Random + Probability Time-Series

Overview

MCH Timeseries contains functions to create simulated times series with MCHammer. Current implementation supports Geometric Brownian Motion, Martingales and Markov Chain Time Series.

Random Walks (Geometric Brownian Motion)

Geometric Brownian Motion is commonly used for simulating financial time series, such as stock prices. It models continuous price paths where changes follow a log-normal distribution.

MCHammer.GBMMfit — Function

GBMMfit(HistoricalData, PeriodsToForecast; rng="none")

GBMMfit uses a vector of historical data to calculate the log returns and use the mean and standard deviation to project a random walk. It the uses the last datapoint in the set as the starting point for the new forecast.

HistoricalData: Vector containing historical data

PeriodsToForecast: integer >1

rng is fully random when set to none. You can specify the rng you want to seed the results e.g. MersenneTwister(1234)

source

using MCHammer, Random, Distributions
rng = MersenneTwister(1)
Random.seed!(1)
historical = rand(Normal(10,2.5),1000)

GBMMfit(historical, 12; rng=rng)

# output
12×1 Matrix{Float64}:
 7.3005613535018785
 9.941620113944857
 7.103000072680243
 3.5244881063798483
 0.5845383089109127
 0.8153566312249856
 1.072169905405429
 1.0457817586351268
 1.1470926126750904
 0.8738792037186909
 1.0021288627898237
 1.7966621213142604

MCHammer.GBMM — Function

GBMM(LastValue, ReturnsMean, ReturnsStd, PeriodsToForecast; rng::Any="none")

GBMM produces a random walk using the last data point and requires a mean and standard deviation to be provided.

LastValue: The most recent data point on which to base your random walk.

ReturnsMean and ReturnsStd : Historical Mean and Standard Deviation of Returns

PeriodsToForecast is an integer >1

rng is fully random when set to none. You can specify the rng you want to seed the results e.g. MersenneTwister(1234)

source

using MCHammer, Random, Distributions
Random.seed!(1)
rng = MersenneTwister(1)
GBMM(100000, 0.05,0.05,12, rng=rng)

# output
12×1 Matrix{Float64}:
 104315.42290790279
 114538.65544311896
 115918.4421778525
 113954.88460165854
 107025.58832570235
 117985.02450091281
 128814.81609339731
 134829.80326014754
 143300.85485171276
 145928.17297856198
 156063.17236356856
 180291.78429354102

MCHammer.GBMA_d — Function

GBMA_d(price_0, t, rf, exp_vol; rng="none")

GBMA_d allows you to forecast the stock price at a given day in the future. This function uses a multiplicative Geometric Brownian Motion to forecast but using the Black-Scholes approach

**price_0**: Stock Price at period 0
**t**: Number of days out to forecast
**rf**: Risk free rate (usually the country's 25 or 30 bond)
**exp_vol**: Expected volatility is the annual volatility
**rng**: Used seed the results to make the forecast reproducible. Set to none by default which means fully random  but can use any of Julia's rngs to seed results. e.g. *GBMA_d(price_0, t, rf, exp_vol; rng=MersenneTwister(1))*

source

using MCHammer, Random #hide
rng = MersenneTwister(1)
Random.seed!(1)
GBMA_d(100, 504,0.03,.3, rng=rng)

# output
88.86175908928719

Simulating a random walk time-series

using Dates, MCHammer, Random, Distributions
Random.seed!(1)

ts_trials =[]

#To setup a TimeSeries simulation with MCHammer
for i = 1:1000
     Monthly_Sales = GBMM(100000, 0.05,0.05,12)
     Monthly_Expenses = GBMM(50000, 0.03,0.02,12)
     MonthlyCOGS = Monthly_Sales .* 0.3
     MonthlyProfit = Monthly_Sales - Monthly_Expenses - MonthlyCOGS
     push!(ts_trials, MonthlyProfit)
end

#You can graph the result using trend_chrt()
dr = collect(Date(2025,1,01):Dates.Month(1):Date(2025,12,31))
trend_chrt(ts_trials,dr)

Martingales

A stochastic time-series modeled as a martingale describes a process where each subsequent value's expected future outcome is equal to the current observed value, conditional on the history of all past values. It characterizes a fair, unbiased random walk without drift, commonly applied in scenarios like fair gambling games, financial markets under risk-neutral conditions, or unbiased forecasting models.

MCHammer.marty — Function

marty(Wager, GamesPlayed; GameWinProb = 0.5, CashInHand = Wager)

In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence, given all prior values, is equal to the present value. (Wikipedia)

The marty function is designed to simulate a Martigale such as that everytime a wager is lost, the next bet doubles the wagered amount to negate the previous loss. The resulting vector is the balance of cash the gambler has in hand at any given point in the Martingale process.

GameWinProb is the estimated probability of winning. CashInHand is the starting balance for the martigale. At times, this parameter can make a difference in whether your survive the process or go home broke.

source

For example a gambler with 50$ making wagers of 50$, 10 times using the double or nothing strategy.

marty(50,10)

10-element Vector{Any}:
 100
  50
 150
 200
 250
 300
 250
 150
  50
 -50

Now let's assume that the gambler knows the odds of winning at the casino are less than 0.5 and decides to bring additional funds to persist until the bet pays off.

println(marty(50,10; GameWinProb=0.45, CashInHand=400))

Any[350, 250, 350, 300, 200, 100, 200, 250, 200, 300]

Using Plots.jl, you can compare outcomes at different win probabilities

using Plots

# Generate the four plots as scatter plots.
Exp11 = scatter(marty(5, 100, GameWinProb = 0.25, CashInHand = 400), title="GameWinProb = 0.25", label="Bets", markerstrokecolor=:white, markercolor=:lightblue, titlefontsize=10)

Exp12 = scatter(marty(5, 100, GameWinProb = 0.33, CashInHand = 400), title="GameWinProb = 0.33", label="Bets", markerstrokecolor=:white, markercolor=:lightblue, titlefontsize=10)

Exp13 = scatter(marty(5, 100, GameWinProb = 0.5,  CashInHand = 400), title="GameWinProb = 0.5", label="Bets", markerstrokecolor=:white, markercolor=:lightblue, titlefontsize=10)

Exp14 = scatter(marty(5, 100, GameWinProb = 0.55, CashInHand = 400), title="GameWinProb = 0.55", label="Bets", markerstrokecolor=:white, markercolor=:lightblue, titlefontsize=10)

#) Combine them into a 2x2 grid layout.
combined = Plots.plot(Exp11, Exp12, Exp13, Exp14, layout=(2,2), legend=:topleft)

Sources & References

Eric Torkia, Decision Superhero Vol. 2, chapter 7 : SuperPower – Modeling Probability and Random Events, 2025
Eric Torkia, Decision Superhero Vol. 3, chapter 5 : Predicting 1000 futures, Technics Publishing, 2025