Abstract Dynamic exposure rules can appear effective simply because they reduce risky participation, not because they time exposure well. This study evaluates Pogi, a recursive FULL/PARTIAL/NONE controller that separates portfolio composition from total risky exposure and uses a non-executed shadow path to observe recovery during defensive states. Using daily cryptocurrency data from 2014–2026, eight chronological test folds, recursive transaction costs, and CRRA certainty-equivalent welfare, Pogi is compared with static scaling, volatility targeting, CPPI, drawdown throttling, moving-average control, fractional Kelly scaling, and an exact ex-post exposure-matched diagnostic. The analysis also tests initialization and memory sensitivity, timing nulls, search capacity, selection-aware inference, and external validation using U.S. industry portfolios and frozen cross-market transfer. The completed evaluation did not establish robust welfare superiority for Pogi or support a broader methodological contribution under the pre-specified evidence criteria. The results instead show why dynamic exposure rules should be judged against exposure-matched benchmarks, model-search controls, recursive-state diagnostics, and genuinely external validation.
The Karhunen–Loève (KL) expansion provides an optimal orthogonal series representation of second-order stochastic processes and random fields. This paper presents a complete mathematical formulation of the KL theory using native Office Math Markup Language (OMML) equations, covering the governing Fredholm integral eigenvalue problem, the expansion and its coefficients, truncation error bounds, and the normalized representation. Building upon this theoretical foundation, we systematically explore practical applications of the KL expansion in contemporary supply-chain research themes: quantum-inspired optimization and uncertainty quantification in logistics networks; construction of resilient and adaptive digital twins for global supply chains; stochastic modelling supporting artificial-intelligence and blockchain integration for transparency, traceability and resilience; and uncertainty-aware modelling in sustainable and green logistics. The KL expansion emerges as a rigorous, computationally tractable tool for dimensionality reduction, random-field generation and risk quantification across these domains, thereby bridging classical stochastic process theory with the emerging requirements of Industry 4.0 and quantum-era logistics systems. Keywords— Karhunen–Loève expansion; stochastic processes; uncertainty quantification; digital twin; supply chain resilience; quantum logistics; green logistics; AI–blockchain integration.
Open access
Supply Chain Resilience and Risk Management
Risk and Portfolio Optimization
Infrastructure Resilience and Vulnerability Analysis
Maximal Extractable Value (MEV) in decentralized finance (DeFi) enables searchers to profit from transaction ordering and arbitrage opportunities across Automated Market Makers (AMMs). Among MEV strategies, atomic triangular arbitrage is widely deployed due to its deterministic execution within a single transaction. However, executing profitable arbitrage under realistic constraints, such as limited wallet balance, pool liquidity, gas costs, and blockchain latency, remains a challenging optimization problem. In this work, we formulate atomic triangular arbitrage as a constrained optimization problem that jointly selects an ordered three-pool path and trade amount to maximize net profit. To solve this non-convex problem, we propose a Deep Reinforcement Learning approach based on Proximal Policy Optimization (PPO). Experimental results show that while exhaustive grid search attains the highest returns, it requires a significantly high amount of inference time, making it infeasible for on-chain execution. In contrast, the proposed PPO agent achieves millisecond-level inference latency while generating consistent positive profit. These findings highlight a fundamental speed–profit trade-off in MEV extraction and demonstrate that PPO provides an effective and practical solution for atomic triangular arbitrage in DeFi.
For a single event with finitely many mutually exclusive outcomes, the full Kelly problem is to maximize expected log wealth over nonnegative stakes together with an optional cash position. The optimal formula is classical, but the support-selection step is often presented via Lagrange multipliers. This note gives a shorter state-price derivation. A cash fraction $c$ acts as an implicit position in every outcome: in terminal-wealth terms, it is equivalent to a baseline stake $cq_i$ on outcome $i$, where $q_i$ is the state price. On any active support, explicit bets therefore only top up favorable outcomes from this baseline $cq_i$ to the optimal total stake $p_i$. This yields the formula $x_i = (p_i - c q_i)_+$, the threshold rule $p_i/q_i > c$, and, after sorting outcomes by $p_i/q_i$, a one-pass greedy algorithm for support selection. The result is standard in substance, but the implicit-position viewpoint gives a compact proof and a convenient way to remember the solution.
