Estimating Greeks Numerically

When closed-form Greeks aren't available—complex payoffs, stochastic volatility models, or path-dependent options—numerical methods compute sensitivities. Each approach balances accuracy, noise, and computational cost differently.

Overview of Methods

Bump and Revalue

Principle: Compute the derivative by finite difference.

Central difference formula: Delta ≈ [V(S + ΔS) - V(S - ΔS)] / (2ΔS)

Gamma ≈ [V(S + ΔS) - 2V(S) + V(S - ΔS)] / (ΔS)²

Vega ≈ [V(σ + Δσ) - V(σ - Δσ)] / (2Δσ)

Step size selection:

Truncation vs. noise trade-off:

• Too large ΔS: Truncation error (derivative approximation is poor)
• Too small ΔS: Noise dominates (Monte Carlo variance amplified)
• Optimal: Balance both errors

For Monte Carlo with 100,000 paths, typical bump sizes work well. With fewer paths, increase bump sizes.

Pathwise (Infinitesimal Perturbation Analysis)

Principle: Differentiate through the path generation, not the payoff.

For European call: dPayoff/dS₀ = ∂/∂S₀ [max(S_T - K, 0)] = 1{S_T > K} × dS_T/dS₀ = 1{S_T > K} × (S_T/S₀)

The indicator function is not differentiated (causes problems at discontinuities).

Advantages:

• Uses same paths as pricing
• No additional variance from bumping
• Unbiased for smooth payoffs

Limitations:

• Fails for discontinuous payoffs (barriers, digitals)
• Must derive differentiated payoff analytically

Example calculation: For each path, multiply payoff derivative by path sensitivity: Delta_path = 1{S_T > K} × (S_T/S₀) Delta = mean(Delta_path) × discount_factor

Likelihood Ratio Method

Principle: Differentiate the probability density rather than the payoff.

For delta: dE[Payoff]/dS₀ = E[Payoff × score_function]

Where score_function = d(log density)/dS₀

For lognormal paths: score_delta = (1/S₀) × [(log(S_T/S₀) - (r-σ²/2)T) / (σ²T)]

Advantages:

• Works for any payoff, including discontinuous
• No need to differentiate payoff
• Unbiased

Limitations:

• Higher variance than pathwise for smooth payoffs
• Score function can be large, increasing noise

When to use: Digital options, barriers, and other discontinuous payoffs where pathwise fails.

Adjoint Algorithmic Differentiation (AAD)

Principle: Compute all derivatives in one backward pass through the pricing algorithm.

Key insight: Forward pass: Compute price Backward pass: Propagate sensitivities (chain rule in reverse)

Cost:

• Memory: O(N) for N operations
• Time: ~3-5× forward pricing time
• Greeks: All sensitivities computed simultaneously

Advantages:

• Scales to thousands of Greeks
• Handles complex models efficiently
• Production-grade accuracy

Limitations:

• Complex implementation
• Requires source-level access to pricing code
• Memory-intensive for long simulations

When to use: Production risk systems computing Greeks for large portfolios.

Do / Don't Guidelines

• Do use central differences: More accurate than one-sided for same cost
• Do scale bump sizes appropriately: 1 bp for rates, 1% for vol, 0.5-1% for spot
• Don't use pathwise for barriers: Discontinuities create bias
• Don't use tiny bumps with Monte Carlo: Noise will dominate

Noise and Step Size Controls

Acceptable noise level: Greeks noise should be < 2% of the Greek value

Example analysis: True delta = 0.55 Monte Carlo delta (100k paths) = 0.548 ± 0.012

Noise = 0.012 / 0.55 = 2.2% → borderline acceptable

To reduce noise:

1. Increase paths (noise ∝ 1/√N)
2. Use variance reduction
3. Increase bump size (if truncation error acceptable)
4. Switch to pathwise/LR if applicable

Step size sensitivity test:

Optimal around $0.50-$1.00 for this example.

Risk Reporting

Numerical Greeks feed into risk reports:

• Position delta exposure by underlying
• Portfolio gamma and vega profiles
• Scenario P/L estimates (delta × ΔS + ½ gamma × ΔS²)

Ensure Greek calculation methodology is documented and consistent across the book. Mismatched bump sizes between desks create reconciliation issues.

Next Steps

For Monte Carlo simulation details, see Monte Carlo Simulation Techniques.

To understand the inputs Greeks are sensitive to, review Black-Scholes Model Inputs and Outputs.