Estimating Greeks with Monte Carlo
Monte Carlo estimates a derivative price as an average of discounted payoffs. A Greek is the derivative of that expectation with respect to spot, volatility, rates, or a model parameter. The numerical difficulty is that a small difference between two noisy averages can have a worse signal-to-noise ratio than either price. A Greek implementation therefore needs an estimator design, not just a pricing routine wrapped in a bump.
Start by stating the derivative: sticky strike or sticky delta implied-volatility convention, fixed or recalibrated model parameters, and treatment of dividends and curves. Many “Greek discrepancies” are actually different risk definitions.
Bump-and-revalue with common paths
The central finite-difference delta is:
Delta_hat = (V_hat(S0+h) - V_hat(S0-h)) / (2h)
Use the same random normals in the up and down valuations. Positive correlation cancels much of the sampling noise in their difference. Independent path sets are almost always a poor baseline.
| Bump size | Dominant error | Symptom |
|---|---|---|
| Too large | finite-difference bias | Greek changes as bump shrinks |
| Too small | Monte Carlo noise | Greek becomes erratic |
| Appropriate | balance | stable plateau across bumps |
def central_delta(price_fn, s0, h, normals):
up = price_fn(s0 + h, normals)
down = price_fn(s0 - h, normals)
return (up - down) / (2.0 * h)
Report the standard error of paired per-path differences, not an error bar inferred from the two price standard errors independently. For gamma, the subtraction is more severe, so path count and variance reduction matter even more.
Pathwise estimators
When the payoff is differentiable almost surely, differentiate each simulated payoff. For a European call in Black-Scholes, the pathwise delta contribution is exp(-rT) 1(ST>K) ST/S0. This avoids a bump choice and usually has lower variance than finite differences.
Vega requires differentiating the terminal state through volatility:
dS_T/dsigma = S_T * (-sigma*T + sqrt(T)*Z)
vega_path = exp(-r*T) * 1(S_T>K) * dS_T/dsigma
The indicator is acceptable for a vanilla under continuous sampling because the strike has zero mass. It is not acceptable for a digital option, where the entire sensitivity is concentrated at the discontinuity.
Likelihood-ratio estimators
The likelihood-ratio method differentiates the sampling density rather than the payoff:
d/dtheta E[f(X)] = E[f(X) * d/dtheta log p_theta(X)]
It can estimate delta for a digital claim because the payoff itself need not be differentiable. In a simple normal setting, a score function can be derived analytically. The trade-off is variance: rare but large payoff realizations can dominate the score-weighted average.
| Payoff feature | Preferred starting estimator |
|---|---|
| smooth vanilla | pathwise |
| discontinuous terminal digital | likelihood ratio or conditioning |
| barrier | conditional expectation / smoothing |
| path-dependent smooth average | tangent pathwise |
| many market inputs | adjoint differentiation |
Combine estimators only after checking correlation and bias. A hybrid pathwise-likelihood estimator can reduce variance, but it is more difficult to audit than a well-tested single estimator.
Conditional Monte Carlo and smoothing
For a discretely monitored barrier or digital, condition analytically on the final normal increment. Replace the last-step discontinuous payoff by its conditional expected value, then differentiate the smooth conditional expression. This often turns an unusably noisy delta into a stable estimator.
For continuously monitored diffusion barriers, Brownian-bridge survival probabilities reduce monitoring bias. They do not apply unchanged when jumps are present: jumps can cross and overshoot barriers. Align the estimator to the actual model rather than applying a familiar correction mechanically.
Variance reduction for sensitivities
Antithetic normals, control variates, stratification, and randomized quasi-Monte Carlo can improve Greek estimates as well as prices. A control must have a known Greek, not merely a known price. For example, the analytic Black-Scholes delta can control a related simulated payoff under a more complex model.
G_controlled = G_target - b (G_control - E[G_control])
Estimate b on a pilot sample or independently; fitting it on the same small sample can distort error estimates. Use confidence intervals based on pathwise Greek contributions and rerun with independent randomized scrambles to check robustness.
Calibration and early-exercise risk
A model Greek differs from a market Greek if model parameters are recalibrated after the bump. Distinguish frozen-parameter, recalibrated, and implied-volatility risks in reports. For American claims, the estimated exercise rule may change under a bump; a fixed-policy Greek understates that effect. The exercise and regression issue is detailed in American options and early exercise.
Use stochastic calculus and Ito's lemma to derive continuous-model sensitivities, then validate the discretized estimator that the production code actually computes.
Key takeaways
- Common random numbers are essential for practical finite-difference Greeks.
- Select bump sizes by a stability study, not a convention.
- Pathwise estimators are efficient for smooth payoffs; likelihood ratios handle discontinuities.
- Conditioning and suitable controls can reduce sensitivity variance substantially.
- State Greek conventions and validate against analytic, bump, and grid-refinement benchmarks.