Many systems map governance and execution power directly to purchasable capital (stake, tokens, shares). This creates structurally unsafe paths to power: influence can be bought, short-window manipulation can become long-lived authority, and low-integrity applications can contaminate system-level decision making. This paper defines Proof of Contribution (POC) as a parent-layer execution-weight reference and constraint layer for contribution-generated assets (ABUE / CGA). POC converts finalized contribution-derived claims into execution weights under strict constraints: Source purity (external purchases do not mint influence), verifiable value caps (weights cannot exceed auditable backing), decay (power requires continued contribution), downward-only normalization (anti-compounding), local negative contributions (risk isolation), and delayed activation (audit windows). Crucially, POC is specified as an audit-executable closed loop: versioned policy bundles with timelocks, deterministic recomputation, public commitments (roots), challenge windows, and automatic consequences (freeze/down-weight/remove; Only-Down). We provide falsifiable hypotheses (H0–H4), trigger playbooks (TRW1–TRW3), Minimum Qualifying Implementation (MQI) boundaries, a parameter ledger, and a reproducible toy simulator framework (ReproPackW/MVDW) intended to validate invariants—not to claim economic optimality. A consensus instantiation is treated as a conditional subset and fully developed in a companion paper.
In recent years, investors have shown growing interest in diversified multi-asset indices that incorporate crypto assets, with Bitcoin at the forefront. The launch of exchange-traded products, the growing acceptance of Bitcoin among institutional investors, and its increasing weight in financial markets all highlight the strategic importance of this asset. Bitcoin has exhibited extraordinary returns in the past, yet as a standalone investment it appears less attractive to risk-averse investors due to extreme volatility and severe drawdowns. The relevant question is therefore whether including Bitcoin in a diversified multi-asset portfolio can enhance performance without deteriorating its overall risk profile, and what allocation methods provide a credible way to achieve this balance. Classical allocation methods offer contrasting perspectives. Mean–variance optimisation explicitly incorporates expected returns, but it is highly sensitive to estimation error, which often results in unstable allocations. Empirical evidence even shows that simple rules such as the naïve 1/N portfolio often outperform mean–variance optimisation out of sample [DeMiguel et al., 2007]. Risk-based approaches, such as minimum variance or risk parity, are more stable but ignore expected returns altogether, which is problematic when dealing with an asset that exhibits an unusually high mean return. Robust optimisation provides a natural way to reconcile these two approaches. By incorporating parameter uncertainty directly into the optimisation problem, robust methods allow expected returns to influence the allocation while penalising excessive reliance on noisy estimates. This framework is particularly well-suited for Bitcoin, whose characteristics amplify estimation risk. The aim of this thesis is to study the construction of a multi-asset index including Bitcoin, with a particular focus on the use of robust optimisation techniques. More specifically, the objective is both to assess whether the inclusion of Bitcoin can enhance the performance of a diversified portfolio without materially worsening its risk profile, and to evaluate whether robust optimisation provides more stable and credible allocations than classical approaches such as mean–variance, risk parity, or equal-weighting. Performance and stability are examined through backtests and Monte Carlo simulations.
