Abstract We develop a continuousâtime control approach to optimal trading in a ProofâofâStake (PoS) blockchain, formulated as a consumptionâinvestment problem that aims to strike the optimal balance between a participant's (or agent's) utility from holding/trading stakes and utility from consumption. We present solutions via dynamic programming and the HamiltonâJacobiâBellman (HJB) equations. When the utility functions are linear or convex, we derive closeâform solutions and show that the bangâbang strategy is optimal (i.e., always buy or sell at full capacity). Furthermore, we bring out the explicit connection between the rate of return in trading/holding stakes and the participant's riskâadjusted valuation of the stakes. In particular, we show when a participant is riskâneutral or riskâseeking, corresponding to the riskâadjusted valuation being a martingale or a subâmartingale, the optimal strategy must be to either buy all the time, sell all the time, or first buy then sell, and with both buying and selling executed at full capacity. We also propose a riskâcontrol version of the consumptionâinvestment problem; and for a special case, the âstakeâparityâ problem, we show a meanâreverting strategy is optimal.
Decentralized Finance (DeFi) aims to use advancements in both computation and cryptography to tackle standard economic problems. It must, therefore, operate within the intersection of constraints required by both the computer science and economic domains. We explore a foundational question at the junction of those fields: is it possible to synthesize variable market-clearing risk-free yield for native tokens via smart contracts? We show using a stylized model representing a large class of existing decentralized consensus algorithms that this is not possible. This places strong bounds on what decentralized financial products can be built and constrains the shape of future developments in DeFi. Among other limitations, our results reveal that markets in DeFi are incomplete.
In this paper, we conduct a fast calibration in the jump-diffusion model to capture the Bitcoin price dynamics, as well as the behavior of some components affecting the price itself, such as the risk of pitfalls and its ambiguous effect on the evolution of Bitcoinâs price. In addition, in our study of the Bitcoin option pricing, we find that the inclusion of jumps in returns and volatilities are significant in the historical time series of Bitcoin prices. The benefits of incorporating these jumps flow over into option pricing, as well as adequately capture the volatility smile in option prices. To the best of our knowledge, this is the first work to analyze the phenomenon of price jump risk and to interpret Bitcoin option valuation as âexceptionally ambiguousâ. Crucially, using hedging options for the Bitcoin market, we also prove some important properties: Bitcoin options follow a convex, but not strictly convex function. This property provides adequate risk assessment for convex risk measure.
Abootaleb Shirvani, Stefan Mittnik, W. Brent Lindquist, Svetlozar T. Rachev
We propose a doubly subordinated Levy process, NDIG, to model the time series properties of the cryptocurrency bitcoin. NDIG captures the skew and fat-tailed properties of bitcoin prices 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 Cboe VIX model to compute an implied volatility; the second uses the volatility of the unit time increment of the NDIG model. Both are compared to a volatility based upon historical standard deviation. With appropriate linear scaling, the NDIG process perfectly captures observed, in-sample, volatility.
This paper investigates a class of unified stochastic linear-quadratic-Gaussian (LQG) social optima problems involving a large number of weakly-coupled interactive agents under a generalized setting. For each individual agent, the control and state process enters both diffusion and drift terms in its linear dynamics, and the control weight might be indefinite in cost functional. This setup is innovative and has great theoretical and realistic significance as its applications in mathematical finance (e.g., portfolio selection in mean-variation model). Using some fully-coupled variational analysis under the person-by-person optimality principle, and the mean-field approximation method, the decentralized social control is derived by a class of new type consistency condition (CC) system for typical representative agent. Such CC system is some mean-field forward-backward stochastic differential equation (MF-FBSDE) combined with embedding representation. The well-posedness of such forward-backward stochastic differential equation (FBSDE) system is carefully examined. The related social asymptotic optimality is related to the convergence of the average of a series of weakly-coupled backward stochastic differential equation (BSDE). They are verified through some Lyapunov equations.
We apply the Markowitz mean-variance framework to assess risk-return benefits of cryptocurrency-portfolios. Using daily data of the three major cryptocurrencies for the time span 1/1/2019 to 27/04/2021, we relate risk and return of different mean-variance portfolio strategies to Bitcoin, Etherium, Ripple and BIST 30 benchmark. We find that combining cryptocurrencies crowds out BIST 30 index to maximize return and Sharpe ratio while cryptocurrencies are crowded out if the optimization problem is changed to a risk minimization problem rather than a return maximization problem. Furthermore, according to rolling-window approach shift from Bitcoin to Etherium is important.
