Abstract Conditional risk measures and their associated risk contribution measures are commonly employed in finance and actuarial science for evaluating systemic risk and quantifying the effects of risk interactions. This paper introduces various types of contribution ratio measures based on the multivariate conditional value-at-risk (MCoVaR), multivariate conditional expected shortfall (MCoES), and multivariate marginal mean excess (MMME) studied in [34] (Ortega-Jiménez, P., Sordo, M., & Suárez-Llorens, A. (2021). Stochastic orders and multivariate measures of risk contagion. Insurance: Mathematics and Economics , vol. 96, 199–207) and [11] (Das, B., & Fasen-Hartmann, V. (2018). Risk contagion under regular variation and asymptotic tail independence. Journal of Multivariate Analysis 165 (1), 194–215) to assess the relative effects of a single risk when other risks in a group are in distress. The properties of these contribution risk measures are examined, and sufficient conditions for comparing these measures between two sets of random vectors are established using univariate and multivariate stochastic orders and statistically dependent notions. Numerical examples are presented to validate these conditions. Finally, a real dataset from the cryptocurrency market is used to analyze the spillover effects through our proposed contribution measures.
Options are fundamental to blockchain-based financial services, offering essential tools for risk management and price speculation, which enhance liquidity, flexibility, and market efficiency in decentralized finance (DeFi). Despite the growing interest in options for blockchain-resident assets, such as cryptocurrencies, current option mechanisms face significant challenges, including a high reliance on trusted third parties, limited asset support, high trading delays, and the requirement for option holders to provide upfront collateral. In this paper, we present a protocol that addresses the aforementioned issues. Our protocol is the first to eliminate the need for holders to post collateral when establishing options in trustless service environments (i.e. without a cross-chain bridge), which is achieved by introducing a guarantee from the option writer. Its universality allows for cross-chain options involving nearly \textit{any} assets on \textit{any} two different blockchains, provided the chains' programming languages can enforce and execute the necessary contract logic. Another key innovation is reducing option position transfer latency, which uses Double-Authentication-Preventing Signatures (DAPS). Our evaluation demonstrates that the proposed scheme reduces option transfer latency to less than half of that in existing methods. Rigorous security analysis proves that our protocol achieves secure option trading, even when facing adversarial behaviors.
The emergence of decentralized finance (DeFi) has revolutionized traditional financial markets, offering greater financial inclusion, transparency, and user autonomy. This paper investigates the creation of a coredao.vip crypto-based liquidity pool, allowing liquidity providers to profit similarly to traditional brokers. By leveraging blockchain-based liquidity pools, coredao.vip enables participants to contribute assets to decentralized platforms and earn rewards based on the trading activity within the pool. This research outlines the mechanism of liquidity pool creation, explores its profitability for liquidity providers, and analyzes the associated risks and benefits. The study further examines how decentralized liquidity protocols can democratize access to financial services while addressing liquidity challenges.
The paper proposes and implements a methodology to fit a seven-parameter Generalized Tempered Stable (GTS) distribution to financial data. The nonexistence of the mathematical expression of the GTS probability density function makes the maximum likelihood estimation (MLE) inadequate for providing parameter estimations. Based on the function characteristic and the fractional Fourier transform (FRFT), we provide a comprehensive approach to circumvent the problem and yield a good parameter estimation of the GTS probability. The methodology was applied to fit two heavily tailed data (Bitcoin and Ethereum returns) and two peaked data (S\&P 500 and SPY ETF returns). For each index, the estimation results show that the six-parameter estimations are statistically significant except for the local parameter, $μ$. The goodness-of-fit was assessed through Kolmogorov-Smirnov, Anderson-Darling, and Pearson's chi-squared statistics. While the two-parameter geometric Brownian motion (GBM) hypothesis is always rejected, the GTS distribution fits significantly with a very high p-value; and outperforms the Kobol, Carr-Geman-Madan-Yor, and Bilateral Gamma distributions.