We present the most complete unified taxonomy of Black-Scholes option price sensitivities (Greeks) available in the literature, encompassing 21 distinct measures through third order: first-order (Delta, Vega, Theta, Rho), second-order (Gamma, Vanna, Charm, Vomma, Veta, Vera, Dual Delta, Dual Gamma), third-order (Speed, Zomma, Color, Ultima, DvannaDvol, DvommaDspot), and portfolio-level (Dollar Delta, Dollar Gamma, Lambda). For each Greek we provide: (i) a fully explicit derivation from the dividend-adjusted Black-Scholes formula showing every application of the chain rule and product rule, (ii) alternative derivation paths including risk-neutral expectation differentiation and heat equation Green’s function representations, (iii) closed-form expressions for both European calls and puts, (iv) identification of put-call parity equivalences, (v) complete asymptotic analysis (deep ITM/OTM, short/long-dated, zero/high vol limits), (vi) monotonicity and convexity properties with extrema locations, (vii) dimensional analysis for practitioner interpretation, and (viii) practical trading context. We derive the Black-Scholes PDE from first principles via Itô’s lemma and the replicating-portfolio argument, establish the risk-neutral pricing connection, and prove the key symmetry lemma that simplifies every Greek derivation. The Gamma-Theta tradeoff is proved directly from the PDE with trading implications. The Vanna-Volga pricing method is derived from smile replication principles. We provide a complete treatment of sticky-strike versus stickydelta hedging conventions, delta hedging theory with continuous and discrete P&L analysis, Greeks under stochastic volatility (Heston model), numerical methods for Greeks computation (finite differences, pathwise, likelihood ratio, and adjoint algorithmic differentiation), and the behavior of Greeks near expiry including pin risk. A differential-geometric interpretation frames the Greeks as gradient, Hessian, and third-order tensor components on the six-dimensional Black-Scholes parameter manifold, with the PDE as a constraint surface. Fourth-order Greeks are derived and a convergence analysis of the Taylor price expansion justifies the third-order truncation for perturbations up to 10% of spot. Publication-quality three-dimensional surface visualizations for all 21 Greeks reveal the topology of each sensitivity across its natural parameter space. Formal proofs of all 12 put-call parity equivalences, a complete 21 Greek formula reference card, and a comprehensive monotonicity and extrema table are provided as appendices. A Python implementation with numerical verification against finite differences accompanies the paper, with over 50 figures. To our knowledge, this constitutes the most comprehensive rigorously derived reference for Black-Scholes Greeks in a single document.
We develop optimal transport stress testing, liquidation cost modeling, and fund-level capital allocation for lending against prediction market collateral. Building on a companion paper that derives first-passage default probabilities under Hawkes-driven jump-discussion dynamics, this paper addresses three challenges that arise when operating the lending protocol at scale. First, we introduce a Wasserstein stress testing methodology that generates synthetic tail scenarios for markets with insufficient historical depth, proving that it achieves strictly higher effective sample sizes than classical Entropy Pooling when the stress region lies outside the empirical support. We further establish an adversarial robustness guarantee: the stressed risk estimate remains bounded even under worst-case perturbations of the empirical distribution within a Wasserstein ball- a formal resilience property that no existing decentralized finance stress testing methodology provides.
Decentralized Finance (DeFi) lending and borrowing protocols enable investors to take leveraged long and short positions on digital assets without centralized intermediaries, but expose them to a distinctive form of risk: on-chain liquidation triggered by debt and collateral value fluctuations. In this work, we provide a detailed formalization of Aave's lending, borrowing, and liquidation mechanisms, grounded in the protocol's open-source implementation. In doing so, we propose a mathematical modeling of the risk of liquidation, including some stochastic approximations with the purpose of efficient analysis, with different applications. Among them, portfolio optimization problem.
This paper proposes a structured decentralized finance (DeFi) strategy designed to accumulate Ethereum (ETH) over time while exploiting market volatility through liquidity provision and controlled directional exposure. The framework combines concentrated liquidity provisioning on Uniswap v3 with a hedge position using low-leverage directional exposure and a reserve of stablecoins for counter-cyclical accumulation during market drawdowns. The strategy is implemented on Layer-2 networks-specifically Arbitrum and Base-to reduce transaction costs and capture diversified trading flows. We present a formal mathematical treatment of impermanent loss under concentrated liquidity, Monte Carlo simulations of ETH price paths under three market regimes, and an optimization framework for liquidity range selection. The proposed system transforms three distinct market conditions-sideways volatility, bullish breakouts, and market downturns-into opportunities for yield generation, directional gains, or asset accumulation. Results indicate that the hedged strategy achieves a superior risk-adjusted profile relative to unhedged liquidity provision across all tested volatility regimes.
