FA-61551 / Options payoff and settlement / Open access

European option value with continuous dividend yield: the call leaves the spot term undiscounted for dividends · case 01

Calls on dividend payers are overpriced.

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

ROOT CAUSE

The call formula uses S*N(d1) without e^{-qT}.

VERIFIED REPAIR

Discount the spot leg by e^{-qT}.

Unsuccessful approach: Discounting the spot leg at the interest rate confuses the carry.

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 * math.sqrt(T)
    if kind == 'C':
        v = S * 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 call spot discounting 1', ['C', 110.0, 105.0, 0.0, 0.015, 0.35, 91], 9.987818], ['regression call spot discounting 2', ['C', 110.0, 100.0, 0.02, 0.03, 0.2, 30], 10.017783], ['partial repair probe 1', ['C', 100.0, 105.0, 0.05, 0.0, 0.1, 730], 8.348306], ['partial repair probe 2', ['C', 100.0, 95.0, 0.02, 0.0, 0.1, 365], 8.228981], ['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', 100.0, 105.0, 0.0, 0.0, 0.2, 7], 5.044601], ['normal control 2', ['P', 110.0, 100.0, 0.02, 0.015, 0.2, 0], 0.0]], [['regression call spot discounting 1', ['C', 110.0, 105.0, 0.0, 0.015, 0.35, 7], 5.441716], ['regression call spot discounting 2', ['C', 110.0, 95.0, 0.0, 0.015, 0.1, 182], 14.246408], ['partial repair probe 1', ['C', 90.0, 100.0, 0.05, 0.0, 0.2, 7], 5.1e-05], ['partial repair probe 2', ['C', 90.0, 95.0, 0.02, 0.0, 0.35, 182], 7.16504], ['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, 100.0, 0.0, 0.03, 0.1, 1], 10.007397], ['normal control 2', ['P', 100.0, 100.0, 0.0, 0.0, 0.2, 1], 0.41763]], [['regression call spot discounting 1', ['C', 110.0, 105.0, 0.05, 0.015, 0.1, 91], 6.250733], ['regression call spot discounting 2', ['C', 110.0, 100.0, 0.0, 0.03, 0.35, 365], 17.981981], ['partial repair probe 1', ['C', 110.0, 95.0, 0.05, 0.0, 0.2, 730], 26.930371], ['partial repair probe 2', ['C', 110.0, 95.0, 0.05, 0.0, 0.1, 91], 16.177886], ['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', 110.0, 100.0, 0.05, 0.0, 0.1, 0], 0.0], ['normal control 2', ['P', 90.0, 95.0, 0.0, 0.0, 0.2, 0], 5.0]], [['regression call spot discounting 1', ['C', 90.0, 95.0, 0.02, 0.015, 0.1, 730], 3.308995], ['regression call spot discounting 2', ['C', 110.0, 95.0, 0.05, 0.03, 0.1, 730], 18.216377], ['partial repair probe 1', ['C', 90.0, 100.0, 0.05, 0.0, 0.35, 730], 17.398625], ['partial repair probe 2', ['C', 110.0, 105.0, 0.02, 0.0, 0.2, 365], 12.482761], ['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, 0.02, 0.0, 0.2, 0], 0.0], ['normal control 2', ['P', 110.0, 105.0, 0.02, 0.015, 0.1, 1], 0.0]], [['regression call spot discounting 1', ['C', 100.0, 105.0, 0.02, 0.015, 0.1, 7], 7.8e-05], ['regression call spot discounting 2', ['C', 90.0, 100.0, 0.02, 0.015, 0.35, 7], 0.02433], ['partial repair probe 1', ['C', 100.0, 105.0, 0.02, 0.0, 0.2, 7], 0.04615], ['partial repair probe 2', ['C', 110.0, 105.0, 0.02, 0.0, 0.1, 182], 6.949255], ['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', 100.0, 100.0, 0.02, 0.015, 0.1, 0], 0.0], ['normal control 2', ['P', 110.0, 105.0, 0.05, 0.015, 0.2, 91], 1.934882]]]
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 call spot discounting 1	`10.246547`	`9.987818`	Failed
regression call spot discounting 2	`10.276`	`10.017783`	Failed
partial repair probe 1	`8.348306`	`8.348306`	Passed
partial repair probe 2	`8.228981`	`8.228981`	Passed
boundary control 1	`10.0`	`10.0`	Passed
boundary control 2	`10.0`	`10.0`	Passed
normal control 1	`5.044601`	`5.044601`	Passed
normal control 2	`0.0`	`0.0`	Passed

