A Monte Carlo Study of the Gambler’s Ruin Problem
DOI:
https://doi.org/10.61173/txzcrb35Keywords:
Gambler’s Ruin, Monte Carlo Simulation, Financial Risk, Random Walk, Stochastic ProcessesAbstract
The Gambler's Ruin problem is one of the classical stochastic models of an agent who gains or loses capital repeatedly, till reaching his target level or till his ruin. In this paper, the problem is theoretically analyzed and solved by Monte Carlo simulation. This paper obtains the closed-form expression for the ruin probability and discuss its sensitivity with respect to the initial capital and the probability bias. The simulation results are close to the theoretical predictions and show how the results of the long-term are sensitive to the small changes of the probability. This paper also carries out an analysis of the real-life applications in the finance and economy field in addition to theoretical and numeric analysis. In particular, this paper looks at the applicability of the Gambler's Ruin problem in the case of grid trading strategies, asymmetric investment decisions, and firm survival dynamics. The applications in the paper show the utility of the model in the areas of long-term risk accumulation, capital sustainability, and decision making under uncertainty. The results show how critical probabilistic reasoning is to assess strategies that might seem to be profitable on the short term but have high risks on the long term.
References
that there are several key takeaways. The first is a strong 2nd ed. Singapore: World Scientific, 2010. (Vol. 14). dependence of the probability of ruin on the initial capital [2] KABANOV Y, PUKHLYAKOV N. Ruin probabilities with and on the probability of success. Secondly, large num- investments: smoothness, integro-differential and ordinary bers of trials can cause a small difference in probabilities differential equations, asymptotic behavior[J]. Journal of Applied
to make a big difference in the long-term result. Thirdly, Probability, 2022, 59(2): 556-570. the length of time that the process will take depends on [3] FERREIRA M, FILIPE J. Gambler ruin random walks and the location of the initial capital with respect to the bound- brownian motions in reserves modeling, application to pensions aries. The results highlight the need for probabilistic rea- funds sustainability[J/OL]. arXiv preprint, arXiv:2109.10313, soning when dealing with uncertainty in a system. Simple 2021. models can yield useful information about a complex [4] KORZENIOWSKI A. Ruin probability for risk model with phenomenon (as long as they retain the key features of the random premiums[J]. Journal of Mathematical Finance, 2023, phenomenon). It is possible to extend the model to more 13(2): 171-179. realistic situations in the future, for example, time varying [5] WANG K, XUN B, GUO X. Finite-time ruin probability of probabilities, asymmetrical payoffs and multi-dimensional a risk model with perturbation and sub-exponential main claims processes. Such extensions would enhance the applicabil-
and by-claims[J]. Stochastics, 2024, 96(8): 2062-2077. ity of the Gambler’s Ruin framework and provide deeper [6] STUDZINSKI PEROTTO F, TRABELSI I, COMBETTES insights into real-world problems. However, this study also has some limitations. The simu- S, et al. Deciding when to quit the gambler's ruin game with lation model used in this paper is based on several simpli- unknown probabilities[J]. International Journal of Approximate
fying assumptions. For example, the probability of success Reasoning, 2021, 137: 16-33. is fixed, the step size is constant, and the boundaries are [7] TARANTO A, KHAN S. Gambler's ruin problem and biclearly defined. These assumptions make the model easier directional grid constrained trading and investment strategies[J].
to analyze, but they also mean that the model cannot fully Investment Management and Financial Innovations, 2020, 17(3): represent complex real-world situations. In financial or 54-66. business applications, probabilities may change over time, [8] DELSING G A, MANDJES M R H, SPREIJ P J C, et al. On gains and losses may be asymmetric, and external factors capital allocation for a risk measure derived from ruin theory[J].
may affect decision making. Therefore, the results should Insurance: Mathematics and Economics, 2022, 104: 76-98. be understood as an illustration of the main principles of [9] WHELAN K. The gambler's ruin with asymmetric the Gambler’s Ruin problem rather than a complete de- payoffs[R]. Dublin: UCD Centre for Economic Research, 2025. scription of all real-world risk processes. (Working Paper WP25/03). Future research could improve this model in several direc- [10] YANG Z. Formalization of Gambler's Ruin Problem in tions. One possible extension is to consider time-varying Isabelle/HOL[J/OL]. probabilities, where the chance of success changes during [11] COAD A, FRANKISH J, ROBERTS R G, et al. Growth the process. Another extension is to study asymmetric paths and survival chances: an application of Gambler's Ruin
payoffs in more detail, so that each win and loss does not theory[J]. Journal of Business Venturing, 2013, 28(5): 615-632.
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