pricing-vanilla-equity-options

name	pricing-vanilla-equity-options
description	Calculates option values with Black-Scholes, binomial, and Monte Carlo methods including Greeks sensitivity analysis. Use when pricing equity options, calculating Greeks, or evaluating option strategies.
tags	["valuation","derivatives-and-structured-products"]
metadata	{"author":"casemark","practice_areas":["Derivatives","Structured Products","Hedging"],"document_types":["Valuation Report"],"skill_modes":["Valuation","Calculation"]}

Pricing Vanilla Equity Options

When To Use

• Pricing European or American calls/puts on single-name equities or equity indices
• Computing Greeks (delta, gamma, vega, theta, rho) for hedging or risk reporting
• Comparing theoretical fair value against quoted market prices to identify mispricings
• Evaluating simple option strategies (spreads, straddles, covered calls) at inception
• Generating sensitivity tables for trade approval memos or client-facing valuation reports

Inputs To Gather

Input	Description	Source Notes
Spot price (S)	Current underlying equity price	Live market feed or last close
Strike price (K)	Contract strike	Term sheet or exchange listing
Time to expiry (T)	Years (or fraction) to expiration date	Contract terms; use ACT/365 unless specified
Risk-free rate (r)	Continuously compounded rate matching tenor	Treasury yield curve or OIS rate [VERIFY tenor match]
Dividend yield / schedule (q)	Continuous yield or discrete dividend dates and amounts	Bloomberg/Refinitiv; confirm ex-dates [VERIFY]
Volatility (σ)	Implied vol from market, or historical/realized vol	Volatility surface, ATM quote, or historical window
Option type	Call or put	Contract terms
Exercise style	European or American	Contract terms — drives model choice

Workflow

1. Validate and Normalize Inputs

• Confirm spot, strike, and expiry are internally consistent (no expired contracts, no negative values).
• Convert discrete dividends to a continuous yield if using Black-Scholes, or keep discrete for binomial/Monte Carlo.
• Select the volatility input: implied vol (preferred for mark-to-market) vs. historical vol (fallback or scenario analysis). Note which is used.
• Match the risk-free rate tenor to option expiry. Interpolate the yield curve if an exact tenor is unavailable. [VERIFY rate convention: continuous vs. simple]

2. Select Pricing Model

• Black-Scholes (closed-form): Default for European options with continuous dividend yield. Fast and analytically tractable. Not suitable for American puts with early exercise value or discrete dividends of material size.
• Binomial tree (CRR or Leisen-Reimer): Use for American options or when discrete dividends must be modeled explicitly. Choose ≥200 steps for convergence; confirm convergence by doubling steps and checking price stability (tolerance < 0.01).
• Monte Carlo simulation: Use when payoff is path-dependent or when extending to multi-asset baskets. For vanilla options it serves as a cross-check. Use ≥100,000 paths with antithetic variates and report standard error.

3. Calculate Option Value and Greeks

• Compute the theoretical price under the chosen model.
• Calculate first-order Greeks: Delta (Δ), Gamma (Γ), Vega (ν), Theta (Θ), Rho (ρ).
• For Black-Scholes, use closed-form Greek formulas. For tree/MC models, use finite-difference bumps:
  • Delta: bump spot ±0.5%; Gamma: second-order from same bumps; Vega: bump vol ±1 vol point; Rho: bump rate ±10 bps.
• Compute implied volatility if a market price is provided (Newton-Raphson or Brent's method on Black-Scholes).

4. Cross-Check and Validate

• Put-call parity: For Europeans, verify C − P = S·e^(−qT) − K·e^(−rT) within a tolerance of ±0.01.
• Boundary conditions: Option price ≥ intrinsic value; call ≤ S; put ≤ K·e^(−rT).
• Early exercise premium: For American options, confirm price ≥ European equivalent.
• Greeks sign checks: Delta in [0,1] for calls, [−1,0] for puts; gamma ≥ 0; vega ≥ 0; theta typically ≤ 0 for long options.
• If two models are run, compare prices and flag divergences > 1%.

5. Build Sensitivity Analysis

• Generate a spot vs. vol matrix showing option price across ±10–20% spot moves and ±5 vol point shifts.
• Produce a time decay profile showing theta bleed at weekly intervals to expiry.
• If relevant, show dividend sensitivity — price impact of ±10% change in dividend assumption.

Output

Structure the deliverable as follows:

1. Summary table — Option type, spot, strike, expiry, vol, rate, dividend assumption, model used, theoretical price.
2. Greeks table — Delta, gamma, vega (per 1 vol point), theta (per day), rho (per 100 bps).
3. Sensitivity matrix — Spot × vol grid of prices (and optionally delta).
4. Model notes — Which model was used, number of steps/paths, convergence checks, and any assumptions or approximations.
5. Flags — Mark any [VERIFY] items, data staleness warnings, or cases where early exercise premium is significant.

Quality Checks

• Put-call parity holds within tolerance for European options
• All Greeks pass sign and magnitude reasonableness checks
• Binomial tree convergence confirmed (step-doubling test)
• Volatility source clearly labeled (implied vs. historical, tenor, date)
• Risk-free rate tenor matches option expiry; interpolation method noted if used
• Dividend treatment documented (continuous yield vs. discrete schedule) [VERIFY ex-dates]
• Sensitivity tables cover a meaningful range of scenarios
• No stale market data — spot, vol, and rate quotes dated within one business day unless noted