This paper aims to leverage Bayesian nonlinear expectations to construct Bayesian lower and upper estimates for prices of Ether options, that is, options written on Ethereum, with conditional heteroscedasticity and model uncertainty. Specifically, a discrete-time generalized conditional autoregressive heteroscedastic (GARCH) model is used to incorporate conditional heteroscedasticity in the logarithmic returns of Ethereum, and Bayesian nonlinear expectations are adopted to introduce model uncertainty, or ambiguity, about the conditional mean and volatility of the logarithmic returns of Ethereum. Extended Girsanov’s principle is employed to change probability measures for introducing a family of alternative GARCH models and their risk-neutral counterparts. The Bayesian credible intervals for “uncertain” drift and volatility parameters obtained from conjugate priors and residuals obtained from the estimated GARCH model are used to construct Bayesian superlinear and sublinear expectations giving the Bayesian lower and upper estimates for the price of an Ether option, respectively. Empirical and simulation studies are provided using real data on Ethereum in AUD. Comparisons with a model incorporating conditional heteroscedasticity only and a model capturing ambiguity only are presented.
Traditional volatility models do not work well when volatility changes rapidly and in the presence of outliers. Therefore, two lines of improvements have been developed separately in the existing literature. Range-based models benefit from efficient volatility estimates based on low and high prices, while robust methods deal with outliers. We propose a range-based GARCH model with a bounded M-estimator, which combines these two improvements with a third new improvement: a modified robust method, which adds elasticity in treating the outliers. We apply this model to Bitcoin , Ethereum Classic, Ethereum, and Litecoin and find that it forecasts variances, value at risk, and expected shortfall more accurately than the standard GARCH model, the standard range-based GARCH model, and the GARCH model with the robust estimation. Utilization of high and low prices joined with a novel treatment of outliers makes our model perform well during extreme periods when traditional volatility models fail.
Abstract Modeling implied volatility (IV) is important for option pricing, hedging, and risk management. Previous studies of deterministic implied volatility functions (DIVFs) propose two parameters, moneyness and time to maturity, to estimate implied volatility. Recent DIVF models have included factors such as a moving average ratio and relative bid-ask spread but fail to enhance modeling accuracy. The current study offers a generalized DIVF model by including a momentum indicator for the underlying asset using a relative strength index (RSI) covering multiple time resolutions as a factor, as momentum is often used by investors and speculators in their trading decisions, and in contrast to volatility, RSI can distinguish between bull and bear markets. To the best of our knowledge, prior studies have not included RSI as a predictive factor in modeling IV. Instead of using a simple linear regression as in previous studies, we use a machine learning regression algorithm, namely random forest, to model a nonlinear IV. Previous studies apply DVIF modeling to options on traditional financial assets, such as stock and foreign exchange markets. Here, we study options on the largest cryptocurrency, Bitcoin, which poses greater modeling challenges due to its extreme volatility and the fact that it is not as well studied as traditional financial assets. Recent Bitcoin option chain data were collected from a leading cryptocurrency option exchange over a four-month period for model development and validation. Our dataset includes short-maturity options with expiry in less than six days, as well as a full range of moneyness, both of which are often excluded in existing studies as prices for options with these characteristics are often highly volatile and pose challenges to model building. Our in-sample and out-sample results indicate that including our proposed momentum indicator significantly enhances the model’s accuracy in pricing options. The nonlinear machine learning random forest algorithm also performed better than a simple linear regression. Compared to prevailing option pricing models that employ stochastic variables, our DIVF model does not include stochastic factors but exhibits reasonably good performance. It is also easy to compute due to the availability of real-time RSIs. Our findings indicate our enhanced DIVF model offers significant improvements and may be an excellent alternative to existing option pricing models that are primarily stochastic in nature.
