API reference

Vibrato(mc, vb_mc)

Vibrato Monte Carlo: mc (a MonteCarloConfiguration) simulates the path up to the last time step, and vb_mc (VibratoMonteCarloGaussLegendre, VibratoMonteCarloAnalytic, VibratoMonteCarloStandard) integrates the payoff against the density of the last step with the likelihood ratio method.

pricer(mcProcess, rfCurve, Vibrato(mc, vb_mc), payoff):

• BlackScholesProcess and HestonProcess: Gaussian last step (for Heston with the simulated variance).
• Lévy processes: FFT density of the last step (requires VibratoMonteCarloAnalytic).
• an AmericanPayoff (AmericanOption, BinaryAmericanOption): exercise at the Nstep grid dates, policy from Longstaff-Schwartz. On BlackScholesProcess vibrato over every exercise date; on Lévy processes and HestonProcess the exercise decisions are frozen when differentiating, so second order sensitivities are biased.

VibratoSaltando(mc, vb_mc)

Vibrato conditioned on the jumps: on the last jump time for finite activity processes (KouProcess, MertonProcess), on the subordinator G_T for VarianceGammaProcess and NormalInverseGaussianProcess. The law of log S_T given that is Gaussian, and vb_mc integrates the payoff against it. Exact for any Nstep. On VG and NIG it also differentiates over κ and T; on Kou and Merton it does not support derivatives over λ or T (it warns).

PureLRM(mc, vb_mc)

Likelihood ratio method on the density at maturity, with no path simulation: Gaussian for Black-Scholes, from the FFT for the Lévy processes and Heston (its marginal density at maturity; requires VibratoMonteCarloAnalytic). mc only provides the seed and, for VibratoMonteCarloStandard, the random numbers.

VibratoMonteCarlo.Conditional(mc)

Same paths as Vibrato on BlackScholesProcess, with the last step priced in closed form (blsprice, blsbin).

VibratoMonteCarlo.ConditionalSaltando(mc)

Same paths as VibratoSaltando, with what the conditioning leaves priced in closed form: the exact inner integration limit of VibratoSaltando.

VibratoMonteCarloGaussLegendre(Npoints = 48)

Inner integration by Gauss-Legendre quadrature on [-8.5, 8.5 + σ] standard deviations, split at the singular points of the payoff (the strikes) with Npoints nodes on each piece. Every piece has a smooth integrand, so the rule converges exponentially: about 1e-14 per path. Gaussian inner laws only (Black and Scholes, Heston, VibratoSaltando, PureLRM on Black and Scholes, American options on Black and Scholes).

VibratoMonteCarloAnalytic(Npoints = 200, x_min = -7.0, x_max = 7.0)

Inner integration by quadrature on a uniform grid of Npoints in [x_min, x_max], in standard deviations of the log increment around its mean. The only inner integration of the FFT density (Lévy processes with Vibrato and PureLRM).

VibratoMonteCarloStandard(Nsim_lrm = 200)

Inner integration by Monte Carlo: Nsim_lrm standard normal samples for every outer path, drawn again for each path from the generator of the MonteCarloConfiguration. With AntitheticMC they come in antithetic pairs, with SobolMC they are a one dimensional Sobol sequence randomly shifted for every path (Cranley-Patterson). Gaussian inner laws only.

VibratoMonteCarlo.VibratoMonteCarloGaussHermite(Npoints = 200)

Inner integration by Gauss-Hermite quadrature on Npoints nodes, Gaussian inner laws only. It converges slowly across the kink or the jump of a payoff: prefer VibratoMonteCarloGaussLegendre.