Pricing Equity Options

When To Use

• Pricing European or American equity options (calls and puts) for valuation reports, trade analysis, or structured product design
• Calculating option Greeks (delta, gamma, theta, vega, rho) for hedging or risk management
• Extracting implied volatility from market prices to assess relative value or calibrate models
• Valuing employee stock options (ESOs) or warrant grants under ASC 718 / IFRS 2
• Building or auditing option pricing models for derivatives desks, fund managers, or corporate treasury

Inputs To Gather

• Underlying price (S): Current spot price of the equity or index
• Strike price (K): Contract strike; confirm currency and adjustment for splits/dividends
• Time to expiration (T): In years; clarify calendar vs. trading days convention used
• Risk-free rate (r): Matching-tenor rate; typically Treasury yield or OIS rate [VERIFY jurisdiction and curve source]
• Volatility (σ): Historical realized vol, implied vol from market quotes, or model-calibrated vol — specify which
• Dividend yield or schedule (q): Continuous yield for indices; discrete dividend dates and amounts for single stocks
• Option style: European (exercise at expiry only) or American (early exercise permitted)
• Exercise features: Bermudan windows, knock-in/knock-out barriers, or other exotic terms if applicable

Workflow

1. Validate inputs and market data

   • Confirm spot price source and timestamp (delayed vs. real-time)
   • Verify strike conventions (percentage of spot, absolute, adjusted for corporate actions)
   • Check dividend assumptions: use ex-dates and declared amounts for near-term; analyst estimates for longer-dated [VERIFY dividend source]
   • Select risk-free rate tenor matching option expiry; document curve date and source

2. Select pricing model

   • Black-Scholes-Merton (BSM): Default for European options on non-dividend-paying or continuous-dividend equities. Closed-form; fast for Greeks computation.
   • BSM with discrete dividends: Subtract PV of expected dividends from spot price, or use the Escrowed Dividend approach for near-term known dividends.
   • Binomial tree (Cox-Ross-Rubinstein): Required for American options where early exercise may be optimal (deep ITM puts, high-dividend calls). Use ≥100 steps for convergence; 500+ for Greeks stability.
   • Trinomial tree: Better convergence properties than binomial for barrier options or when computing smooth Greeks.
   • Monte Carlo: Use for path-dependent exotics (Asian, lookback) or when payoff cannot be handled by lattice methods. Minimum 100,000 paths with variance reduction (antithetic variates, control variates).

3. Run pricing calculations

   • Compute theoretical value (TV) under the selected model
   • Calculate first- and second-order Greeks:
     • Delta (Δ): Sensitivity to underlying price; hedge ratio
     • Gamma (Γ): Rate of change of delta; convexity risk
     • Theta (Θ): Time decay per day (specify calendar or trading day convention)
     • Vega (ν): Sensitivity to 1-point change in implied vol
     • Rho (ρ): Sensitivity to interest rate shift
   • For American options, identify the early exercise boundary and report whether early exercise is optimal at current levels

4. Implied volatility extraction

   • If market price is available, back-solve for implied vol using Newton-Raphson or Brent's method on the BSM formula
   • Report bid/ask implied vol separately when spread is material
   • Construct vol smile or surface if multiple strikes/expiries are available — note skew and term structure features
   • Compare implied vol to realized vol (20-day, 60-day, 252-day) to flag rich/cheap assessment

5. Cross-validate results

   • Put-call parity check: C − P = S·e^(−qT) − K·e^(−rT) (European). Deviations beyond 1–2 cents indicate input or model error.
   • Compare model price to market mid-price; explain residual as model risk, liquidity premium, or data staleness
   • Sensitivity/scenario analysis: reprice at ±1σ underlying move, ±5 vol points, ±50 bps rate shift
   • For American options, confirm binomial price ≥ BSM price (early exercise premium must be non-negative)

Output

Structure the deliverable as a Valuation Report containing:

• Summary table: Option terms, model used, theoretical value, market price (if available), implied vol
• Greeks table: All first- and second-order Greeks with units clearly labeled
• Assumptions & inputs: Spot source/date, vol type and lookback, rate curve, dividend treatment
• Sensitivity matrix: Option value across a grid of spot prices (rows) and volatilities (columns)
• Model commentary: Why the selected model is appropriate; known limitations (e.g., BSM assumes constant vol, log-normal returns, no jumps)
• Early exercise analysis (American only): Boundary level and whether current conditions favor early exercise

Quality Checks

• Greeks satisfy finite-difference consistency: e.g., delta from bumping spot ±$0.01 matches analytical delta within tolerance
• Implied vol solver converges (residual < $0.001); flag if vol is negative or unreasonably high (>200%)
• Put-call parity holds within bid-ask spread for European options
• Binomial tree price converges as steps increase (compare 100 vs. 500 vs. 1000 steps)
• All rates, yields, and times are on consistent day-count and compounding conventions [VERIFY ACT/365 vs. ACT/360 vs. 30/360]
• Dividend assumptions match between pricing model and Greeks computation
• Report carries appropriate disclaimer: model output is not a trade recommendation; actual execution prices depend on liquidity, market conditions, and counterparty terms