The NNStopping algorithm

NNStopping(models, opt)

Deep Optimal Stopping, S Becker, P Cheridito, A Jentzen3, and T Welti.

Arguments

• models::Vector{Flux.Chain}: A vector of Flux.Chain where each model corresponds to a specific timestep from the timespan (tspan). The overall length of the vector should be length(timesteps) - 1.
• opt: the optimization algorithm to be used to optimize the neural networks. Defaults to ADAM(0.1).

Example

d = 3 # Number of assets in the stock r = 0.05 # interest rate beta = 0.2 # volatility T = 3 # maturity u0 = fill(90.0, d) # initial stock value delta = 0.1 # delta f(du, u, p, t) = du .= (r - delta) * u # drift sigma(du, u, p, t) = du .= beta * u # diffusion tspan = (0.0, T) N = 9 # discretization parameter dt = T / (N) K = 100.00 # strike price

payoff function

function g(x, t) return exp(-r * t) * (max(maximum(x) - K, 0)) end

prob = PIDEProblem(f, sigma, u0, tspan; payoff = g) models = [Chain(Dense(d + 1, 32, tanh), BatchNorm(32, tanh), Dense(32, 1, sigmoid)) for i in 1:N]

opt = Flux.Optimisers.Adam(0.01) alg = NNStopping(models, opt)

sol = solve(prob, alg, SRIW1(); dt = dt, trajectories = 1000, maxiters = 1000, verbose = true) ```

solve(
    prob::ParabolicPDEProblem,
    pdealg::NNStopping,
    sdealg;
    verbose,
    maxiters,
    trajectories,
    dt,
    ensemblealg,
    kwargs...
) -> NamedTuple{(:payoff, :stopping_time), <:Tuple{Any, Any}}

Returns a NamedTuple with payoff and stopping_time

Arguments:

• sdealg: a SDE solver from DifferentialEquations.jl. If not provided, the plain vanilla DeepBSDE method will be applied. If provided, the SDE associated with the PDE problem will be solved relying on methods from DifferentialEquations.jl, using Ensemble solves via sdealg. Check the available sdealg on the DifferentialEquations.jl doc.
• maxiters: The number of training epochs. Defaults to 300
• trajectories: The number of trajectories simulated for training. Defaults to 100
• Extra keyword arguments passed to solve will be further passed to the SDE solver.