The stability of markets hosting leveraged exchange-traded products is governed not by any single product's loop gain but by the spectral radius of a loop-gain matrix, and scalar per-product monitoring underestimates system feedback by construction. Recent work measures the self-reinforcement of a leveraged fund's daily close rebalancing through a scalar loop gain and treats cross-asset spillovers as bias. We model complexes on correlated underlyings as a coupled feedback system with matrix gain L and show that scalar monitoring has two blind spots: (i) cycle amplification, since rho(L) >= max_i l_ii for nonnegative coupling, strict under two-way coupling; and (ii) transmitted displacement, which arises already under one-way coupling and is invisible to the receiver's own gain. We give a reduced-form estimator of L requiring only prices and public fund assets -- no signed order flow -- via cross-asset overnight reversals, reporting its measurement-convention sensitivity explicitly. In simulation the spectral radius is recovered with RMSE 0.005 at T=250, a lead-lag confounder yields a 2% false-alarm rate, and in a calibrated blind-spot configuration the scalar monitor reports "safe" and the matrix monitor "unsafe" on 100% of paths. In the 2026 Korean single-stock LETF episode we detect transmission from the SK Hynix complex into Samsung Electronics' closing price (DiD z=-2.82; exact randomization p=0.0055 against 182 control pairs), scaling with the sender's rebalancing capital; conservatively, about 41% of Samsung's closing displacement variance is imported -- invisible to its own "moderate" gain of 0.24. The same estimator returns nulls for the U.S. MSTR-Bitcoin-Coinbase complex, whose capital is comparable but whose closing venue is far deeper. Monitoring should be organized around the (complex x venue) matrix, not around products.
Proof-of-stake protocols lock tokens into staking positions, shrinking the tradable float. We develop a continuous-time model in which price-impact volatility is a convex decreasing function of the float. Two results emerge. Conditional return variance amplifies as the float contracts, with amplification accelerating in high-staking regimes. Protocol changes that shift the long-run staking target produce persistent volatility regime transitions, with convergence speed governed by protocol adjustment capacity. Liquid-staking tokens attenuate both effects, with attenuation increasing in their liquidity parameter.
We study permissionless spot--perpetual basis trading in decentralized finance as a collateral control problem. The strategy holds spot inventory, hedges directional exposure with a short perpetual, and allocates capital between spot inventory and derivative margin under on-chain liquidity and execution frictions. The paper delivers three results. First, it solves a static control problem for the collateral share and shows that the risk-constrained formulation provides a more robust operating benchmark relative to the economic optimum. In comparative calibration, the required collateral rises monotonically under volatility stress. The collateral is the lowest for BTC and increases significantly for long tail assets such as LINK and DOGE. Second, the paper derives an asymmetric dynamic extension in which the lower boundary of intervention is solvency driven, and the upper boundary is determined by a trade-off between carry-loss and the cost of rebalancing. Monte Carlo simulation shows that the lower boundary remains structurally relevant, whereas meaningful interior upper triggers survive mainly in the regimes with high carry and low costs. Third, the paper validates an execution-aware implementation with live routed execution and historical backtests. The execution layer shows that the realized wedges are significant, but become worse in the case of selling the basis. This justifies a minimum effective rebalancing size and a positive execution buffer. The historical validation shows that in the case of a fixed control rule the realized performance is predominantly explained by the funding environment.
