black-scholes

Black-Scholes Option Pricing

Compute the theoretical price of a European call or put under the Black-Scholes-Merton model with optional continuous dividend yield.

The model assumes log-normal underlying dynamics, constant volatility and rates, no early exercise, no transaction costs, and continuous trading.

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Step 1: Extract Inputs

Field	Notes	Default if missing
S — spot price	Current underlying price (NOT strike)	Required — ask user if missing
K — strike	The contract strike	Required
T — time to expiry	In years (DTE / 365 for calendar days)	30/365
r — risk-free rate	Annualized, continuous compounding	0.043
q — dividend yield	Continuous; 0 for non-dividend names	0
σ — volatility	Annualized standard deviation of log-returns	0.20 if no quote
type	"call" or "put"	Required

Critical: σ is annualized. If user says "20% IV" that means σ = 0.20, not 20.

Current risk-free reference (3M T-bill):

!`python3 -c "import yfinance as yf; r = yf.Ticker('^IRX').fast_info['lastPrice']/100; print(f'r ≈ {r:.4f}')" 2>/dev/null || echo "r unavailable — default to 0.043"`

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Step 2: Compute d1, d2

d1 = [ ln(S/K) + (r − q + σ²/2)·T ] / (σ·√T)
d2 = d1 − σ·√T

Use scipy.stats.norm.cdf for N(·). If scipy is unavailable, use the Horner approximation in references/bs_formulas.md.

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Step 3: Compute Price

Call:

C = S·e^(−qT)·N(d1) − K·e^(−rT)·N(d2)

Put:

P = K·e^(−rT)·N(−d2) − S·e^(−qT)·N(−d1)

Edge cases — handle explicitly, never silently:

• T ≤ 0 → return intrinsic: max(S−K, 0) for call, max(K−S, 0) for put. Print a warning.
• σ ≤ 0 → BS undefined. Return intrinsic and warn.
• S, K ≤ 0 → reject input with a clear error.

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Step 4: Sanity Checks

Run BOTH checks and report the result before quoting the price:

1. Bounds: A call must satisfy max(0, S·e^(−qT) − K·e^(−rT)) ≤ C ≤ S·e^(−qT). Same shape for puts. If violated, the inputs are inconsistent.
2. Put-call parity: C − P = S·e^(−qT) − K·e^(−rT). Price both legs from the same σ; the residual must be ≤ 1e-6 in absolute terms.

If either fails, stop and report the failure with the residual — do not "round to fix it."

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Step 5: Respond to User

Report:

1. The theoretical price (round to 4 decimals for premium, 2 decimals if > $10).
2. Intrinsic vs. extrinsic split: intrinsic = max(S−K, 0) (call), extrinsic = price − intrinsic.
3. The exact assumption set used (S, K, T, r, q, σ).
4. A one-line comparison to the user's market quote if provided ("model 4.12 vs quoted 4.35 → 5.6% rich").

For deeper diagnostics (e.g., why model and market disagree), suggest the greeks-calculator or iv-surface skills.

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Reference Files

• references/bs_formulas.md — Full derivation, scipy-free normCDF, dividend handling, batch pricing template