casemark/pricing-vanilla-equity-options

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