Decentralized finance has introduced new forms of market making through automated market makers, where users provide liquidity to decentralized exchanges such as Uniswap. In Uniswap v3, liquidity providers (LPs) can concentrate their liquidity within custom price ranges, improving capital efficiency but introducing exposure to impermanent loss and nonlinear portfolio risk. Managing this exposure dynamically poses a challenge as the LP position’s token composition changes continuously with market prices. This thesis investigates whether a Deep Reinforcement Learning (DRL) agent can effectively hedge a Uniswap v3 LP position using cryptocurrency futures. A simulated Uniswap environment is developed to model concentrated liquidity providing positions and the hedge positions tied to it. The hedging problem is framed as a sequential decision process, where the agent seeks to minimize downside portfolio variance while maintaining upside exposure. The Proximal Policy Optimization algorithm is applied to learn an adaptive hedging policy, which is evaluated against baseline strategies such as fixed-frequency rebalancing. The results show that the DRL-based hedges can outperform the baseline strategies, achieving higher average portfolio returns, with similar average drawdowns. However, the learned policies varied between agents, and although they reduced downside variance in many cases, they were also more likely to experience larger maximum drawdowns. These findings indicate that reinforcement learning offers a promising but complex approach for managing the nonlinear risks of concentrated liquidity provision in decentralized exchanges.
Nicholas Brandt, Miguel Cueto Noval, Christoph U. Günther, Akın Ünal · 5 authors
CVRFs are PRFs that unify the properties of verifiable and constrained PRFs. Since they were introduced concurrently by Fuchsbauer and Chandran-Raghuraman-Vinayagamurthy in 2014, it has been an open problem to construct CVRFs without using heavy machinery such as multilinear maps, obfuscation or functional encryption. We solve this problem by constructing a prefix-constrained verifiable PRF that does not rely on the aforementioned assumptions. Essentially, our construction is a verifiable version of the Goldreich-Goldwasser-Micali PRF. To achieve verifiability we leverage degree-2 algebraic PRGs and bilinear groups. In short, proofs consist of intermediate values of the Goldreich-Goldwasser-Micali PRF raised to the exponents of group elements. These outputs can be verified using pairings since the underlying PRG is of degree 2. We prove the selective security of our construction under the Decisional Square Diffie-Hellman (DSDH) assumption and a new assumption, which we dub recursive Decisional Diffie-Hellman (recursive DDH). We prove the soundness of recursive DDH in the generic group model assuming the hardness of the Multivariate Quadratic (MQ) problem and a new variant thereof, which we call MQ+. Last, in terms of applications, we observe that our CVRF is also an exponent (C)VRF in the plain model. Exponent VRFs were recently introduced by Boneh et al. (Eurocrypt’25) with various applications to threshold cryptography in mind. In addition to that, we give further applications for prefix-CVRFs in the blockchain setting, namely, stake-pooling and compressible randomness beacons.
This review explores the intersection of probability theory and data visualization in the domain of financial risk prediction. It examines how probabilistic models—such as Value-at-Risk (VaR), Conditional Value-at-Risk (CVaR), Monte Carlo simulation, stochastic processes, and Bayesian inference—serve as the backbone of uncertainty modeling in finance by reviewing previous studies. Simultaneously, it highlights the role of visualization in transforming abstract probability distributions into interpretable insights through dashboards, heatmaps, clustering, and interactive visual frameworks. Drawing on over 20 open-access sources, the review synthesizes applications across corporate profitability, systemic risk, portfolio optimization, credit default, exchange rate forecasting, ESG sustainability, start-up financing, and decentralized finance (DeFi). It concludes by identifying limitations—including data quality issues, computational complexity, interpretability challenges, and ethical/regulatory concerns—and proposes future research directions in robust probabilistic modeling, scalable explainable AI, standardized visualization practices, and fairness-aware risk systems. Together, probability and visualization provide complementary tools that are indispensable for navigating financial uncertainty in the 21st century.