SHA-256 / c256e0b9426ba9ed2cba234aad3c7a225019a4f99efa6d068270a35846fad820

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 * math.sqrt(T)
    if kind == 'C':
        v = S * math.exp(-r * 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 call spot discounting 1', ['C', 110.0, 105.0, 0.0, 0.015, 0.35, 91], 9.987818], ['regression call spot discounting 2', ['C', 110.0, 100.0, 0.02, 0.03, 0.2, 30], 10.017783], ['partial repair probe 1', ['C', 100.0, 105.0, 0.05, 0.0, 0.1, 730], 8.348306], ['partial repair probe 2', ['C', 100.0, 95.0, 0.02, 0.0, 0.1, 365], 8.228981], ['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', 100.0, 105.0, 0.0, 0.0, 0.2, 7], 5.044601], ['normal control 2', ['P', 110.0, 100.0, 0.02, 0.015, 0.2, 0], 0.0]], [['regression call spot discounting 1', ['C', 110.0, 105.0, 0.0, 0.015, 0.35, 7], 5.441716], ['regression call spot discounting 2', ['C', 110.0, 95.0, 0.0, 0.015, 0.1, 182], 14.246408], ['partial repair probe 1', ['C', 90.0, 100.0, 0.05, 0.0, 0.2, 7], 5.1e-05], ['partial repair probe 2', ['C', 90.0, 95.0, 0.02, 0.0, 0.35, 182], 7.16504], ['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, 100.0, 0.0, 0.03, 0.1, 1], 10.007397], ['normal control 2', ['P', 100.0, 100.0, 0.0, 0.0, 0.2, 1], 0.41763]], [['regression call spot discounting 1', ['C', 110.0, 105.0, 0.05, 0.015, 0.1, 91], 6.250733], ['regression call spot discounting 2', ['C', 110.0, 100.0, 0.0, 0.03, 0.35, 365], 17.981981], ['partial repair probe 1', ['C', 110.0, 95.0, 0.05, 0.0, 0.2, 730], 26.930371], ['partial repair probe 2', ['C', 110.0, 95.0, 0.05, 0.0, 0.1, 91], 16.177886], ['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', 110.0, 100.0, 0.05, 0.0, 0.1, 0], 0.0], ['normal control 2', ['P', 90.0, 95.0, 0.0, 0.0, 0.2, 0], 5.0]], [['regression call spot discounting 1', ['C', 90.0, 95.0, 0.02, 0.015, 0.1, 730], 3.308995], ['regression call spot discounting 2', ['C', 110.0, 95.0, 0.05, 0.03, 0.1, 730], 18.216377], ['partial repair probe 1', ['C', 90.0, 100.0, 0.05, 0.0, 0.35, 730], 17.398625], ['partial repair probe 2', ['C', 110.0, 105.0, 0.02, 0.0, 0.2, 365], 12.482761], ['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, 0.02, 0.0, 0.2, 0], 0.0], ['normal control 2', ['P', 110.0, 105.0, 0.02, 0.015, 0.1, 1], 0.0]], [['regression call spot discounting 1', ['C', 100.0, 105.0, 0.02, 0.015, 0.1, 7], 7.8e-05], ['regression call spot discounting 2', ['C', 90.0, 100.0, 0.02, 0.015, 0.35, 7], 0.02433], ['partial repair probe 1', ['C', 100.0, 105.0, 0.02, 0.0, 0.2, 7], 0.04615], ['partial repair probe 2', ['C', 110.0, 105.0, 0.02, 0.0, 0.1, 182], 6.949255], ['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', 100.0, 100.0, 0.02, 0.015, 0.1, 0], 0.0], ['normal control 2', ['P', 110.0, 105.0, 0.05, 0.015, 0.2, 91], 1.934882]]]
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 call spot discounting 1	`10.246547`	`9.987818`	Failed
regression call spot discounting 2	`10.103785`	`10.017783`	Failed
partial repair probe 1	`1.9969`	`8.348306`	Failed
partial repair probe 2	`6.689926`	`8.228981`	Failed
boundary control 1	`10.0`	`10.0`	Passed
boundary control 2	`10.0`	`10.0`	Passed
normal control 1	`5.044601`	`5.044601`	Passed
normal control 2	`0.0`	`0.0`	Passed

