CaseMark 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
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
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).
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
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
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