Modeling FX Derivative Pricing

Prices FX options and exotic structures using Garman-Kohlhagen, local volatility, and stochastic volatility frameworks. Covers vanilla European/American FX options, barrier options, digital options, and cross-currency swaps.

When To Use

• Pricing vanilla FX calls/puts (European or American exercise) on spot or forward
• Valuing exotic FX structures: barriers (knock-in/knock-out), digitals, range accruals, quanto options
• Marking an FX options book to market or generating independent price verification (IPV)
• Evaluating hedge costs for corporate treasury FX exposures
• Structuring cross-currency basis swaps or dual-currency notes
• Stress-testing FX derivative portfolios under volatility surface shifts

Inputs To Gather

• Currency pair and notional: e.g., EUR/USD, 10M EUR notional; confirm quoting convention (domestic/foreign)
• Spot rate: current mid-market spot from a reliable source (Bloomberg, Reuters, central bank fix)
• Interest rates: domestic and foreign risk-free rates matching the option tenor; use OIS or deposit rates as appropriate [VERIFY: confirm curve source and day-count convention per currency]
• Volatility data: at-the-money implied vol at minimum; for exotics, the full vol surface (delta or strike pillars × tenors), smile/skew parameters, or calibrated model parameters
• Option terms: strike, expiry date, cut time (NY 10am or Tokyo 3pm), settlement convention (cash or physical), premium currency
• Exotic features (if applicable): barrier levels and monitoring frequency (continuous vs. discrete), rebate amounts, averaging dates, correlation inputs for quanto/basket
• Model choice justification: Garman-Kohlhagen for vanilla European; local vol (Dupire) for barrier/digital sensitivity to skew; stochastic vol (Heston, SABR) for smile dynamics and vega hedging

Workflow

1. Select the pricing model

   • Vanilla European FX option → Garman-Kohlhagen (modified Black-Scholes with foreign rate as continuous dividend yield)
   • Barrier, digital, or path-dependent → local volatility surface calibrated via Dupire's formula; verify fit to market smile at relevant strikes
   • Products sensitive to vol-of-vol or forward smile → stochastic volatility (Heston or SABR); calibrate to market-quoted vols across strikes and tenors
   • American exercise → use finite-difference or binomial lattice with early-exercise boundary

2. Calibrate model inputs

   • Build or import the implied volatility surface; interpolate via SVI, SABR, or cubic spline
   • Verify put-call parity holds across the surface (arbitrage-free check)
   • For local vol: invert the Dupire equation numerically; check that local vol values are positive and smooth
   • For Heston: fit parameters (v₀, κ, θ, σ_v, ρ) to market vols using least-squares or differential evolution; report calibration error per strike/tenor

3. Price the derivative

   • Compute closed-form price (Garman-Kohlhagen) or run Monte Carlo / PDE solver
   • For Monte Carlo: specify number of paths (≥100k for vanillas, ≥500k for barriers), time steps, and variance reduction technique (antithetic, control variate)
   • Report price in both premium currencies (e.g., USD pips and % of notional)

4. Compute Greeks and risk sensitivities

   • Delta (spot and forward), gamma, vega (parallel and pillar), theta, rho (domestic and foreign)
   • For barriers: report pin risk near barrier, vanna, volga
   • Express Greeks in standard market conventions (e.g., delta as % of foreign notional)

5. Run scenario and sensitivity analysis

   • Spot bump: ±1%, ±5%, ±10%
   • Vol bump: ±1 vol point parallel shift; ±25-delta risk reversal shift
   • Rate bump: ±25 bps domestic and foreign independently
   • Time decay: price at T−1d, T−1w, T−1m
   • For barriers: sensitivity to barrier monitoring frequency (continuous vs. daily)

6. Document and deliver

   • State model choice and rationale
   • List all inputs with sources and value dates
   • Present pricing results in a summary table
   • Attach Greeks and scenario grids
   • Flag any inputs that were assumed or interpolated with [VERIFY]

Output

• Pricing summary table: option terms, model used, theoretical price (bid/mid/ask if spread is modeled), premium in both currencies
• Greeks table: delta, gamma, vega, theta, rho per leg or per option
• Volatility surface snapshot: implied vol at key pillars; local vol or Heston parameter set if used
• Scenario matrix: P&L impact across spot/vol/rate shifts
• Calibration diagnostics (for local/stochastic vol): calibration error by strike/tenor, parameter stability notes
• Assumptions and limitations log: model limitations (e.g., constant correlation assumption, no jumps), data staleness, interpolation choices

Quality Checks

• Verify put-call parity: C − P = e^(−r_f T) S − e^(−r_d T) K; flag deviations > 0.5 bps of notional
• Confirm Greeks sum correctly for structured packages (e.g., risk reversal delta = call delta + put delta)
• Cross-check vanilla prices against Garman-Kohlhagen closed-form even when using Monte Carlo or PDE methods
• Ensure barrier option prices converge as barrier moves far from spot to the corresponding vanilla price
• Validate that digital option prices equal the negative of the derivative of the vanilla price with respect to strike (call spread approximation)
• Check that local vol surface produces no negative variances or calendar spread arbitrage
• [VERIFY] Interest rate day-count conventions match market standard for each currency (ACT/360 for USD, ACT/365 for GBP, etc.)
• [VERIFY] Vol surface source date matches trade date; stale vol data invalidates pricing
• [VERIFY] Settlement convention (T+2 for most G10 pairs; exceptions for CAD, TRY, RUB)