Solving the 100-dimensional Black-Scholes-Barenblatt Equation

Black Scholes equation is a model for stock option price. In 1973, Black and Scholes transformed their formula on option pricing and corporate liabilities into a PDE model, which is widely used in financing engineering for computing the option price over time. [1.] In this example, we will solve a Black-Scholes-Barenblatt equation of 100 dimensions. The Black-Scholes-Barenblatt equation is a nonlinear extension to the Black-Scholes equation, which models uncertain volatility and interest rates derived from the Black-Scholes equation. This model results in a nonlinear PDE whose dimension is the number of assets in the portfolio.

To solve it using the TerminalPDEProblem, we write:

d = 100 # number of dimensions
X0 = repeat([1.0f0, 0.5f0], div(d,2)) # initial value of stochastic state
tspan = (0.0f0,1.0f0)
r = 0.05f0
sigma = 0.4f0
f(X,u,σᵀ∇u,p,t) = r * (u - sum(X.*σᵀ∇u))
g(X) = sum(X.^2)
μ_f(X,p,t) = zero(X) #Vector d x 1
σ_f(X,p,t) = Diagonal(sigma*X) #Matrix d x d
prob = TerminalPDEProblem(g, f, μ_f, σ_f, X0, tspan)

As described in the API docs, we now need to define our NNPDENS algorithm by giving it the Flux.jl chains we want it to use for the neural networks. u0 needs to be a d-dimensional -> 1-dimensional chain, while σᵀ∇u needs to be d+1-dimensional to d dimensions. Thus we define the following:

hls  = 10 + d #hide layer size
opt = Flux.ADAM(0.001)
u0 = Flux.Chain(Dense(d,hls,relu),
                Dense(hls,hls,relu),
                Dense(hls,1))
σᵀ∇u = Flux.Chain(Dense(d+1,hls,relu),
                  Dense(hls,hls,relu),
                  Dense(hls,hls,relu),
                  Dense(hls,d))
pdealg = NNPDENS(u0, σᵀ∇u, opt=opt)

And now we solve the PDE. Here, we say we want to solve the underlying neural SDE using the Euler-Maruyama SDE solver with our chosen dt=0.2, do at most 150 iterations of the optimizer, 100 SDE solves per loss evaluation (for averaging), and stop if the loss ever goes below 1f-6.

ans = solve(prob, pdealg, verbose=true, maxiters=150, trajectories=100,
                            alg=EM(), dt=0.2, pabstol = 1f-6)

Reference

  1. Shinde, A. S., and K. C. Takale. "Study of Black-Scholes model and its applications." Procedia Engineering 38 (2012): 270-279.