Part 1 — Pricing · BS · SPX call capstone

S_{0}

= 7,414 SPX index

K

= 7,900 strike

T

= 0.523 yr time to expiry

r

= 0.036 13-wk T-bill

C_{mkt}

= $180.30 market mid

§1.1 #

Estimating σ from history

Eight-year window, daily log-returns, annualized.

lookback_days = 2018 (≈ 8 calendar years). Long enough for statistical stability (n ≈ 2,017 daily returns) and to span the calm of 2019, the COVID crash, the 2022 bear market, and the 2023–2026 rally. Shorter windows are dominated by the current regime; much longer windows contaminate the estimate with structural conditions no longer in force.

r_{t} = ln (S_{t} / S_{t - 1}), σ = s_{daily} \cdot 252 .

	Value
Calendar days requested	2,018
Daily returns computed	2,017
Daily std. dev. of log-returns	0.012257
Annualized volatility σ	0.1946 (19.46%)

Annualized volatility

σ = 0.1946

§1.2 #

Taylor series for Φ

Build the series for eˣ, substitute, integrate term by term, then evaluate at d₁ and d₂.

(i)

Taylor series for eˣ at x = 0

e^{x} = 1 + x + \frac{x ^{2}}{2 !} + \frac{x ^{3}}{3 !} + \frac{x ^{4}}{4 !} + \dots = n = 0 \sum \infty \frac{x ^{n}}{n !} .

(ii)

Substitute x = −t²/2

e^{- t^{2} /2} = n = 0 \sum \infty \frac{( - 1 ) ^{n} t ^{2 n}}{2 ^{n} n !} = 1 - \frac{t ^{2}}{2} + \frac{t ^{4}}{8} - \frac{t ^{6}}{48} + \frac{t ^{8}}{384} - \dots

(iii)

Integrate term-by-term to get Φ(z)

Φ (z) = \frac{1}{2} + \frac{1}{2 π} \int_{0}^{z} e^{- t^{2} /2} d t = \frac{1}{2} + \frac{1}{2 π} n = 0 \sum \infty \frac{( - 1 ) ^{n} z ^{2 n + 1}}{2 ^{n} n ! ( 2 n + 1 )} .

Φ (z) = \frac{1}{2} + \frac{1}{2 π} [z - \frac{z ^{3}}{6} + \frac{z ^{5}}{40} - \frac{z ^{7}}{336} + \frac{z ^{9}}{3456} - \dots] .

(iv)

Compute d₁, d₂, and evaluate Φ

σ^{2} σ^{2} /2 r + σ^{2} /2 (r + σ^{2} /2) T S_{0} / K ln (S_{0} / K) T σ T = (0.1946)^{2} = 0.0379, = 0.0190, = 0.0550, = 0.0550 \times 0.523 = 0.0288, = 7414/7900 = 0.9385, = - 0.0635, = 0.7232, = 0.1407.

d_{1} = \frac{ln ( S _{0} / K ) + ( r + σ ^{2} /2 ) T}{σ T} = \frac{- 0.0635 + 0.0288}{0.1407} = - 0.2466.

d_{2} = d_{1} - σ T = - 0.2466 - 0.1407 = - 0.3873.

T_{0} T_{1} T_{2} = - 0.2466, ∣ T_{0} ∣ = 0.2466 = - z^{3} /6 = + 0.0025, ∣ T_{1} ∣ = 0.0025 = + z^{5} /40 = - 0.0000, ∣ T_{2} ∣ = 0.000023 < 0.0001 \Rightarrow stop (2 terms)

Φ (d_{1}) = \frac{1}{2} + 0.3989 \times (- 0.2441) = 0.4026.

T_{0} T_{1} T_{2} T_{3} = - 0.3873, ∣ T_{0} ∣ = 0.3873 = + 0.0097, ∣ T_{1} ∣ = 0.0097 = - 0.0002, ∣ T_{2} ∣ = 0.0002 = + 0.0000, ∣ T_{3} ∣ = 3.9 \times 1 0^{- 6} < 0.0001 \Rightarrow stop (3 terms)

Φ (d_{2}) = \frac{1}{2} + 0.3989 \times (- 0.3778) = 0.3493.

Φ(d₁)

0.4026

Φ(d₂)

0.3493

§1.3 #

Hand-computed call price

C = S_{0} Φ (d_{1}) - K e^{- r T} Φ (d_{2}) .

Discount factor

e^{-rT} via Taylor series

r T x e^{- 0.0188} = 0.036 \times 0.523 = 0.0188, = 0.0188, x^{2} = 0.0004, x^{2} /2 = 0.0002, \approx 1 - 0.0188 + 0.0002 - 0.0000 = 0.9814.

Term 1

S₀ · Φ(d₁)

S_{0} Φ (d_{1}) = 7414 \times 0.4026 = 2984.8764.

Term 2

K · e^{-rT} · Φ(d₂)

K e^{- r T} K e^{- r T} Φ (d_{2}) = 7900 \times 0.9814 = 7753.0600, = 7753.0600 \times 0.3493 = 2708.1439.

C = 2984.8764 - 2708.1439 = 276.7325.

Hand-computed call price

C = $276.73

§1.4 #

Python verification

scipy.stats.norm.cdf for Φ, full precision.

import numpy as np
from scipy.stats import norm

def black_scholes_call(S0, K, r, sigma, T):
    d1 = (np.log(S0/K) + (r + sigma**2/2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)
    return S0 * norm.cdf(d1) - K * np.exp(-r*T) * norm.cdf(d2)

# Sanity check (expect ~10.45)
print(black_scholes_call(100, 100, 0.05, 0.20, 1))   # -> 10.4506

# Project parameters
S0, K, r, sigma, T = 7414, 7900, 0.036, 0.1946, 0.523
print(black_scholes_call(S0, K, r, sigma, T))         # -> 277.29

	Taylor (hand)	norm.cdf (Python)	Δ
Φ(d₁)	0.4026	0.4025	0.0001
Φ(d₂)	0.3493	0.3491	0.0002

	Hand	Python
C	$276.7325	$277.2896
difference		$0.5571

§1.5 #

Market gap — model vs $180.30

	Value
Model C (hand)	$276.7325
Market C	$180.30
Gap (model − market)	+$96.4325

The market price sits $96.43 below the model — a discount of roughly 35%. Three factors plausibly account for the gap. First, $σ = 19.5%$ was estimated from eight years of realized daily returns, which include the COVID crash and the 2022 bear market; the option market is forward-looking and is currently pricing closer to 13–14% implied vol for OTM SPX calls at this maturity. Second, the formula assumes the index pays no dividends, but SPX carries a ~1.3% effective dividend yield from its constituents; including dividends shaves several dollars off the model price. Third, the volatility skew/smile: out-of-the-money calls (the $7,900 strike is $486 above spot) trade at lower implied vols than at-the-money options, because hedging demand concentrates in puts and ATM calls.