (r - q + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * T
    if kind == 'C':
        v = S * math.exp(-q * T) * N(d1) - K * math.exp(-r * T) * N(d2)
    else:
        v = K * math.exp(-r * T) * N(-d2) - S * math.exp(-q * T) * N(-d1)
    return round(v, 6)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression d2 volatility term 1', ['C', 90.0, 95.0, 0.02, 0.03, 0.35, 30], 1.69956], ['regression d2 volatility term 2', ['C', 90.0, 100.0, 0.02, 0.015, 0.2, 182], 1.805544], ['partial repair probe 1', ['C', 100.0, 100.0, 0.02, 0.03, 0.35, 365], 13.067027], ['partial repair probe 2', ['C', 100.0, 100.0, 0.02, 0.0, 0.1, 365], 5.016981], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 100.0, 95.0, 0.05, 0.0, 0.1, 1], 0.0], ['normal control 2', ['P', 90.0, 100.0, 0.0, 0.0, 0.2, 0], 10.0]], [['regression d2 volatility term 1', ['P', 90.0, 95.0, 0.05, 0.0, 0.35, 91], 8.491395], ['regression d2 volatility term 2', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 30], 0.035336], ['partial repair probe 1', ['C', 100.0, 95.0, 0.05, 0.0, 0.2, 365], 13.346465], ['partial repair probe 2', ['C', 110.0, 95.0, 0.0, 0.015, 0.35, 365], 21.78191], ['boundary control 1', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 90.0, 95.0, 0.02, 0.03, 0.2, 0], 5.0], ['normal control 2', ['P', 90.0, 105.0, 0.0, 0.03, 0.35, 1], 15.007397]], [['regression d2 volatility term 1', ['C', 100.0, 105.0, 0.0, 0.03, 0.35, 7], 0.399252], ['regression d2 volatility term 2', ['P', 110.0, 100.0, 0.02, 0.03, 0.1, 91], 0.063955], ['partial repair probe 1', ['C', 110.0, 100.0, 0.05, 0.015, 0.1, 365], 13.698166], ['partial repair probe 2', ['P', 110.0, 95.0, 0.02, 0.0, 0.1, 365], 0.200779], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['C', 100.0, 100.0, 0.05, 0.015, 0.35, 0], 0.0], ['normal control 2', ['C', 90.0, 105.0, 0.0, 0.0, 0.35, 0], 0.0]], [['regression d2 volatility term 1', ['C', 110.0, 105.0, 0.02, 0.03, 0.2, 91], 7.012004], ['regression d2 volatility term 2', ['P', 90.0, 95.0, 0.05, 0.03, 0.35, 30], 6.596833], ['partial repair probe 1', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 365], 1.30802], ['partial repair probe 2', ['C', 90.0, 100.0, 0.0, 0.03, 0.1, 365], 0.378908], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 0], 0.0], ['normal control 2', ['C', 100.0, 105.0, 0.05, 0.03, 0.35, 0], 0.0]], [['regression d2 volatility term 1', ['P', 110.0, 105.0, 0.02, 0.015, 0.2, 182], 3.745196], ['regression d2 volatility term 2', ['P', 100.0, 95.0, 0.05, 0.03, 0.2, 7], 0.032572], ['partial repair probe 1', ['C', 90.0, 95.0, 0.02, 0.0, 0.2, 365], 5.838977], ['partial repair probe 2', ['P', 110.0, 95.0, 0.02, 0.03, 0.2, 365], 2.923685], ['boundary control 1', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['C', 90.0, 100.0, 0.02, 0.015, 0.35, 1], 0.0], ['normal control 2', ['C', 100.0, 105.0, 0.05, 0.03, 0.1, 1], 0.0]]]
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 d2 volatility term 1	`-0.613446`	`1.69956`	Failed
regression d2 volatility term 2	`0.594633`	`1.805544`	Failed
partial repair probe 1	`13.067027`	`13.067027`	Passed
partial repair probe 2	`5.016981`	`5.016981`	Passed
boundary control 1	`10.0`	`10.0`	Passed
boundary control 2	`10.0`	`10.0`	Passed
normal control 1	`0.0`	`0.0`	Passed
normal control 2	`10.0`	`10.0`	Passed

