mschauer

# Monte Carlo pricing for Asian options


function montecarlo_price_asian(payoff, S0, T, r, sigma, N)
    # Generate N random samples from a standard normal distribution
    mc_average = 0.0
    for i in 1:N
        Z = randn()
    
       

mschauer

###############################################################################
# Martingales by simulation (Julia)
#
# This script illustrates several facts discussed in lecture:
#
# (1) Brownian motion W_t has independent increments and is a martingale.
# (2) Additive Brownian stock model S_t = S_0 + μ t + σ W_t.
# (3) The conditional expectation (filtered value)
#         V_t = E[(S_T - K)^+ | 𝔽_t]
#     is a martingale in t (tower property).

mschauer

    using Random
    rng = MersenneTwister(0)
    n = 2000

    # True SEM: X -> Y -> Z
    true_adj = Bool[
        0 1 0
        0 0 1
        0 0 0]

mschauer

using DelimitedFiles, GLMakie, Distributions

# Black-Scholes formula
normcdf(x) = cdf(Normal(), x)
function blcall(S0, K, r, T, σ)
    B = exp(r*T)
    F = S0 * B
    d1 = log(F/K) / (σ*sqrt(T)) + σ*sqrt(T)/2
    (F*normcdf(d1) - K*normcdf((d1 - σ*sqrt(T))))/B
end

mschauer

# Solving the problem in https://bsky.app/profile/p-hunermund.com/post/3lci6xojlmt25

using CausalInference, Graphs
# defining the graphical do-operator we need here
function do!(g, v)
    for u in collect(inneighbors(g, v))
        rem_edge!(g, u, v)
    end
end

mschauer

using CausalInference
using Associations: CorrTest, PC, Associations
using Test
using Graphs: SimpleDiGraph, Graphs, complete_graph
using StableRNGs
using LinearAlgebra, Random, Distributions
using CausalInference: pcalg, gausscitest, CausalInference
using Combinatorics
using Tables: table, istable
rng = StableRNG(123)

mschauer

using CausalInference, Graphs
V = [:U, :T, :P, :O]
ι = Dict(v=>i for (i,v) in enumerate(V))
g = digraph([1=>3, 2=>3, 3=>4, 2=>4, 1=>4])


# Can estimate total effect T=>O without observing U?
u = ι[:T]
v = ι[:O]
∅ = Set{Int}()

mschauer

using Kalman
using Kalman.GaussianDistributions
using Statistics, LinearAlgebra

# prior for time 0
x0 = 0.0
P0 = floatmax(x0)

# observation operator
H = 1.0

mschauer

using GLMakie
using Random 

# mixture distribution
X(c) = rand() < c ? sqrt(rand()) :  1 - sqrt(rand())

# sample
Random.seed!(1);
A = [0:0.01:1;; X.(0:0.01:1)]; 

mschauer

 0.0 0.0
 -0.032984387220723596 0.030216400626819087
 0.005532055967160396 -0.06303492443179701
 0.04716487617405366 0.06151822897388728
 -0.08816371845816047 -0.01559492041364278
 0.0844812855043274 -0.05374007858781831
 -0.02848102330289795 0.10594798706320867
 -0.054631054889518714 -0.10518877181328493
 0.1190544569384441 0.04347849672678073
 -0.12429436989984163 0.05130690904254036