Portfolio optimization is a cornerstone of modern financial decision-making, tradition-ally based on the mean–variance model introduced by Markowitz. However, this framework relies on restrictive assumptions—such as normally distributed returns and symmetric risk preferences—that often fail in real-world markets, particularly in volatile and non-Gaussian environments such as cryptocurrencies. To address these limitations, this paper proposes a novel multi-objective model that combines expected return max-imization, mean absolute deviation (MAD) minimization, and entropy-based diversifi-cation into a unified optimization structure: the Mean–Deviation–Entropy (MDE) model. The MAD metric offers a robust alternative to variance by capturing the average mag-nitude of deviations from the mean without inflating extreme values, while entropy serves as an information-theoretic proxy for portfolio diversification and uncertainty. Three entropy formulations are considered—Shannon entropy, Tsallis entropy, and cumulative residual Sharma–Taneja–Mittal entropy (CR-STME)—to explore different notions of uncertainty and structural diversity. The MDE model is formulated as a tri-objective optimization problem and solved via scalarization techniques, enabling flexible trade-offs between return, deviation, and en-tropy. The framework is empirically tested on a cryptocurrency portfolio composed of Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB), using daily data over a 12-month period. The empirical setting reflects a high-volatility, high-skewness regime, ideal for testing entropy-driven diversification. Comparative outcomes reveal that entropy-integrated models yield more robust weightings, particularly when tail risk and regime shifts are present. Comparative results against classical mean–variance and mean–MAD models indicate that the MDE model achieves improved di-versification, enhanced allocation stability, and greater resilience to volatility clustering and tail risk. This study contributes to the literature on robust portfolio optimization by integrating entropy as a formal objective within a scalarized multi-criteria framework. The proposed approach offers promising applications in sustainable investing, algorithmic asset allo-cation, and decentralized finance, especially under high-uncertainty market conditions.
Traditional portfolio optimization models, rooted in the mean–variance framework of Markowitz, rely heavily on variance as a risk measure. Although theoretically elegant, this approach becomes fragile in volatile and structurally unstable markets such as cryptocurrencies, where return distributions deviate significantly from normality, cor-relations are unstable, and concentration risk emerges. These limitations have motivated the search for alternative frameworks capable of capturing uncertainty in a more flexible and distribution-free manner. Entropy, originally introduced by Shannon as a measure of information, has gradually been recognized in the financial literature as a suitable proxy for diversification and systemic uncertainty. To address the shortcomings of variance-based models, this paper introduces the Weighted Shannon Entropy (WSE) model as a diversification-oriented alternative. By extending the classical Shannon entropy with asset-specific informational weights, the WSE framework provides additional flexibility for modeling heterogeneous asset char-acteristics, such as liquidity, informational value, or perceived reliability. Using the principle of maximum entropy and the method of Lagrange multipliers, we derive ex-ponential-form solutions for portfolio weights that naturally discourage concentration, ensure balanced allocations, and remain analytically tractable. The methodology is validated empirically on a portfolio of four leading cryptocurren-cies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB)—using market data from January to March 2025. The results demonstrate that the entropy-based optimization framework produces well-diversified portfolios, robust to volatility and structural instability, and provides a distribution-free alternative to the classical mean–variance model. Beyond its empirical performance, the WSE formulation highlights the conceptual advantage of entropy in integrating return, risk, and diversification into a single unified framework. The paper contributes both theoretically and practically: it strengthens the mathematical foundation of entropy-based portfolio selection, extends its applicability to digital asset markets, and illustrates how weighting schemes can enrich the classical Shannon measure. Future research may extend this approach to multi-period optimization, gen-eralized entropies such as Tsallis and Kaniadakis, or integration with machine learning models for dynamic portfolio management.