A model is proposed for Bitcoin prices that takes into account market attention. Market attention, modeled by a mean-reverting Cox-Ingersoll-Ross processes, affects the volatility of Bitcoin returns, with some delay. The model is affine and tractable, with closed formulae for the conditional characteristic functions with respect to both the conventional and a delayed filtration. This leads to semi-closed formulae for European call and put prices. A maximum likelihood estimation procedure is provided, as well as a method for changing to a risk-neutral measure. The model compares very well against classical and attention-based models when tested on real data.
In this paper, the HestonâNandi futures option pricing model is applied to Bitcoin futures options. The model prices are compared to market prices to give an indication of the pricing performance. In addition, a multivariate Bitcoin futures option pricing methodology based on a multivatiate GARCH model is developed. The empirical results show that a symmetric model is a better fit when applied to Bitcoin futures returns, and also produces more accurate option prices compared to market prices for two out of three expiry dates considered.
Unter Verwendung eines neuen Datensatzes von Deribit, einer der gröĂten Bitcoin Derivate Börsen, werden Bitcoin Pricing Kernel berechnet. Diese ermöglichen die arbitragefreie Bepreisung neuer Instrumente. State Price Densities werden mit Rookleys Methode geschĂ€tzt. Der zugrundeliegende Bitcoin Asset Prozess wird als ein SVCJ Modell betrachtet. Die geschĂ€tzten Pricing Kernel werden in einem forminvarianten Kernel zusammengefasst. Mit der Struktur der Pricing Kernel werden Marktineffizienzen gefunden, denen mit einer simulierten Handelsstrategie entgegengewirkt werden kann.
This paper studies the dynamics of cryptocurrency volatility using a stochastic volatility model with simultaneous and correlated jumps in returns and volatility. We estimate the model using an efficient sequential learning algorithm that allows for learning about multiple unknown model parameters simultaneously, with daily data on four popular cryptocurrencies. We find that these cryptocurrencies have quite different volatility dynamics. In particular, they exhibit different return-volatility relationships: While Ethereum and Litecoin show a negative relationship, Chainlink displays a positive one and interestingly, Bitcoinâs one changes from negative to positive in June 2016. We also provide evidence that the sequential learning algorithm helps better detect large jumps in the cryptocurrency market in real time. Overall, incorporating volatility jumps helps better capture the dynamic behavior of highly volatile cryptocurrencies.
Weihao Han, David Newton, Emmanouil Platanakis, Charles Sutcliffe · 5 authors
Cryptocurrency returns are highly non-normal, casting doubt on the standard performance metrics. We apply almost stochastic dominance (ASD), which does not require any assumption about the return distribution or degree of risk aversion. From 29 long-short cryptocurrency factor portfolios, we find eight that dominate our four benchmarks. Their returns cannot be fully explained by the three-factor coin model of Liu et al. (2022). So we develop a new three-factor model where momentum is replaced by a mispricing factor based on size and risk-adjusted momentum, which significantly improves pricing performance.
In the current paper, we develop a methodology to price lookback options for cryptocurrencies. We propose a discretely monitored window average lookback option, whose monitoring frequencies are randomly selected within the time to maturity, and whose monitoring price is the average asset price in a specified window surrounding the instant. We price these options whose underlying asset is the CCI30 index of various Cryptocurrencies, as opposed to a single cryptocurrency, with the intention of reducing volatility, and thus, the option price. We employ the Normal Inverse Gaussian (NIG) and Rough Fractional Stochastic Volatility (RFSV) models to the cryptocurrency market and using the Black-Scholes as the benchmark model. In doing so, we intend to capture the extreme characteristics such as jumps and volatility roughness for cryptocurrency price fluctuations. Since there is no availability of a closed-form solution for lookback option prices under these models, we utilize the Monte Carlo simulation for pricing and augment it using the antithetic method for variance reduction. Finally, we present the simulation results for the lookback options and compare the prices resulting from using the NIG model, RFSV model with those from the Black-Scholes model. We found that the option price is indeed lower for our proposed window average lookback option than for a traditional lookback option. We found the Hurst parameter to be H = 0.09 which confirms that the cryptocurrencies market is indeed rough.
The objective of this study is, to show the importance of incorporating jumps in both returns and volatility dynamics for Bitcoin. For that purpose, we introduce the Double Exponential Jump-Diffusion model with Stochastic Volatility (DEJDSVJ) that contains asymmetric jumps. The use of the Markov Chain Monte Carlo methods for estimation has proved the meaningful presence of jumps in Bitcoin price and volatility. Moreover, based on the Bitcoin options market, a comparison between the underlying model, the Double Exponential Jump Diffusion model (DEJD) with Stochastic Volatility (no Jumps) and the Stochastic Volatility (SV) shows the goodness of the DEJDSVJ modelâs calibration over others for pricing Bitcoin options.