This paper develops a deep reinforcement learning framework for cryptocurrency portfolio management in which transaction costs are derived from the Riemannian geometry of the underlying volatility model rather than assumed constant. A Proximal Policy Optimisation agent is trained on a reward function grounded in non-equilibrium thermodynamics: we use the free-energy Bellman equation, in which transaction costs are the geodesic slippage on the Fisher information manifold of a maximum-entropy Markov-switching GARCH model, and regime-transition costs are the Wasserstein-2 distance between the calm and turbulent return distributions. A thermodynamic Carnot bound on portfolio efficiency is established and empirically validated. Five hypotheses are tested across Bitcoin, Ethereum, Ripple, Litecoin, and Bitcoin Cash over January 2017 to March 2026. The geometric-cost agent achieves statistically superior Sharpe ratios relative to flat-fee baselines on four of five assets; portfolio turnover is reduced by 56 to 83 percent relative to signal-following; the thermodynamic friction point at which the agent prefers no-trade is asset-specific and ordered by turbulent half-life; a joint topological and geometric circuit breaker reduces Maximum Drawdown by 28 to 38 percent; and ablation confirms that every component of the observation vector contributes a statistically significant performance gain. The framework requires liquid cryptocurrency markets with validated parametric volatility models; transferability to other asset classes requires upstream recalibration.
Using a comprehensive dataset from Deribit, we show that Bitcoin options trading activity is concentrated around two distinct intraday periods: 8:00â9:00 GMT and 14:00â15:00 GMT, relative to other hours of the day. The latter peak coincides with the opening of the New York Stock Exchange and is largely absent on weekends, suggesting spillovers from traditional equity markets to the Bitcoin options market. In contrast, the concentration of trading activity around the 8:00â9:00 GMT period appears to be driven by investors rolling over and re-establishing expiring options around the 8:00 GMT settlement, as this effect persists on both weekdays and weekends, and is stronger on days with more expiring contracts and for contracts with shorter maturities. These findings highlight how institutional trading conventions shape intraday activity in cryptocurrency derivatives and provide the first systematic evidence of intraday patterns in Bitcoin options trading.
Rizqi Akbar Makarim, Desinta Maheswari, Aqila Dina Pramustiwi, Kartika Ayu Rahmawati · 5 authors
The volatility of cryptocurrency markets has increased substantially in recent years, particularly for Ethereum (ETH), which exhibits fat-tailed distributions and persistent volatility clustering that traditional linear models are unable to capture. This study aims to analyze and model the volatility of ETH/USD using high-frequency hourly data to determine the most appropriate volatility model for describing Ethereumâs intraday market dynamics. The dataset consists of 8,760 hourly closing prices from October 31, 2024 to October 31, 2025, obtained through the CryptoCompare API. The methodological framework includes data preprocessing, log-return transformation, stationarity analysis using the Augmented DickeyâFuller test, detection of heteroskedasticity via the ARCHâLM test, and estimation of several ARCH and GARCH model specifications. The results show that ETH/USD returns are stationary, non-normally distributed, and exhibit clear volatility clustering. Among the ARCH models, only ARCH(1) adequately captures short-term fluctuations, while ARCH(2) provides no additional benefit. In contrast, GARCH models demonstrate superior performance in capturing both short-term shocks and long-term persistence. Based on AIC, BIC, and log-likelihood values, GARCH(1,2) emerges as the best-performing model, offering the highest flexibility in representing Ethereumâs persistent and reactive volatility patterns. These findings confirm that ETH/USD volatility is predictable and can be modeled statistically. Future research may incorporate asymmetric GARCH extensions or external explanatory variables to improve predictive performance.
P Praveen Kumar, Dudimetla Pravalika, B. S. Dileep Kumar, Gattu Akshitha · 5 authors
The rapid advancement of blockchain technology has introduced new possibilities for secure digital ownership and transparent fundraising through Non-Fungible Tokens (NFTs).However, most existing charity platforms remain centralized, limiting transparency, accountability, and verifiable proof of donations.Donors often lack visibility into how funds are utilized, while reliance on intermediaries increases risks such as data manipulation, reduced auditability, and decreased trust.To address these issues, this work proposes a decentralized charity auction framework that leverages blockchain technology and NFT-based asset representation.The system is developed using the Django web framework integrated with Web3 infrastructure and smart contracts.In this model, each auction item is tokenized as a unique NFT, ensuring authenticity, traceability, and immutable ownership.The platform allows users to act as donors or auctioneers, enabling participation in NFT-based charity auctions.Users can place bids or contribute funds, with all transactions securely recorded on a blockchain ledger.At the end of each auction, NFT ownership is automatically transferred to the highest bidder or contributor, providing verifiable proof of participation.By removing intermediaries and incorporating a transparent, incentive-driven mechanism, the proposed system enhances donor trust and engagement.It ensures tamper-proof record-keeping and clear fund flow, strengthening accountability within charitable ecosystems.This framework demonstrates a scalable and efficient approach to modern fundraising, showcasing the potential of blockchain and NFTs in improving trust and transparency in charity applications.
