Kolmogorov Backward Equations

Ashutosh Bharambe
using Flux, StochasticDiffEq
using NeuralPDE
using Plots
using CUDA
ERROR: LoadError: LoadError: LoadError: LoadError: UndefVarError: CUDA not defined
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in expression starting at /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea/packages/Zygote/i1R8y/src/forward/Forward.jl:1
in expression starting at /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea/packages/Zygote/i1R8y/src/Zygote.jl:1
in expression starting at /root/.cache/julia-buildkite-plugin/depots/a6029d3a-f78b-41ea-bc97-28aa57c6c6ea/packages/Flux/qp1gc/src/Flux.jl:1

Introduction on Backward Kolmogorov Equations

The backward Kolmogorov Equation deals with a terminal condtion. The one dimensional backward kolmogorov equation that we are going to deal with is of the form :

\[ \frac{\partial p}{\partial t} = -\mu(x)\frac{\partial p}{\partial x} - \frac{1}{2}{\sigma^2}(x)\frac{\partial^2 p}{\partial x^2} ,\hspace{0.5cm} p(T , x) = \varphi(x) \]

for all $ t \in{ [0 , T] } $ and for all $ x \in R^d $

The Black Scholes Model

The Black-Scholes Model governs the price evolution of the European put or call option. In the below equation V is the price of some derivative , S is the Stock Price , r is the risk free interest rate and σ the volatility of the stock returns. The payoff at a time T is known to us. And this makes it a terminal PDE. In case of an European put option the PDE is:

\[ \frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}{\sigma^2}{S^2}\frac{\partial^2 V}{\partial S^2} -rV = 0 ,\hspace{0.5cm} V(T , S) = max\{\mathcal{K} - S , 0 \} \]

for all $ t \in{ [0 , T] } $ and for all $ S \in R^d $

In order to make the above equation in the form of the Backward - Kolmogorov PDE we should substitute

\[ V(S , t) = e^{r(t-T)}p(S , t) \]

and thus we get

\[ e^{r(t-T)}\frac{\partial p}{\partial t} + re^{r(t-T)}p(S , t) = -\mu(x)\frac{\partial p}{\partial x}e^{r(t-T)} - \frac{1}{2}{\sigma^2}(x)\frac{\partial^2 p}{\partial x^2}e^{r(t-T)} + re^{r(t-T)}p(S , t) \]

And the terminal condition

\[ p(S , T) = max\{ \mathcal{K} - x , 0 \} \]

We will train our model and the model itself will be the solution of the equation

Defining the problem and the solver

We should start defining the terminal condition for our equation:

function phi(xi)
    y = Float64[]
    K = 100
    for x in eachcol(xi)
        val = max(K - maximum(x) , 0.00)
        y = push!(y , val)
    end
    y = reshape(y , 1 , size(y)[1] )
    return y
end
phi (generic function with 1 method)

Now we shall define the problem : We will define the σ and μ by comparing it to the orignal equation. The xspan is the span of initial stock prices.

d = 1
r = 0.04
sigma = 0.2
xspan = (80.00 , 115.0)
tspan = (0.0 , 1.0)
σ(du , u , p , t) = du .= sigma.*u
μ(du , u , p , t) = du .= r.*u
prob = KolmogorovPDEProblem(μ , σ , phi , xspan , tspan, d)
ERROR: UndefVarError: KolmogorovPDEProblem not defined

Now once we have defined our problem it is necessary to define the parameters for the solver.

sdealg = EM()
ensemblealg = EnsembleThreads()
dt = 0.01
dx = 0.01
trajectories = 100000
ERROR: UndefVarError: EM not defined

Now lets define our model m and the optimiser

m = Chain(Dense(d, 64, elu),Dense(64, 128, elu),Dense(128 , 16 , elu) , Dense(16 , 1))
use_gpu = false
if CUDA.functional() == true
  m = fmap(CUDA.cu , m)
  use_gpu = true
end
opt = Flux.ADAM(0.0005)
ERROR: UndefVarError: Dense not defined

And then finally call the solver

@time sol = solve(prob, NeuralPDE.NNKolmogorov(m, opt, sdealg, ensemblealg), verbose = true, dt = dt,
            dx = dx , trajectories = trajectories , abstol=1e-6, maxiters = 1000 , use_gpu = use_gpu)
ERROR: UndefVarError: NeuralPDE not defined

Analyzing the solution

Now let us find a Monte-Carlo Solution and plot the both:

monte_carlo_sol = []
x_out = collect(85:2.00:110.00)
for x in x_out
  u₀= [x]
  g_val(du , u , p , t) = du .= 0.2.*u
  f_val(du , u , p , t) = du .= 0.04.*u
  dt = 0.01
  tspan = (0.0,1.0)
  prob = SDEProblem(f_val,g_val,u₀,tspan)
  output_func(sol,i) = (sol[end],false)
  ensembleprob_val = EnsembleProblem(prob , output_func = output_func )
  sim_val = solve(ensembleprob_val, EM(), EnsembleThreads() , dt=0.01, trajectories=100000,adaptive=false)
  s = reduce(hcat , sim_val.u)
  mean_phi = sum(phi(s))/length(phi(s))
  global monte_carlo_sol = push!(monte_carlo_sol , mean_phi)
end
ERROR: UndefVarError: SDEProblem not defined