This paper is concerned with a class of linear-quadratic stochastic large-population problems with partial information, where the individual agent only has access to a noisy observation process related to the state. The dynamics of each agent follows a linear stochastic differential equation driven by individual noise, and all agents are coupled together via the control average term. Using the mean-field game approach and the backward separation principle with a state decomposition technique, the decentralized optimal control can be obtained in the open-loop form through a forward-backward stochastic differential equation with the conditional expectation. The optimal filtering equation is also provided. By the decoupling method, the decentralized optimal control can also be further presented as the feedback of state filtering via the Riccati equation. The explicit solution of the control average limit is given, and the consistency condition system is discussed. Moreover, the related $\varepsilon$-Nash equilibrium property is verified. To illustrate the good performance of theoretical results, an example in finance is studied.
Reducing financial risk is of paramount importance to investors, financial institutions, and corporations. Since the pioneering contribution of Johnson (1960), the optimal hedge ratio based on futures is regularly utilized. The current paper suggests an explicit and efficient method for testing the null hypothesis of a symmetric optimal hedge ratio against an asymmetric alternative one within a multivariate setting. If the null is rejected, the position dependent optimal hedge ratios can be estimated via the suggested model. This approach is expected to enhance the accuracy of the implemented hedging strategies compared to the standard methods since it accounts for the fact that the source of risk depends on whether the investor is a buyer or a seller of the risky asset. An application is provided using spot and futures prices of Bitcoin. The results strongly support the view that the optimal hedge ratio for this cryptocurrency is position dependent. The investor that is long in Bitcoin has a much higher conditional optimal hedge ratio compared to the one that is short in the asset. The difference between the two conditional optimal hedge ratios is statistically significant, which has important repercussions for implementing risk management strategies.
This study recovers the Bitcoin option-implied risk aversion by jointly estimating a cross-sectional dataset of option prices and time-series data of realized returns on underlying asset prices. The empirical analysis of Bitcoin options on Deribit shows that the risk aversion function exhibits a peak shape, with the level of implied risk aversion of Bitcoin options ranging from −0.2 to 0.05, which is significantly lower than that of the traditional options market; furthermore, maturity affects the level of option-implied risk aversion, with shorter maturity implying higher risk-aversion levels. Moreover, our research indicates that after halving of Bitcoin, investors’ risk aversion function becomes higher and steeper than before.
Di Zhang, Youzhou Zhou, Hengyan Liu, Bintao Hu · 6 authors
The goal of cryptocurrencies is decentralization, but it is impractical to set up a trading market between every two currencies. To solve this optimization problem, we use a two-stage process: 1) Fill in missing values based on a regularized, truncated eigenvalue decomposition, where the regularization term is used to control what extent missing values should be limited to zero. 2) Search for the optimal trading pairs, based on a branch and bound process, with heuristic search and pruning strategies.The experimental results show that: 1) If the number of denominated coins is not limited, we will get a more decentralized trading pair setting, which advocates the establishment of trading pairs directly between large currency pairs. 2) There is a certain room for optimization in all exchanges. The setting of inappropriate trading pairs is mainly caused by subjectively setting small coins to quote or failing to track emerging big coins in time. 3) Too few trading pairs will lead to low coverage; too many trading pairs will need to be adjusted with markets frequently. Exchanges should consider striking an appropriate balance between them.
The Financial Risk Meter (FRM) employs Quantile-LASSO regression to identify systemic financial risk and dependencies among tail events across financial assets. This paper establishes, both theoretically and empirically, a meaningful economic relationship between the FRM index, derived from the penalization parameter in quantile LASSO regression, and the volatility of assets' pricing kernels, the attainable maximal Sharpe ratio, and market volatility. Despite the rapid growth of the crypto market and its increasing integration with traditional financial markets, there remains a dearth of risk measures in this space. FRM@Crypto exhibits robust predictive capabilities in anticipating future market risk, potentially filling a critical void in this market.