This paper develops a deep reinforcement learning (DRL) framework for cryptocurrency portfolio management in which transaction costs are derived from the Riemannian geometry of the underlying volatility model rather than assumed constant. A Proximal Policy Optimisation (PPO) agent is trained on a reward function derived from non-equilibrium thermodynamics: the free-energy Bellman equation, in which (i) transaction costs are the geodesic slippage Sâ on the Fisher information manifold of a maximum-entropy Markov-switching GARCH model, and (ii) regime-transition costs are the Wasserstein-2 distance Wt between the calm and turbulent return distributions. The agent is embedded in the WOW-E-W quadrilogy, a four-paper research programme that integrates statistical mechanics, fluid dynamics, Riemannian information geometry, and thermodynamic control into a unified cryptocurrency risk architecture. The PPO agent observes an 11-dimensional state vector ot that combines turbulent-regime probabilities \( \hat{\xi}_t(2) \) and parameter estimates \( \hat{\theta}_t \) from a maximum-entropy Markov-switching GARCH model, a viscosity-filtered velocity signal ht and gate states zt, rt from a GRU viscosity filter, and the Fisher curvature Gt, Ricci scalar Îșt, Betti numbers ÎČ0,t, ÎČ1,t,Wasserstein dissipation Wt, and topological alarm dI(t) from the Riemannian execution geometry layer. The framework establishes a thermodynamic Carnot bound on portfolio efficiency: η †1 â Hturb/Hcalm, where Hturb and Hcalm are the maximum-entropy values of the turbulent and calm regime distributions. Five hypotheses are tested across Bitcoin, Ethereum, Ripple, Litecoin, and Bitcoin Cash over January 2017 to March 2026: the geometric-cost PPO agent achieves higher Sharpe ratio than Buy-and-Hold, Greedy signal-following, and flat-fee PPO baselines (bootstrap p < 0.05 for four of five assets); portfolio turnover is reduced by 56 to 83 percent relative to signal-following; the thermodynamic friction point at which the agent prefers no-trade is asset-specific and ranges from 0.6 percent (Bitcoin) to 1.8 percent (Ethereum), ordered by turbulent half-life (Spearman Ï = 0.94, p = 0.017); a joint topological and geometric circuit breaker reduces Maximum Drawdown by 28 to 38 percent; and ablation confirms that every component of ot contributes a statistically significant performance gain (Diebold-Mariano p < 0.05 for at least four of five assets per component). The framework requires liquid cryptocurrency markets with validated parametric volatility models; transferability to other asset classes requires upstream recalibration and is an explicitly bounded limitation.
We study passive scalar mixing by parallel shear flows in the presence of weak molecular diffusion. We recover the sharp uniform-in-diffusivity mixing rate for shear flows with finitely many critical points, recently proven in [1]. Our approach is based on the stochastic representation formula of the associated advection-diffusion equation and yields two short proofs. The first uses a stochastic integration-by-parts argument and gives optimal mixing under the weakest regularity assumption required in the zero-diffusion case, answering Question II in [1, Section 4]. The second adopts a dynamical systems perspective and provides a proof of shear-induced mixing that, to our knowledge, is new even in the zero-diffusivity setting.
