Accurate Greeks for Non-smooth Payoffs

The Challenge

Traditional finite differencing for Greeks calculation:

High computational cost (bump-and-revalue for each risk factor)
Poor accuracy for discontinuous payoffs (digitals, barriers, autocalls)
Especially problematic for complex derivatives with multiple barriers
Real Impact: Out-of-money options, correlation risk particularly challenging
High model parametrization granularity yields small changes in payoffs → more numerical noise from Monte Carlo methods

Bottom line: Traditional methods struggle with modern structured products

Why AAD Fails for Non-smooth Payoffs

Pathwise AAD Limitation: Only works for continuous payoffs

Digital functions → Zero gradients everywhere
Example: European payoff ✓, digitals ✗

// This gives zero gradients with AAD:
Real digitalPayoff = (S > K) ? 1.0 : 0.0;

Need for smoothing to make payoffs differentiable

Less Trivial Example: sqrt(|X_t|)

Watch out for less obvious problems:

Path-wise MC method can randomly fail for sqrt(abs(X_t))
Finite Differencing gives noisy but reasonable derivatives
AAD can fail catastrophically!

// Problematic for AAD when X_t crosses zero:
Real payoff = sqrt(abs(underlying_path[t]));

Case Study: Phoenix Autocallable Notes

Two-asset autocallable structure:

Contingent coupons (2.55% quarterly if worst performer > 70%)
Early redemption if both assets > 100% initial
Principal protection with barrier at 70%

Multiple discontinuities:

Autocall triggers
Barrier events
Worst-of conditions

Perfect test case for smoothing + AAD approach

Smoothing Methodology

Mathematical Foundation:

Replace Heaviside functions H(x) with smooth sigmoids:

contLess (a, b, h) = {\begin{cases} 1_{a < b} & if h = 0 \\ \frac{1}{2} (\frac{(b - a) / (0.02 h)}{\sqrt{1 + ((b - a) / (0.02 h))^{2}}} + 1) & if h > 0 \end{cases}

$\text{contLess}(a, b, h) = \begin{cases} \mathbf{1}_{a < b} & \text{if } h = 0 \\ \frac{1}{2}\left(\frac{(b-a)/(0.02h)}{\sqrt{1 + ((b-a)/(0.02h))^2}} + 1\right) & \text{if } h > 0 \end{cases}$

Key Properties:

Parameter selection: Balance accuracy vs. differentiability
Economic preservation: Area under smoothed curve ≈ original digital
Smooth transitions: Enable AAD computation

Smoothing Implementation Details

Explain how smoothing works with not_autocalled_so_far:

Change pricing from digital events to “proportion” of called contract
Specific path can enter multiple smoothed call regions
At maturity T, total contract value is weighted sum of all possible digital events

Real not_autocalled_so_far = 1.0;
// ... for each observation date:
Real autocalled_on_this_coupon = contLess(barrier, price, h);
total_amount += not_autocalled_so_far * (
    autocall_amount + coupon_amount
);
not_autocalled_so_far *= contNot(autocalled_on_this_coupon);

Implementation Approach

Three calculation methods compared:

Base: No smoothing, finite differencing (bump-and-revalue)
Smoothed: Smoothing + finite differencing
AADC: Smoothing + Automatic Adjoint Differentiation

Technical details:

Central differencing (not forward!)
Optimized bump sizes (1-5%)

Source code available: github.com/matlogica/QuantBench

Benchmark Results: Key Findings

Performance improvements:

Smoothed payoffs converge faster than non-smoothed
AADC with smoothing: orders of magnitude improvement
Correlation Greeks: Challenging even at 10M paths without smoothing

CPU Time vs. Path Count:

Base case (non-smoothed): 10M paths for reasonable stability
Smoothed AAD: comparable accuracy with <1M paths
Orders of magnitude improvement in computational efficiency

Live demo: Interactive results viewer

Practical Considerations

Smoothing parameter selection:

Trade-off between stability and accuracy
Similar concept to finite difference bump size
Applied at return rate level → similar magnitude for all assets
Different units/volatilities need individual scaling

Advanced Features & Extensions

Based on Peter Jaeckel’s feedback - Future work on this benchmark:

Quasi-Monte Carlo: Sobol sequences + Brownian Bridge
Low-discrepancy numbers: Not “quasi-random” - they’re deterministic!
Open source - contributions welcome

Generalization with PayOff Language:

AADC-xVA-prototype

Compositional approach to financial contracts
Longstaff-Schwartz regression for callable products
XVA calculations via regression and AAD
Framework ready for AAD integration

Q&A

Resources:

Thank You

Contact:

Dmitri Goloubentsev
Head of Automatic Adjoint Differentiation
Matlogica LTD

Email: dmitri@matlogica.com
Web: matlogica.com
Phone: +447378414528