Bitcoin treasury companies have taken stock markets by storm amassing billions of dollars worth of tokens in hundreds of entities. The paper discusses, how leverage - whether created through corporate debt or investors using stock as loan collateral - fuels this trend. The extension of the binary-choice Kelly criterion to incorporate uncertainty in the form of the Kullback-Leibler divergence or more generally Bregman divergence is also briefly discussed.
This paper introduces a novel multi-objective optimization framework for sustainable portfolio rebalancing under uncertainty. The model simultaneously targets return maximization, downside risk control, and liquidity preservation, addressing the complex trade-offs faced by investors in volatile markets. Unlike traditional static approaches, the framework allows for dynamic asset reallocation and explicitly incorporates nonlinear transaction costs, offering a more realistic representation of trading frictions. Key financial parameters—including expected returns, volatility, and liquidity—are modeled using interval arithmetic, enabling a flexible, distribution-free depiction of uncertainty. Risk is measured through semi-absolute deviation, providing a more intuitive and robust assessment of downside exposure compared to classical variance. A core innovation lies in the behavioral modeling of investor preferences, operationalized through three strategic configurations, pessimistic, optimistic, and mixed, implemented via convex combinations of interval bounds. The framework is empirically validated using a diversified cryptocurrency portfolio consisting of Bitcoin, Ethereum, Solana, and Binance Coin, observed over a six-month period. The simulation results confirm the model’s adaptability to shifting market conditions and investor sentiment, consistently generating stable and diversified allocations. Beyond its technical rigor, the proposed framework aligns with sustainability principles by enhancing portfolio resilience, minimizing systemic concentration risks, and supporting long-term decision-making in uncertain financial environments. Its integrated design makes it particularly suitable for modern asset management contexts that require flexibility, robustness, and alignment with responsible investment practices.
This paper presents a robust multi-period portfolio optimization framework that integrates interval analysis, entropy-based diversification, and downside risk control. In contrast to classical models relying on precise probabilistic assumptions, our approach captures uncertainty through interval-valued parameters for asset returns, risk, and liquidity—particularly suitable for volatile markets such as cryptocurrencies. The model seeks to maximize terminal portfolio wealth over a finite investment horizon while ensuring compliance with return, risk, liquidity, and diversification constraints at each rebalancing stage. Risk is modeled using semi-absolute deviation, which better reflects investor sensitivity to downside outcomes than variance-based measures, and diversification is promoted through Shannon entropy to prevent excessive concentration. A nonlinear multi-objective formulation ensures computational tractability while preserving decision realism. To illustrate the practical applicability of the proposed framework, a simulated case study is conducted on four major cryptocurrencies—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), and Binance Coin (BNB). The model evaluates three strategic profiles based on investor risk attitude: pessimistic (lower return bounds and upper risk bounds), optimistic (upper return bounds and lower risk bounds), and mixed (average values). The resulting final terminal wealth intervals are [1085.32, 1163.77] for the pessimistic strategy, [1123.89, 1245.16] for the mixed strategy, and [1167.42, 1323.55] for the optimistic strategy. These results demonstrate the model’s adaptability to different investor preferences and its empirical relevance in managing uncertainty under real-world volatility conditions.
Abstract Decentralized finance is upending the financial system through innovative, open, and interoperable financial solutions. Decentralized finance is a rapidly emerging field based on distributed ledger technology. Decentralized banking protocols are witnessing a perfect storm in terms of growth. However, because these financial innovations pose specific risks to consumers, creators, regulators, and other stakeholders, this emerging subject demands careful investigation. The current study tries to categorize and rate the risks connected with decentralized finance. The current study seeks to identify the multiple risks associated with decentralized finance through a thorough literature analysis. Data gathered from specialists in prior research were incorporated into the study used for empirical analysis. As MCDM techniques, IVFF-based DEMATEL, AHP, and TOPSIS are first used, and then the IVFF-ARAS method and sensitivity analysis are used for performance evaluation and verification. The findings of this study have several ramifications for legislators, businesspeople, technologists, and practitioners. These stakeholders can concentrate on these weaknesses in the future and provide longer-lasting solutions.