##Plotting the Solutions We should reshape the inputs and outputs to make it compatible with our model. This is the most important part. The algorithm gives a distributed function over all initial prices in the xspan.

x_model = reshape(x_out, 1 , size(x_out)[1])
if use_gpu == true
  m = fmap(cpu , m)
end
y_out = m(x_model)
y_out = reshape(y_out , 13 , 1)
ERROR: UndefVarError: use_gpu not defined

And now finally we can plot the solutions

plot(x_out , y_out , lw = 3 ,  xaxis="Initial Stock Price", yaxis="Payoff" , label = "NNKolmogorov")
plot!(x_out , monte_carlo_sol , lw = 3 ,  xaxis="Initial Stock Price", yaxis="Payoff" ,label = "Monte Carlo Solutions")
ERROR: UndefVarError: plot not defined

Appendix

These tutorials are a part of the SciMLTutorials.jl repository, found at: https://github.com/SciML/SciMLTutorials.jl. For more information on high-performance scientific machine learning, check out the SciML Open Source Software Organization https://sciml.ai.

To locally run this tutorial, do the following commands:

using SciMLTutorials
SciMLTutorials.weave_file("tutorials/advanced","03-kolmogorov_equations.jmd")

Computer Information:

Julia Version 1.6.2
Commit 1b93d53fc4 (2021-07-14 15:36 UTC)
Platform Info:
  OS: Linux (x86_64-pc-linux-gnu)
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  LLVM: libLLVM-11.0.1 (ORCJIT, znver2)
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  [0c0b7dd1] Xorg_libXau_jll v1.0.9+4
  [935fb764] Xorg_libXcursor_jll v1.2.0+4
  [a3789734] Xorg_libXdmcp_jll v1.1.3+4
  [1082639a] Xorg_libXext_jll v1.3.4+4
  [d091e8ba] Xorg_libXfixes_jll v5.0.3+4
  [a51aa0fd] Xorg_libXi_jll v1.7.10+4
  [d1454406] Xorg_libXinerama_jll v1.1.4+4
  [ec84b674] Xorg_libXrandr_jll v1.5.2+4
  [ea2f1a96] Xorg_libXrender_jll v0.9.10+4
  [14d82f49] Xorg_libpthread_stubs_jll v0.1.0+3
  [c7cfdc94] Xorg_libxcb_jll v1.13.0+3
  [cc61e674] Xorg_libxkbfile_jll v1.1.0+4
  [12413925] Xorg_xcb_util_image_jll v0.4.0+1
  [2def613f] Xorg_xcb_util_jll v0.4.0+1
  [975044d2] Xorg_xcb_util_keysyms_jll v0.4.0+1
  [0d47668e] Xorg_xcb_util_renderutil_jll v0.3.9+1
  [c22f9ab0] Xorg_xcb_util_wm_jll v0.4.1+1
  [35661453] Xorg_xkbcomp_jll v1.4.2+4
  [33bec58e] Xorg_xkeyboard_config_jll v2.27.0+4
  [c5fb5394] Xorg_xtrans_jll v1.4.0+3
  [8f1865be] ZeroMQ_jll v4.3.2+6
  [3161d3a3] Zstd_jll v1.5.0+0
  [0ac62f75] libass_jll v0.14.0+4
  [f638f0a6] libfdk_aac_jll v0.1.6+4
  [b53b4c65] libpng_jll v1.6.38+0
  [a9144af2] libsodium_jll v1.0.20+0
  [f27f6e37] libvorbis_jll v1.3.6+6
  [1270edf5] x264_jll v2020.7.14+2
  [dfaa095f] x265_jll v3.0.0+3
  [d8fb68d0] xkbcommon_jll v0.9.1+5
  [0dad84c5] ArgTools
  [56f22d72] Artifacts
  [2a0f44e3] Base64
  [ade2ca70] Dates
  [8bb1440f] DelimitedFiles
  [8ba89e20] Distributed
  [f43a241f] Downloads
  [7b1f6079] FileWatching
  [9fa8497b] Future
  [b77e0a4c] InteractiveUtils
  [4af54fe1] LazyArtifacts
  [b27032c2] LibCURL
  [76f85450] LibGit2
  [8f399da3] Libdl
  [37e2e46d] LinearAlgebra
  [56ddb016] Logging
  [d6f4376e] Markdown
  [a63ad114] Mmap
  [ca575930] NetworkOptions
  [44cfe95a] Pkg
  [de0858da] Printf
  [9abbd945] Profile
  [3fa0cd96] REPL
  [9a3f8284] Random
  [ea8e919c] SHA
  [9e88b42a] Serialization
  [1a1011a3] SharedArrays
  [6462fe0b] Sockets
  [2f01184e] SparseArrays
  [10745b16] Statistics
  [4607b0f0] SuiteSparse
  [fa267f1f] TOML
  [a4e569a6] Tar
  [8dfed614] Test
  [cf7118a7] UUIDs
  [4ec0a83e] Unicode
  [e66e0078] CompilerSupportLibraries_jll
  [deac9b47] LibCURL_jll
  [29816b5a] LibSSH2_jll
  [c8ffd9c3] MbedTLS_jll
  [14a3606d] MozillaCACerts_jll
  [4536629a] OpenBLAS_jll
  [bea87d4a] SuiteSparse_jll
  [83775a58] Zlib_jll
  [8e850ede] nghttp2_jll
  [3f19e933] p7zip_jll