Gambler's Ruin Models

A study of the classical Gambler's Ruin problem using absorbing Markov chains, Monte Carlo simulation, and an application to historical financial data.

About the project

The Gambler's Ruin problem describes a player whose capital changes by one unit after each independent round:

• the player gains one unit with probability p;
• the player loses one unit with probability q = 1 - p;
• the process ends when the capital reaches 0 or a predefined target N.

The two absorbing states are:

• 0: ruin;
• N: victory.

This project compares theoretical results with Monte Carlo simulations and applies the same framework to daily price movements of PETR4 shares between 2020 and 2024.

The objective is to explore:

• random walks;
• absorbing Markov chains;
• ruin and victory probabilities;
• expected absorption time;
• Monte Carlo validation;
• hypothesis testing;
• a simplified application to financial data.

Project components

The repository contains two main analyses.

Script	Description	Generated figures
simulacao_ruina.R	Simulates random trajectories and compares empirical ruin probabilities and expected durations with their theoretical values.	fig_trajetorias.png, fig_ruin_prob.png, fig_duration.png
aplicacao_real.R	Applies the model to PETR4 daily returns from 2020 to 2024, estimates the probability of a positive return, tests the fair-game hypothesis, and compares empirical and theoretical ruin probabilities.	fig_real_ruin.png, fig_real_traj.png

Mathematical framework

Let:

• x be the initial capital, with 0 < x < N;
• N be the target capital;
• p be the probability of gaining one unit;
• q = 1 - p be the probability of losing one unit;
• r = q / p.

Victory probability

For a fair random walk, where p = q = 0.5, the probability of reaching N before ruin is:

$$ P(\text{victory} \mid X_0 = x) = \frac{x}{N} $$

For p ≠ q:

$$ P(\text{victory} \mid X_0 = x)

\frac{1-r^x}{1-r^N} $$

Ruin probability

The probability of reaching 0 before N is the complement of the victory probability.

For p = q = 0.5:

$$ P(\text{ruin} \mid X_0 = x)

1-\frac{x}{N} $$

For p ≠ q:

$$ P(\text{ruin} \mid X_0 = x)

\frac{r^x-r^N}{1-r^N} $$

Expected duration

For a fair random walk:

$$ E[T \mid X_0=x] = x(N-x) $$

For p ≠ q:

$$ E[T \mid X_0=x]

\frac{N \cdot P(\text{victory} \mid X_0=x)-x}{p-q} $$

Here, T represents the number of rounds required for the process to reach either absorbing state.

Monte Carlo simulation

The simulation repeatedly generates random walks until each trajectory reaches:

• ruin at 0; or
• victory at N.

The empirical estimates are then compared with the theoretical formulas for:

• probability of ruin;
• probability of victory;
• expected absorption time.

The comparison helps verify that the simulation converges toward the analytical results as the number of repetitions increases.

Financial application

The second script applies the framework to PETR4 daily adjusted prices between 2020 and 2024.

Daily returns are simplified into two possible outcomes:

• positive return: +1;
• negative return: -1.

The empirical probability of a positive return is estimated as:

$$ \hat{p}

\frac{\text{number of positive returns}} {\text{number of non-zero returns}} $$

The analysis then:

1. downloads PETR4 historical market data;
2. calculates daily returns;
3. converts each return into an upward or downward movement;
4. estimates p;
5. tests whether the process is compatible with a fair game, where p = 0.5;
6. simulates capital trajectories using the estimated probability;
7. compares empirical and theoretical ruin probabilities.

Important interpretation

The financial application is a simplified illustration of the Gambler's Ruin framework.

It assumes that:

• daily movements are independent;
• the probability of an increase remains constant;
• all positive and negative returns have the same magnitude;
• transaction costs and volatility magnitude are ignored.

These assumptions do not represent the full behavior of financial markets.

Therefore, the project should be interpreted as an academic application of stochastic-process concepts, not as an investment strategy or financial forecasting model.

Running the project

Requirements

• R 4.1 or later;
• internet access for the PETR4 analysis.

Required packages include:

• ggplot2;
• dplyr;
• scales;
• quantmod;
• zoo.

Missing packages are installed automatically by the scripts.

Run the simulated study

Rscript simulacao_ruina.R

Run the financial application

Rscript aplicacao_real.R

The second script requires internet access to download PETR4 historical data.

Outputs

The simulated study generates:

Figure	Description
fig_trajetorias.png	Examples of simulated capital trajectories until ruin or victory.
fig_ruin_prob.png	Comparison between theoretical and empirical ruin probabilities.
fig_duration.png	Comparison between theoretical and empirical expected durations.

The financial application generates:

Figure	Description
fig_real_ruin.png	Comparison between theoretical and empirical ruin probabilities using the estimated PETR4 movement probability.
fig_real_traj.png	Simulated trajectories based on the empirical PETR4 probability.

Implementation details

Several improvements were applied to make the scripts more efficient and reusable:

• ruin and duration are calculated during the same Monte Carlo simulation;
• theoretical functions are vectorized over different initial-capital values;
• plotting logic is organized into reusable helper functions;
• simulation results are stored in structured data frames;
• theoretical and empirical results are displayed together for direct comparison;
• random seeds are used where appropriate to improve reproducibility.

Technologies and concepts

• R
• Monte Carlo simulation
• Markov chains
• Random walks
• Absorbing states
• Hypothesis testing
• Financial time series
• Data visualization
• quantmod
• ggplot2

Limitations

The main limitations of the project are:

• the random walk uses fixed movements of +1 and -1;
• observations are treated as independent;
• the probability p is assumed to remain constant;
• the financial application ignores return magnitude;
• transaction costs, dividends, liquidity, and market regimes are not considered;
• theoretical results rely on the assumptions of the classical Gambler's Ruin model.

Possible improvements

Future versions could include:

• confidence intervals for Monte Carlo estimates;
• convergence analysis for different simulation sizes;
• sensitivity analysis for p, x, and N;
• unequal gain and loss sizes;
• time-varying transition probabilities;
• comparison across multiple financial assets;
• bootstrap estimation of the movement probability;
• regime-dependent models;
• an interactive application for changing simulation parameters.

Academic context

This repository documents an applied study of stochastic processes, combining analytical probability results, numerical simulation, and a simplified application to real-world data.

The project was designed to connect the theoretical behavior of absorbing Markov chains with empirical results obtained through Monte Carlo simulation.

Author

Felipe Cordeiro

Data & BI Analyst and Statistics undergraduate focused on analytics, statistical modeling, machine learning, and data science.

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