Ovaj rad istražuje ulogu kriptovaluta u optimizaciji portfelja koristeći Markowitzev model moderne teorije portfelja. Cilj je bio analizirati u kojoj mjeri uključivanje kriptovaluta može poboljšati omjer rizika i prinosa u odnosu na portfelje koji se sastoje isključivo od tradicionalnih instrumenata. Analiza je provedena na odabranim kriptovalutama (Bitcoin, Ethereum, Solana, Binance Coin i XRP) te na dva tradicionalna tržišna indeksa (S&P 500 (SPY) i CROBEX), u razdoblju od 2023. do 2025. godine. Metodološki pristup uključivao je izračun logaritamskih tjednih prinosa, varijanci i standardnih devijacija, te konstrukciju matrice kovarijanci i korelacija između instrumenata. Na temelju tih podataka definirano je pet različitih portfelja s različitim razinama izloženosti kriptovalutama: portfelj sastavljen isključivo od kriptovaluta, kombinirani portfelj kriptovaluta i indeksa, defenzivni portfelj s većinskim udjelom indeksa, portfelj temeljen na indeksima i Bitcoinu, te diverzificirani portfelj u kojem svaki instrument ima minimalni udio. Za svaki portfelj konstruirana je efikasna granica kako bi se prikazao odnos rizika i prinosa. Rezultati pokazuju da kriptovalute, zbog svoje visoke volatilnosti, značajno povećavaju rizik portfelja, ali istodobno omogućuju postizanje viših očekivanih prinosa. Najveći potencijal za smanjenje rizika i postizanje uravnoteženog odnosa rizika i prinosa pokazali su portfelji koji uključuju kombinaciju tradicionalnih indeksa i Bitcoina, dok su diverzificirani portfelji ponudili najbolju stabilnost. Portfelj sastavljen isključivo od kriptovaluta, iako je donosio najviše prinose, bio je izložen i najvećem riziku, što potvrđuje da kriptovalute same po sebi nisu dovoljne za optimalnu strategiju ulaganja. Zaključno, rad pokazuje da kriptovalute mogu igrati važnu ulogu u optimizaciji portfelja, ali prvenstveno u kombinaciji s tradicionalnim instrumentima. Na taj način mogu pridonijeti diversifikaciji i poboljšanju efikasnosti portfelja, dok istodobno smanjuju ekstremne rizike povezane s isključivim oslanjanjem na kriptovalute.
The cryptocurrency market offers attractive but risky investment opportunities, characterized by rapid growth, extreme volatility, and uncertainty. Traditional risk management models, which rely on probabilistic assumptions and historical data, often fail to capture the market’s unique dynamics and unpredictability. In response to these challenges, this paper introduces a novel portfolio optimization model tailored for the cryptocurrency market, leveraging a credibilistic CVaR framework. CVaR was chosen as the primary risk measure because it is a downside risk measure that focuses on extreme losses, making it particularly effective in managing the heightened risk of significant downturns in volatile markets like cryptocurrencies. The model employs credibility theory and trapezoidal fuzzy variables to more accurately capture the high levels of uncertainty and volatility that characterize digital assets. Unlike traditional probabilistic approaches, this model provides a more adaptive and precise risk management strategy. The proposed approach also incorporates practical constraints, including cardinality and floor and ceiling constraints, ensuring that the portfolio remains diversified, balanced, and aligned with real-world considerations such as transaction costs and regulatory requirements. Empirical analysis demonstrates the model’s effectiveness in constructing well-diversified portfolios that balance risk and return, offering significant advantages for investors in the rapidly evolving cryptocurrency market. This research contributes to the field of investment management by advancing the application of sophisticated portfolio optimization techniques to digital assets, providing a robust framework for managing risk in an increasingly complex financial landscape.