SHA-256 / f7c806a5fb21e7c321d24b52fc5cbd34e476c23e4a8cf711dd5acc6462385249

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 call spot discounting 1', ['C', 110.0, 105.0, 0.0, 0.015, 0.35, 91], 9.987818], ['regression call spot discounting 2', ['C', 110.0, 100.0, 0.02, 0.03, 0.2, 30], 10.017783], ['partial repair probe 1', ['C', 100.0, 105.0, 0.05, 0.0, 0.1, 730], 8.348306], ['partial repair probe 2', ['C', 100.0, 95.0, 0.02, 0.0, 0.1, 365], 8.228981], ['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', 100.0, 105.0, 0.0, 0.0, 0.2, 7], 5.044601], ['normal control 2', ['P', 110.0, 100.0, 0.02, 0.015, 0.2, 0], 0.0]], [['regression call spot discounting 1', ['C', 110.0, 105.0, 0.0, 0.015, 0.35, 7], 5.441716], ['regression call spot discounting 2', ['C', 110.0, 95.0, 0.0, 0.015, 0.1, 182], 14.246408], ['partial repair probe 1', ['C', 90.0, 100.0, 0.05, 0.0, 0.2, 7], 5.1e-05], ['partial repair probe 2', ['C', 90.0, 95.0, 0.02, 0.0, 0.35, 182], 7.16504], ['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, 100.0, 0.0, 0.03, 0.1, 1], 10.007397], ['normal control 2', ['P', 100.0, 100.0, 0.0, 0.0, 0.2, 1], 0.41763]], [['regression call spot discounting 1', ['C', 110.0, 105.0, 0.05, 0.015, 0.1, 91], 6.250733], ['regression call spot discounting 2', ['C', 110.0, 100.0, 0.0, 0.03, 0.35, 365], 17.981981], ['partial repair probe 1', ['C', 110.0, 95.0, 0.05, 0.0, 0.2, 730], 26.930371], ['partial repair probe 2', ['C', 110.0, 95.0, 0.05, 0.0, 0.1, 91], 16.177886], ['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', 110.0, 100.0, 0.05, 0.0, 0.1, 0], 0.0], ['normal control 2', ['P', 90.0, 95.0, 0.0, 0.0, 0.2, 0], 5.0]], [['regression call spot discounting 1', ['C', 90.0, 95.0, 0.02, 0.015, 0.1, 730], 3.308995], ['regression call spot discounting 2', ['C', 110.0, 95.0, 0.05, 0.03, 0.1, 730], 18.216377], ['partial repair probe 1', ['C', 90.0, 100.0, 0.05, 0.0, 0.35, 730], 17.398625], ['partial repair probe 2', ['C', 110.0, 105.0, 0.02, 0.0, 0.2, 365], 12.482761], ['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, 0.02, 0.0, 0.2, 0], 0.0], ['normal control 2', ['P', 110.0, 105.0, 0.02, 0.015, 0.1, 1], 0.0]], [['regression call spot discounting 1', ['C', 100.0, 105.0, 0.02, 0.015, 0.1, 7], 7.8e-05], ['regression call spot discounting 2', ['C', 90.0, 100.0, 0.02, 0.015, 0.35, 7], 0.02433], ['partial repair probe 1', ['C', 100.0, 105.0, 0.02, 0.0, 0.2, 7], 0.04615], ['partial repair probe 2', ['C', 110.0, 105.0, 0.02, 0.0, 0.1, 182], 6.949255], ['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', 100.0, 100.0, 0.02, 0.015, 0.1, 0], 0.0], ['normal control 2', ['P', 110.0, 105.0, 0.05, 0.015, 0.2, 91], 1.934882]]]
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 call spot discounting 1	`9.987818`	`9.987818`	Passed
regression call spot discounting 2	`10.017783`	`10.017783`	Passed
partial repair probe 1	`8.348306`	`8.348306`	Passed
partial repair probe 2	`8.228981`	`8.228981`	Passed
boundary control 1	`10.0`	`10.0`	Passed
boundary control 2	`10.0`	`10.0`	Passed
normal control 1	`5.044601`	`5.044601`	Passed
normal control 2	`0.0`	`0.0`	Passed

SHA-256 / 69bd2d500a054859024158f498bb72a772aa3e3a60e4cc7176ef150171cbc0f2

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 2026-09-29T14:46:56.242394+00:00.

Case digest / 4bd75a0b2e682f39bded591cda5166a50fa0b9695b8fc6e2a6b5005d1fa71d25