Currently, the most liquidly traded options on the crypto underlying are the so-called inverse options. An inverse option contract is quoted and traded in the units of the underlying cryptocurrency. The main economic reason for the popularity of inverse contracts in the crypto exchanges (such as Deribit) is that inverse contracts enable traders to operate without maintaining fiat cash accounts. For the theoretical part, we show that inverse options are just regular vanilla options considered under the martingale measure using the forward of the underlying as the numéraire. This measure requires an adjustment to option delta. For the empirical part, we use Deribit options data of past five years to backtest delta-hedged option strategies. We introduce USD and Coin accounting of trading Profit&Loss (P&L) which is important for designing strategies in crypto options. We show empirically that USD and Coin accounting rules are equivalent when performance is measured is Coin and USD units, respectively. We establish that the risk-premia observed in options on Deribit is negative and significant so that strategies selling volatility are expected to generate positive risk-adjusted performance in the long-term.
Current digital payment solutions are fragile and offer less privacy than traditional cash. Their critical dependency on an online service used to perform and validate transactions makes them void if this service is unreachable. Moreover, no transaction can be executed during server malfunctions or power outages. Due to climate change, the likelihood of extreme weather increases. As extreme weather is a major cause of power outages, the frequency of power outages is expected to increase. The lack of privacy is an inherent result of their account-based design or the use of a public ledger. The critical dependency and lack of privacy can be resolved with a Central Bank Digital Currency that can be used offline. This thesis proposes a design and a first implementation for an offline-first digital euro. The protocol offers complete privacy during transactions using zero-knowledge proofs. Furthermore, transactions can be executed offline without third parties and retroactive double-spending detection is facilitated. To protect the users' privacy, but also guard against money laundering, we have added the following privacy-guarding mechanism. The bank and trusted third parties for law enforcement must collaborate to decrypt transactions, revealing the digital pseudonym used in the transaction. Importantly, the transaction can be decrypted without decrypting prior transactions attached to the digital euro. The protocol has a working initial implementation showcasing its usability and demonstrating functionality.
This study introduces closed-form formulas for valuing European call options, assuming that Bitcoin follows a compound Poisson process. Additionally, instantaneous forward interest rates are considered in the Heath-Jarrow-Morton model, which includes a jump component. To address the impacts of systematic risk on Bitcoin price and interest rate, we model two stochastic processes using a correlated bivariate jump-diffusion model to capture individual jumps and systematic co-jumps. This study provides analytic formulas for pricing Bitcoin call options and zero-coupon bonds under the correlated jump-diffusion Heath-Jarrow-Morton model. Numerical analysis shows how co-jump intensity affects the prices of both zero-coupon bonds and Bitcoin call options. We specifically look at how these prices change in response to co-jump intensity across three different instantaneous forward rate term structures. The findings show that the prices of Bitcoin call options are contingent on the term structure types of zero-coupon bonds. In addition, the interaction of co-jump intensity and types of term structure also affects Bitcoin option prices. The practical significance of this study is to provide a comprehensive model to evaluate Bitcoin call options and enhance risk management strategies in the Bitcoin market when the Bitcoin market encounters changes in monetary policy or changes in macroeconomic conditions.
Srisht Fateh Singh, Panagiotis Michalopoulos, Andreas Veneris
This paper investigates the current state of option trading platforms for cryptocurrencies, encompassing both centralized and decentralized exchanges. Option contracts in cryptocurrency markets offer functionalities akin to traditional markets, providing investors with tools to mitigate risks, particularly those arising from price volatility. The paper discusses these applications of option contracts in the context of decentralized finance, emphasizing their utility in managing market uncertainties. Despite a recent surge in the trading volume of option contracts on cryptocurrencies, decentralized platforms account for less than $1 \%$ of this total volume. Hence, this paper takes a closer look by examining the design choices of these platforms to understand the challenges hindering their growth and adoption. It identifies technical, financial, and adoption-related challenges faced by decentralized exchanges. Subsequently, the paper provides commentary on existing platform responses.
