Unbiased Monte Carlo Greeks for Discontinuous Payoffs

RESUMO

Pathwise differentiation of Monte Carlo estimators fails at payoff discontinuities, producing zero or biased sensitivities for barriers, autocallables, and digital options. The industry workaround---smoothing the indicator functions---introduces bias and requires per-product calibration. We derive a correction formula that restores unbiased Greeks without smoothing. For F(Z,θ) that is piecewise smooth with discontinuities on surfaces {g_i=0}, we show sensitivity decomposes into a pathwise term (computed by standard AAD) plus a sum of boundary corrections, each involving the jump, Gaussian density at boundary, and boundary derivative to parameter. The correction is computed by Newton root-finding in normal-random space, with jump evaluated by two forward replays of the pricing kernel. Implementation uses AADC (pip install aadc), whose tape replay automatic discontinuity tracking make the method fully automatic. We prove arbitrary compositions of payoff functions, covering real autocallable structures with recursive alive/dead logic. Benchmarks against QuantLib models (GBM, Heston, Hull-White) show agreement within 0.1--4% of analytic or bump-and-revalue references.