Abstract With the introduction of spot Ethereum ETFs, Ethereum plays an increasingly important role in the cryptocurrency market. In this paper, we propose a Bayesian modelling framework incorporating a mixture copula for co-modelling Ethereum returns with Bitcoin or FTSE 100 returns. The mixture copula is designed as a combination of the Clayton copula and its three rotations, Frank, and Gaussian copulas. It provides substantial flexibility for handling a variety of dependency structures. The Bayesian approach offers the advantage of jointly estimating both the margins and copulas and simulating future returns in a coherent procedure. Using 10 different risk or risk-return measures, we provide updated empirical evidence on Ethereumâs role in both cryptocurrency and mixed portfolios. The analysis not only evaluates its diversification potential numerically but also sheds light on how the optimal allocations vary across distinct risk preferences and portfolio objectives. Moreover, based on the data of 2017â2024, we estimate that Ethereum futures has a hedging effectiveness on Bitcoin of about 30â40% across different risk preferences. Beyond these findings, the Bayesian mixture copula framework represents a methodological contribution to the modelling of complex dependence structures between financial returns. Taken together, our study delivers new insights that are particularly relevant in light of the evolving cryptocurrency landscape and the increasing integration of digital assets into mainstream investment practice.
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. Meanâvariance 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 naĂŻve 1/N portfolio often outperform meanâvariance 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 meanâvariance, 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 Greenâs 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 ItĂŽâs 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.
Cascading liquidations across decentralized finance (DeFi) lending protocols represent a systemic risk that standard empirical models often fail to capture. To quantify this phenomenon, we apply a 3-variate Hawkes process to model crossprotocol liquidation clustering among Aave V3, Compound V3, and Morpho on Ethereum. Using 7,500 on-chain liquidation events spanning 2023-01-01 through 2025-12-31, we estimate exponential triggering kernels via maximum likelihood estimation and validate the approach against nonparametric spectral estimates. The results indicate a stable, subcritical regime (Ï = 0.725) characterized by statistically significant off-diagonal excitation. The strongest cross-protocol channel runs from Morpho to Compound V3 (branching ratio Î = 0.418), while selfexcitation ratios range from 0.28 to 0.32. Directional predictive dependence tests confirm asymmetric spillover effects. Furthermore, likelihood-based comparisons demonstrate that crossprotocol excitation significantly outperforms self-excitation-only and common-factor baselines, including models with ETH return controls. Placebo permutations verify that this off-diagonal structure is not an artifact of shared timing. Ultimately, while the findings document robust cross-protocol clustering consistent with spillover channels, we emphasize that Hawkes crossexcitation captures directional predictive dependence rather than strict structural causation.
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.
We introduce a model-free structural framework to value liabilities of firms whose primary assets are digital assets typically Bitcoin. Our no-arbitrage approach prices convertible debts and extracts the risk-neutral probabilities of their terminal states using the market information of Bitcoin options, thereby bypassing the restrictive assumptions of traditional structural models. We perform comparative statics analysis to demonstrate how the resulting bond spreads and option values are structurally determined. We then test the framework in a real-world case study of MicroStrategy's convertible bonds, finding that it generates accurate, market-consistent valuations in an out-of-sample setting. Together, our theoretical and empirical results establish a robust, market-based blueprint for pricing the emerging class of crypto-backed credit in general.
We formulate and solve stochastic control problems that model the core yield-generating strategy of the Ethena protocol, a decentralized finance (DeFi) stablecoin that earns yield by combining a long position in staked Ethereum (stETH) with an equal-sized short position in ETH perpetual futures. The combined position is delta-neutral with respect to the ETH spot price, yet earns carry from two sources: staking rewards on the stETH leg, and funding-rate payments received from long perpetual holders when the perpetual trades at a premium to spot. A key feature of our model is that the control -- the rate of simultaneously buying stETH and shorting the perpetual -- exerts two distinct types of price impact. \textit{Permanent} impact shifts the mid-market prices of both legs, compressing the basis and permanently eroding future funding income. \textit{Temporary} impact reflects execution slippage on each leg. We study both an infinite-horizon discounted problem and a finite-horizon problem in which the protocol maximizes total wealth up to a fixed date $T$, subject to a terminal cost for liquidating any remaining position. In both cases the optimal control is obtained explicitly.