SHA-256 / ac056318a97d31c94b2b1425c4d8ef2ded928b829e3dfab254cefb970a48de56

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, q, sigma, days):
    def N(x):
        return 0.5 * (1 + math.erf(x / math.sqrt(2)))
    T = days / 365
    if days == 0:
        return round(max(S - K, 0.0) if kind == 'C' else max(K - S, 0.0), 6)
    d1 = (math.log(S / K) + (r - q + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma ** 2 * math.sqrt(T)
    if kind == 'C':
        v = S * math.exp(-q * T) * N(d1) - K * math.exp(-r * T) * N(d2)
    else:
        v = K * math.exp(-r * T) * N(-d2) - S * math.exp(-q * T) * N(-d1)
    return round(v, 6)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression d2 volatility term 1', ['C', 90.0, 95.0, 0.02, 0.03, 0.35, 30], 1.69956], ['regression d2 volatility term 2', ['C', 90.0, 100.0, 0.02, 0.015, 0.2, 182], 1.805544], ['partial repair probe 1', ['C', 100.0, 100.0, 0.02, 0.03, 0.35, 365], 13.067027], ['partial repair probe 2', ['C', 100.0, 100.0, 0.02, 0.0, 0.1, 365], 5.016981], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 100.0, 95.0, 0.05, 0.0, 0.1, 1], 0.0], ['normal control 2', ['P', 90.0, 100.0, 0.0, 0.0, 0.2, 0], 10.0]], [['regression d2 volatility term 1', ['P', 90.0, 95.0, 0.05, 0.0, 0.35, 91], 8.491395], ['regression d2 volatility term 2', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 30], 0.035336], ['partial repair probe 1', ['C', 100.0, 95.0, 0.05, 0.0, 0.2, 365], 13.346465], ['partial repair probe 2', ['C', 110.0, 95.0, 0.0, 0.015, 0.35, 365], 21.78191], ['boundary control 1', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 90.0, 95.0, 0.02, 0.03, 0.2, 0], 5.0], ['normal control 2', ['P', 90.0, 105.0, 0.0, 0.03, 0.35, 1], 15.007397]], [['regression d2 volatility term 1', ['C', 100.0, 105.0, 0.0, 0.03, 0.35, 7], 0.399252], ['regression d2 volatility term 2', ['P', 110.0, 100.0, 0.02, 0.03, 0.1, 91], 0.063955], ['partial repair probe 1', ['C', 110.0, 100.0, 0.05, 0.015, 0.1, 365], 13.698166], ['partial repair probe 2', ['P', 110.0, 95.0, 0.02, 0.0, 0.1, 365], 0.200779], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['C', 100.0, 100.0, 0.05, 0.015, 0.35, 0], 0.0], ['normal control 2', ['C', 90.0, 105.0, 0.0, 0.0, 0.35, 0], 0.0]], [['regression d2 volatility term 1', ['C', 110.0, 105.0, 0.02, 0.03, 0.2, 91], 7.012004], ['regression d2 volatility term 2', ['P', 90.0, 95.0, 0.05, 0.03, 0.35, 30], 6.596833], ['partial repair probe 1', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 365], 1.30802], ['partial repair probe 2', ['C', 90.0, 100.0, 0.0, 0.03, 0.1, 365], 0.378908], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 0], 0.0], ['normal control 2', ['C', 100.0, 105.0, 0.05, 0.03, 0.35, 0], 0.0]], [['regression d2 volatility term 1', ['P', 110.0, 105.0, 0.02, 0.015, 0.2, 182], 3.745196], ['regression d2 volatility term 2', ['P', 100.0, 95.0, 0.05, 0.03, 0.2, 7], 0.032572], ['partial repair probe 1', ['C', 90.0, 95.0, 0.02, 0.0, 0.2, 365], 5.838977], ['partial repair probe 2', ['P', 110.0, 95.0, 0.02, 0.03, 0.2, 365], 2.923685], ['boundary control 1', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['C', 90.0, 100.0, 0.02, 0.015, 0.35, 1], 0.0], ['normal control 2', ['C', 100.0, 105.0, 0.05, 0.03, 0.1, 1], 0.0]]]
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 d2 volatility term 1	`-0.404446`	`1.69956`	Failed
regression d2 volatility term 2	`-1.581095`	`1.805544`	Failed
partial repair probe 1	`4.225537`	`13.067027`	Failed
partial repair probe 2	`1.564997`	`5.016981`	Failed
boundary control 1	`10.0`	`10.0`	Passed
boundary control 2	`10.0`	`10.0`	Passed
normal control 1	`0.0`	`0.0`	Passed
normal control 2	`10.0`	`10.0`	Passed

