Optimal Stopping Times of American Options

Here, we will aim to solve an optimal stopping problem using the NNStopping algorithm.

Let us consider standard American options. Unlike European options, American options can be exercized before their maturity and thus the problem reduces to finding an optimal stopping time.

As stated above, since we can execute the option at any optimal time before the maturity of the option, the standard Black-Scholes model gets modified to:

\[ \frac{∂V}{∂t} + rS\frac{∂V}{∂S} + \frac{1}{2}{\σ^2}{S^2}\frac{∂^2 V}{\∂S^2} -rV ≤ 0\]

The stock price will follow a standard geometric brownian motion given by:

\[ dS_t = rS_tdt + σS_tdW_t\]

And thus our final aim will be to calculate:

american_option

We will be using a SDEProblem to denote a problem of this type. We can define this as a SDEProblem and add a terminal condition g in order to price the American Options.

We will take the case of an American max put option with strike price K, constant volatility β, a risk-free rate r, the initial stock price u0 = 80.00, the maturity T, and number of steps N. The forcing function f and noise function sigma are defined for the type of model. See StochasticDiffEq documentation.

d = 1 #Dimensions of initial stock price
r = 0.04f0
beta = 0.2f0
K = 100.00
T = 1.0
u0 = fill(80.00 , d , 1) #Initial Stock Price
#Defining the drift (f) and diffusion(sigma)
f(du,u,p,t) = (du .= r*u)
sigma(du,u,p,t)  = (du .= Diagonal(beta*u))

tspan = (0.0 , T)
N = 50
dt = tspan[2]/(N - 1)

The final part is the payoff function:

payoff_func

The discounted payoff function is:

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

Now, in order to define an optimal stopping problem, we will use the SDEProblem and pass the discounted payoff function g as an kwarg.

prob  = SDEProblem(f , sigma , u0 , tspan ; g = g)

Finally, let's build our neural network model using Flux.jl. Note that the final layer should be the softmax (Flux.softmax) function as we need the sum of probabilities at all stopping times to be 1. And then add an optimizer function.

m = Chain(Dense(d , 5, tanh), Dense(5, 16 , tanh)  , Dense(16 , N ), softmax)
opt = Flux.ADAM(0.1)

We add algorithms to solve the SDE and the Ensemble. These are the algorithms required to solve the SDEProblem (we use the Euler-Maruyama algorithm in this case) and the EnsembleProblem to run multiple simulations. See Ensemble Algorithms.

sdealg = EM()
ensemblealg = EnsembleThreads()

Finally, we call the solve function:

sol = solve(prob, NeuralPDE.NNStopping( m, opt , sdealg , ensemblealg), verbose = true, dt = dt,
            abstol=1e-6, maxiters = 20 , trajectories = 200)