Abstract In the FinTech era, we contribute to the literature by studying the pricing of Bitcoin options, which is timely and important given that both Nasdaq and the CME Group have started to launch a variety of Bitcoin derivatives. We find pricing errors in the presence of market smiles in Bitcoin options, especially for short-maturity ones. Long-maturity options display more of a “smirk” than a smile. Additionally, the ARJI-EGARCH model provides a better overall fit for the pricing of Bitcoin options than the other ARJI-GARCH type models. We also demonstrate that the ARJI-GARCH model can provide more precise pricing of Bitcoin and its options than the SVCJ model in term of the goodness-of-fit in forecasting. Allowing for jumps is crucial for modeling Bitcoin options as we find evidence of time-varying jumps. Our empirical results demonstrate that the realized jump variation can describe the volatility behavior and capture the jump risk dynamics in Bitcoin and its options.
Abootaleb Shirvani, Stefan Mittnik, W. Brent Lindquist, Svetlozar T. Rachev
We propose a doubly subordinated Lévy process, the normal double inverse Gaussian (NDIG), to model the time series properties of the cryptocurrency bitcoin. By using two subordinated processes, NDIG captures both the skew and fat-tailed properties of, as well as the intrinsic time driving, bitcoin returns and gives rise to an arbitrage-free option pricing model. In this framework, we derive two bitcoin volatility measures. The first combines NDIG option pricing with the Chicago Board Options Exchange VIX model to compute an implied volatility; the second uses the volatility of the unit time increment of the NDIG model. Both volatility measures are compared to the volatility based on the historical standard deviation. With appropriate linear scaling, the NDIG process perfectly captures the observed in-sample volatility.
Abstract The growing interest in cryptocurrencies has brought this new means of exchange to the attention of the financial world. This study aims to investigate the effects that a cryptocurrency can have when it is considered as a financial asset. The analysis is carried out from an ex‐post perspective, evaluating the performance achieved in a certain period by three different portfolios. These are the one composed only of equities, bonds and commodities, the second one only of cryptocurrencies, and the third one is a combination of these both ones and thus made up of all considered “traditional” assets and the most performing cryptocurrency of the second portfolio. For these purposes, the classic variance‐covariance approach is applied where the calculation of the risk structure is done via the GARCH‐Copula and GARCH‐Vine Copula approaches. The optimal weights of the assets in the optimized portfolios are determined through Markowitz optimization problem. The analysis mainly showed that the portfolio composed of cryptocurrency and traditional assets has a higher Sharpe index, from an ex‐post perspective, and more stable performances, from an ex‐ante perspective. We justify our selection of the Markowitz approach over conditional VaR and expected shortfall due to their heightened sensitivity to unsystematic extreme events in crypto markets.
This study examines the dynamics between treasury and market capitalization in two Decentralized Autonomous Organization (DAO) projects: OlympusDAO and KlimaDAO. This research examines the relationship between market capitalization and treasuries in these projects using vector autoregression (VAR), Granger causality, and Vector Error Correction models (VECM), incorporating an exogenous variable to account for the comovement of decentralized finance assets. Additionally, a Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model is employed to assess the impact of carbon offset tokens on KlimaDAO’s market capitalization returns’ conditional variance. The findings suggest a connection between market capitalization and treasuries in the analyzed projects, underscoring the importance of the treasury and carbon offset tokens in impacting a DAO’s market capitalization and variance. Additionally, the results suggest significant implications for predictive modeling, highlighting the distinct behaviors observed in OlympusDAO and KlimaDAO. Investors and policymakers can leverage these results to refine investment strategies and adjust treasury allocation strategies to align with market trends. Furthermore, this study addresses the importance of responsible investing, advocating for including sustainable investment assets alongside a foundational framework for informed investment decisions and future studies in the field, offering novel insights into decentralized finance dynamics and tokenized assets’ role within the crypto-asset ecosystem.