modeling-exotic-option-payoffs

# Modeling Exotic Option Payoffs

Builds pricing models for barrier, Asian, lookback, and other path-dependent options with Monte Carlo simulation.

## When To Use

- Pricing path-dependent options (barriers, Asians, lookbacks, cliquets, autocallables) where Black-Scholes closed forms are insufficient or unavailable
- Evaluating embedded option components within structured notes or equity-linked products
- Comparing fair value across exotic payoff structures for trade desk or risk management
- Stress-testing exotic book positions under alternative volatility surface or correlation assumptions
- Building hedging models that require Greeks derived from simulation (pathwise or likelihood-ratio methods)

## Inputs To Gather

- **Option term sheet**: payoff formula, barrier levels, observation schedule (continuous vs. discrete), averaging method (arithmetic vs. geometric), settlement type (cash vs. physical)
- **Underlying specifications**: spot price, dividend schedule (discrete or continuous yield), repo/borrow cost
- **Volatility data**: implied vol surface (strike × tenor), local vol grid, or calibrated stochastic vol parameters (Heston: v₀, κ, θ, σ_v, ρ; SABR: α, β, ρ, ν) [VERIFY calibration date and source]
- **Rate curve**: risk-free discount curve, funding spread if applicable
- **Correlation matrix**: for multi-asset exotics (worst-of, basket, rainbow) [VERIFY correlation estimation window and method]
- **Simulation parameters**: number of paths, time-step granularity, random number generator seed, variance reduction technique preferences

## Workflow

1. **Classify the payoff type and select methodology**
   - Map the payoff to a category: barrier (knock-in/knock-out, single/double), Asian (fixed/floating strike, arithmetic/geometric), lookback (fixed/floating), cliquet/ratchet, autocallable, or hybrid
   - Determine if a closed-form or semi-analytic solution exists (e.g., geometric Asian via adjusted Black-Scholes, continuous barrier via reflection principle) — use as a benchmark even when Monte Carlo is the primary method
   - Select volatility model: flat vol for quick indicative pricing, local vol for barrier-sensitive products, stochastic vol (Heston/SABR) for smile-dependent payoffs [VERIFY appropriateness of vol model for the specific exotic]

2. **Configure the Monte Carlo engine**
   - Set path count (10,000 minimum for indicative; 100,000–1,000,000 for production pricing) and time steps (daily or matching observation frequency)
   - Implement variance reduction: antithetic variates (standard), control variates (use vanilla option or geometric Asian as control), importance sampling for deep out-of-the-money barriers
   - For discrete barriers, apply Broadie-Glasserman-Kou continuity correction to reduce bias from discrete monitoring approximation
   - Generate correlated Brownian increments via Cholesky decomposition for multi-asset structures

3. **Implement the payoff function**
   - Code the exact contractual payoff, matching the term sheet precisely — barrier observation dates, averaging windows, memory features for autocallables, participation rates, caps/floors
   - Handle boundary cases: what happens at exact barrier touch, rounding conventions on averaging, early termination cashflows
   - For lookbacks, track running max/min along each path; for Asians, accumulate price observations at specified fixing dates

4. **Price and compute Greeks**
   - Discount expected payoff to present value using the appropriate curve
   - Compute delta, gamma, vega via finite-difference bumps on spot, vol surface; use pathwise differentiation where payoff is continuous, likelihood-ratio method where payoff has discontinuities (barriers)
   - Calculate vanna (∂²V/∂S∂σ) and volga (∂²V/∂σ²) for vol-of-vol sensitive structures
   - Report standard error of the Monte Carlo estimate alongside the price

5. **Validate and stress-test**
   - Benchmark against closed-form where available (geometric Asian, continuous barrier) — deviation should be within Monte Carlo standard error
   - Run convergence test: verify price stabilizes as path count increases
   - Stress-test key parameters: ±5 vol points, ±20% spot move, barrier shift by 1–2%, correlation ±0.1 for multi-asset
   - Verify put-call parity or other no-arbitrage relationships hold within tolerance

## Output

- **Pricing summary table**: fair value (mid), bid/ask spread estimate, standard error, number of paths, vol model used
- **Greeks table**: delta, gamma, vega, theta, rho; higher-order Greeks (vanna, volga) where relevant
- **Payoff diagram**: terminal payoff vs. underlying at expiry (and vs. path-dependent variable where applicable)
- **Sensitivity matrix**: price under spot × vol grid (at minimum 5×5), barrier distance sensitivity, correlation sensitivity for multi-asset
- **Methodology notes**: vol model choice and calibration details, variance reduction techniques applied, continuity corrections, any simplifying assumptions
- **Convergence evidence**: price vs. path count chart or table showing standard error reduction

## Quality Checks

- Payoff function matches term sheet exactly — verify barrier direction (up/down, in/out), averaging convention, observation schedule, and settlement terms against the contract
- Monte Carlo standard error is below 1% of the option price for production runs; flag if convergence is slow
- Greeks are internally consistent: delta integrates to approximately 1 for ATM calls, gamma is non-negative for long option positions
- Barrier options show expected price behavior: knock-out price ≤ corresponding vanilla price; knock-in + knock-out = vanilla (within MC noise) [VERIFY for discrete monitoring]
- Stochastic vol parameters produce a smile that matches market-observed implied vols at relevant strikes — report calibration error
- All market data inputs carry timestamps; flag any data older than T+1 for production pricing
- For structured product components, verify that the sum of decomposed parts reconciles to the total product price