This paper studies the benefits of timing Bitcoin returns by upside and downside volatilities. Standard volatility management implicitly treats volatility spikes as signals of adverse states, reducing exposure when total volatility increases. However, in Bitcoin, volatility spikes are frequently due to price rallies, which typically indicate subsequent positive returns. We show that semivolatility timing rules that account for both downside and upside risk concerns yield substantially stronger risk-adjusted performance than buy-and-hold and volatility-managed strategies. This stems from the fact that high upside-driven volatility states in Bitcoin are disproportionately associated with positive returns in the next period.
This paper measures price differences between Hegic option quotes on Arbitrum and a model-based benchmark built on Black--Scholes model with regime-sensitive volatility estimated via a two-regime MS-AR-(GJR)-GARCH model. Using option-level feasible GLS, we find benchmark prices exceed Hegic quotes on average, especially for call options. The price spread rises with order size, strike, maturity, and estimated volatility, and falls with trading volume. By underlying, wrapped Bitcoin options show larger and more persistent spreads, while Ethereum options are closer to the benchmark. The framework offers a data-driven analysis for monitoring and calibrating on-chain option pricing logic.
Abstract This paper considers option valuation under finite mixture models in a discrete-time economy. Specifically, the Esscher transform is employed to select a pricing kernel. Novel finite mixture models with negative-shifted Gamma and negative-shifted inverse Gaussian distributions are developed. A hybrid finite mixture model that allows different parametric forms for component distributions is introduced to incorporate model uncertainty. An empirical characteristic function estimation method is employed to estimate the finite mixture models. Closed-form pricing formulas for a European call option are obtained for some finite mixture models. Empirical examples using data on the Bitcoin-USD prices are provided to illustrate an application of the proposed models to value Bitcoin options.
The aim of this thesis is to examine the pricing and efficiency of Bitcoin options. It reviews theories of market efficiency and considers how effectively these frameworks apply to cryptocurrency markets. The thesis examines multiple option pricing models by comparing their performance for pricing Bitcoin options. Bitcoinâs high volatility and the relatively young age of its market development highlight the need to analyze how these characteristics influence both option pricing and overall market efficiency. In addition, the thesis examines the characteristics of Bitcoin options. The study provides guidelines for future research and market development, helping to build trust and support the integration of cryptocurrency derivatives into the broader financial system. TĂ€mĂ€n opinnĂ€ytetyön tavoitteena on tarkastella Bitcoin-optioiden hinnoittelua ja markkinoiden tehokkuutta. TyössĂ€ kĂ€ydÀÀn lĂ€pi markkinatehokkuuden teorioita ja arvioidaan, kuinka hyvin nĂ€mĂ€ viitekehykset soveltuvat kryptovaluuttamarkkinoihin. OpinnĂ€ytetyössĂ€ tarkastellaan useita optioiden hinnoittelumalleja vertailemalla niiden toimi- vuutta Bitcoin-optioiden hinnoittelussa. Bitcoinin korkea volatiliteetti ja sen markkinoiden suhteellisen varhaisessa kehitysvaiheessa oleva tila korostavat tarvetta analysoida, miten nĂ€mĂ€ ominaisuudet vaikuttavat sekĂ€ optioiden hinnoitteluun ettĂ€ markkinoiden yleiseen tehokkuuteen. LisĂ€ksi työssĂ€ tarkastellaan Bitcoin-optioiden erityispiirteitĂ€. Tutkimus tarjoaa suuntaviivoja tu- levalle tutkimukselle ja markkinoiden kehittĂ€miselle, ja sen tavoitteena on lisĂ€tĂ€ luottamusta sekĂ€ tukea kryptovaluuttajohdannaisten integroitumista laajempaan finanssijĂ€rjestelmÀÀn.