Exotic Option Greeks and Cross-Gamma

Exotic option Greeks are sensitivities of a path-dependent or multi-factor payoff to market and model inputs. They are harder than vanilla Greeks because the payoff may be discontinuous, the state includes the path, and a price can depend on several correlated assets, rates, volatility factors, barriers, or fixings. Cross-gamma captures the interaction between two risk factors and is essential whenever hedging one factor changes exposure to another.

Cross-gamma definition

For price V(S1, S2), the mixed second derivative is:

Gamma_12 = d2V / (dS1 dS2)

It measures how the delta to asset one changes when asset two moves. A two-asset worst-of, spread option, basket option, or quanto structure may have large cross-gamma even when each standalone gamma appears modest. In a second-order P&L approximation:

dV ≈ sum_i Delta_i*dS_i
   + 0.5*sum_i Gamma_ii*(dS_i)^2
   + sum_(i<j) Gamma_ij*dS_i*dS_j

The cross term is weighted by realized co-movement. Ignoring it effectively assumes the assets never move together.

Product feature	Important non-vanilla risks
Barrier	spot gamma near barrier, discontinuous delta, monitoring
Asian	fixing schedule, remaining-average delta
Basket/worst-of	correlation and cross-gamma
Quanto	FX delta, equity-FX correlation
Callable	rates, volatility, exercise-boundary risk

Bump design matters

Central finite differences estimate cross-gamma as:

Gamma_12 ≈ [V(++ ) - V(+-) - V(-+) + V(--)] / (4*h1*h2)

def cross_gamma(price, x1, x2, h1, h2):
    return (price(x1+h1, x2+h2) - price(x1+h1, x2-h2)
            - price(x1-h1, x2+h2) + price(x1-h1, x2-h2)) / (4*h1*h2)

For Monte Carlo pricing, reuse common random numbers in all four valuations. Independent draws make the subtraction extremely noisy. Select bump sizes after a convergence study: tiny bumps expose simulation and interpolation noise; large bumps measure curvature over a range rather than a local Greek. Store bump convention, model parameters held fixed, and random seed policy with the result.

Path dependence and discontinuity

A barrier option's delta can jump as spot nears a barrier, especially with discrete monitoring. A vanilla-style gamma evaluated at one spot may understate the gap loss when the barrier is crossed overnight. Brownian-bridge corrections improve continuous-barrier simulation, but actual contractual monitoring must still be modeled exactly.

For an Asian option, today's delta depends on accrued average and future fixing weights. A ten-day delay in a daily fixing schedule is not a small date-input error. For autocallables and callable notes, the exercise boundary makes bumping sensitive to numerical regression, mesh resolution, or policy approximation. Validate the Greek itself, not only the price.

Correlation risk is not cross-gamma

Correlation sensitivity is:

CorrVega_12 = dV/drho_12

Cross-gamma multiplies realized product moves, while correlation vega reprices the distribution before the move. Both matter in multi-asset exotics. A worst-of option can be short correlation vega and have a cross-gamma sign that changes with spot region. Report them separately and run joint shocks rather than inferring one from the other.

Model-consistent versus sticky risk

For a vanilla surface, “sticky strike,” “sticky delta,” and recalibrated-model Greeks are distinct hedge assumptions. Exotics add another layer: bumping spot may change local volatility, barrier survival probability, stochastic-volatility state, and hedge portfolio weights. Define whether a market bump holds a calibrated Heston model fixed, preserves market implied vols, or recalibrates the model. All can be useful; none is universally correct.

Risk controls

Use PDE/analytic benchmarks for simplified cases, compare bump Greeks with adjoint or pathwise estimators where available, and require stability across bump sizes and numerical tolerances. Stress discrete jumps, correlation breaks, barrier hits, volatility-surface twists, and missing fixings. Limit reliance on a single scalar Greek around known discontinuities.

Key takeaways

• Cross-gamma measures interacting spot risks and contributes when assets co-move.
• Monte Carlo cross-gamma needs common random numbers and bump-convergence tests.
• Barriers, fixings, and exercise boundaries can make local Greeks unreliable near events.
• Correlation vega and cross-gamma are different risks and require joint scenarios.
• Specify calibration and sticky-surface assumptions with every exotic risk report.
• Preserve conventions for reproducible independent risk review.