Automatic differentiation

Prices are generic functions of the inputs of the model, the curve, and the payoff, so they can be differentiated by the following back-ends, loaded with the corresponding package:

Method	DualNumbers	HyperDualNumbers	TaylorSeries	ForwardDiff	ReverseDiff	Zygote	Mooncake
`LewisMethod`	yes	yes	yes	yes	yes	yes	yes
`CosMethod`	yes	yes	yes	yes	yes	yes	yes
`CarrMadanLewisMethod`	yes	yes	yes	yes	yes	yes	yes
`CarrMadanMethod`	yes	yes	yes	yes	yes	yes	yes
`FFTDensityInverter`	yes	yes	yes	yes	yes	yes	yes
`CosDensityMethod`	yes	yes	yes	yes	yes	yes	yes

The test suite checks each entry against the closed formulas of Black and Scholes, for European and binary options and their smiles, with all of the inputs S0, K, r, T, σ, d. Gradients can be taken through the Heston model too. TaylorSeries covers both Taylor1 and TaylorN, which gives the mixed derivatives of several inputs at once:

using FinancialFFT, FinancialMonteCarlo, ForwardDiff

price(x) = pricer(BlackScholesProcess(x[3], Underlying(x[1], x[4])), ZeroRate(x[2]), LewisMethod(600.0, 20000), EuropeanOption(1.0, 100.0))
ForwardDiff.gradient(price, [100.0, 0.02, 0.2, 0.01])

4-element Vector{Float64}:
   0.554049403294252
  47.055534562328404
  39.05541959828385
 -55.40494032942518

TaylorN works in the same way; its variables are declared with variables!, or set_variables before TaylorSeries 0.23:

using TaylorSeries

δS, δσ = TaylorSeries.variables!("δS δσ"; order = 2)
pricer(BlackScholesProcess(0.2 + δσ, Underlying(100.0 + δS, 0.01)), ZeroRate(0.02), LewisMethod(600.0, 20000), EuropeanOption(1.0, 100.0))

Densities are differentiated in the same way, through log(f(y)) (see Densities):

Density	DualNumbers	HyperDualNumbers	TaylorSeries	ForwardDiff	ReverseDiff	Zygote	Mooncake
`FFTDensityInverter`	yes	yes	yes	yes	yes	yes	yes
`CosDensityMethod`	yes	yes	yes	yes	yes	yes	yes

Known limitations

ReverseDiff.jl requires version 1.18 or later.
On the GPU only ForwardDiff.jl and Zygote.jl are supported, see GPU.