SHA-256 / 487be230165a12e20e5177df8db31e6565cfb00c03f80ff26087e730e2eb3730

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, q, sigma, days):
    def N(x):
        return 0.5 * (1 + math.erf(x / math.sqrt(2)))
    T = days / 365
    if days == 0:
        return round(max(S - K, 0.0) if kind == 'C' else max(K - S, 0.0), 6)
    d1 = (math.log(S / K) + (r - q + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * math.sqrt(T)
    if kind == 'C':
        v = S * math.exp(-q * T) * N(d1) - K * math.exp(-r * T) * N(d2)
    else:
        v = K * math.exp(-r * T) * N(-d2) - S * math.exp(-q * T) * N(-d1)
    return round(v, 6)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression d2 volatility term 1', ['C', 90.0, 95.0, 0.02, 0.03, 0.35, 30], 1.69956], ['regression d2 volatility term 2', ['C', 90.0, 100.0, 0.02, 0.015, 0.2, 182], 1.805544], ['partial repair probe 1', ['C', 100.0, 100.0, 0.02, 0.03, 0.35, 365], 13.067027], ['partial repair probe 2', ['C', 100.0, 100.0, 0.02, 0.0, 0.1, 365], 5.016981], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 100.0, 95.0, 0.05, 0.0, 0.1, 1], 0.0], ['normal control 2', ['P', 90.0, 100.0, 0.0, 0.0, 0.2, 0], 10.0]], [['regression d2 volatility term 1', ['P', 90.0, 95.0, 0.05, 0.0, 0.35, 91], 8.491395], ['regression d2 volatility term 2', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 30], 0.035336], ['partial repair probe 1', ['C', 100.0, 95.0, 0.05, 0.0, 0.2, 365], 13.346465], ['partial repair probe 2', ['C', 110.0, 95.0, 0.0, 0.015, 0.35, 365], 21.78191], ['boundary control 1', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 90.0, 95.0, 0.02, 0.03, 0.2, 0], 5.0], ['normal control 2', ['P', 90.0, 105.0, 0.0, 0.03, 0.35, 1], 15.007397]], [['regression d2 volatility term 1', ['C', 100.0, 105.0, 0.0, 0.03, 0.35, 7], 0.399252], ['regression d2 volatility term 2', ['P', 110.0, 100.0, 0.02, 0.03, 0.1, 91], 0.063955], ['partial repair probe 1', ['C', 110.0, 100.0, 0.05, 0.015, 0.1, 365], 13.698166], ['partial repair probe 2', ['P', 110.0, 95.0, 0.02, 0.0, 0.1, 365], 0.200779], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['C', 100.0, 100.0, 0.05, 0.015, 0.35, 0], 0.0], ['normal control 2', ['C', 90.0, 105.0, 0.0, 0.0, 0.35, 0], 0.0]], [['regression d2 volatility term 1', ['C', 110.0, 105.0, 0.02, 0.03, 0.2, 91], 7.012004], ['regression d2 volatility term 2', ['P', 90.0, 95.0, 0.05, 0.03, 0.35, 30], 6.596833], ['partial repair probe 1', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 365], 1.30802], ['partial repair probe 2', ['C', 90.0, 100.0, 0.0, 0.03, 0.1, 365], 0.378908], ['boundary control 1', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['P', 100.0, 95.0, 0.05, 0.03, 0.1, 0], 0.0], ['normal control 2', ['C', 100.0, 105.0, 0.05, 0.03, 0.35, 0], 0.0]], [['regression d2 volatility term 1', ['P', 110.0, 105.0, 0.02, 0.015, 0.2, 182], 3.745196], ['regression d2 volatility term 2', ['P', 100.0, 95.0, 0.05, 0.03, 0.2, 7], 0.032572], ['partial repair probe 1', ['C', 90.0, 95.0, 0.02, 0.0, 0.2, 365], 5.838977], ['partial repair probe 2', ['P', 110.0, 95.0, 0.02, 0.03, 0.2, 365], 2.923685], ['boundary control 1', ['P', 90.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['boundary control 2', ['C', 110.0, 100.0, 0.05, 0.0, 0.2, 0], 10.0], ['normal control 1', ['C', 90.0, 100.0, 0.02, 0.015, 0.35, 1], 0.0], ['normal control 2', ['C', 100.0, 105.0, 0.05, 0.03, 0.1, 1], 0.0]]]
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 d2 volatility term 1	`1.69956`	`1.69956`	Passed
regression d2 volatility term 2	`1.805544`	`1.805544`	Passed
partial repair probe 1	`13.067027`	`13.067027`	Passed
partial repair probe 2	`5.016981`	`5.016981`	Passed
boundary control 1	`10.0`	`10.0`	Passed
boundary control 2	`10.0`	`10.0`	Passed
normal control 1	`0.0`	`0.0`	Passed
normal control 2	`10.0`	`10.0`	Passed

SHA-256 / 114e7c1c57a42438643791e0e6dbd0f5552c22b33bdf5af4561cf2e83502bcc9

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 2026-09-29T14:46:56.174003+00:00.

Case digest / 2aa84aecad2b144bf52e15b9ab31d79c721878ff013aff54c0211d8848bbb570