Yizhou Cao, Min Dai, Steven Kou, Lewei Li · 5 authors
Abstract Existing cryptocurrencies are too volatile to be used as currencies for daily payments. Stablecoins, which are cryptocurrencies pegged to other stable financial assets such as the US dollar, are desirable for payments within blockchain networks, whereby being often called the âHoly Grail of cryptocurrency.â By using the option pricing theory and the Ethereum platform that allows running smart contracts, we design several dualâclass structures that are written on the ETH cryptocurrency and offer a fixedâincome crypto asset (Class A coin), a stablecoin (Class AâČ coin) pegged to a traditional currency, and leveraged investment instruments (Class B and BâČ coins). Our investigation of the values of stablecoins in the presence of jump risk and black swanâtype events shows the robustness of the design. The design has been implemented on the Ethereum platform.
Lili Matic, Natalie Packham, Wolfgang Karl HĂ€rdle
The cryptocurrency market is volatile, non-stationary and non-continuous. Together with liquid derivatives markets, this poses a unique opportunity to study risk management, especially the hedging of options, in a turbulent market. We study the hedge behaviour and effectiveness for the class of affine jump diffusion models and infinite activity Levy processes. First, market data is calibrated to stochastic volatility inspired (SVI)-implied volatility surfaces to price options. To cover a wide range of market dynamics, we generate Monte Carlo price paths using an SVCJ model (stochastic volatility with correlated jumps), a close-to-actual-market GARCH-filtered kernel density estimation as well as a historical backtest. In all three settings, options are dynamically hedged with Delta, Delta-Gamma, Delta-Vega and Minimum Variance strategies. Including a wide range of market models allows to understand the trade-off in the hedge performance between complete, but overly parsimonious models, and more complex, but incomplete models. The calibration results reveal a strong indication for stochastic volatility, low jump frequency and evidence of infinite activity. Short-dated options are less sensitive to volatility or Gamma hedges. For longer-dated options, tail risk is consistently reduced by multiple-instrument hedges, in particular by employing complete market models with stochastic volatility.
Diese Masterarbeit greift die Theorie der Portfoliooptimierung auf: die mathematische Formulierung des Problems, seine Ableitungen (Risikominimierungsformulierung) und Annahmen, seine EinschrÀnkungen sowie einige Verbesserungen und Erweiterungen des bestehenden Frameworks. Ziel der Arbeit ist es auch, in Python zu simulieren und zu implementieren: Markowitz (Global Varianzminimal, Maximum Sharpe), Hierarchical Risk Parity und drei naiv Portfolios: gleichgewichtete, inverse VolatilitÀt und inverse Varianz in der neuartigen Anlageklasse der KryptowÀhrungen. Als Benchmark wird die CRyptocurrency IndeX, CRIX, verwendet. Die Portfoliooptimierung wird anhand von 120 Tagen tÀglicher historischer Daten berechnet, wobei die Portfolio-Anpassung alle 7 Tage und 30 Tage erfolgt. Portfolios sind Long-Short Strategien ohne Hebelwirkung und Verbesserungen in der Kovarianzmatrix werden mithilfe von Eigenwert-Clipping der Zufallsmatrixtheorie angewendet.
The semi-nonparametric (SNP) modeling of the return distribution has been proved to be a flexible and accurate methodology for portfolio risk management that allows two-step estimation of the dynamic conditional correlation (DCC) matrix. For this SNP-DCC model, we propose a stepwise procedure to compute pairwise conditional correlations under bivariate marginal SNP distributions, overcoming the curse of dimensionality. The procedure is compared to the assumption of Dynamic Equicorrelation (DECO), which is a parsimonious model when correlations among the assets are not significantly different but requires joint estimation of the multivariate SNP model. The risk assessment of both methodologies is tested for a portfolio on cryptocurrencies by implementing backtesting techniques and for different risk measures: Value-at-Risk, Expected Shortfall and Median Shortfall. The results support our proposal showing that the SNP-DCC model has better performance for a smaller confidence level than the SNP-DECO model, although both models perform similarly for higher confidence levels.
We study portfolio optimization of four major cryptocurrencies. Our time series model is a generalized autoregressive conditional heteroscedasticity (GARCH) model with multivariate normal tempered stable (MNTS) distributed residuals used to capture the non-Gaussian cryptocurrency return dynamics. Based on the time series model, we optimize the portfolio in terms of Foster-Hart risk. Those sophisticated techniques are not yet documented in the context of cryptocurrency. Statistical tests suggest that the MNTS distributed GARCH model fits better with cryptocurrency returns than the competing GARCH-type models. We find that Foster-Hart optimization yields a more profitable portfolio with better risk-return balance than the prevailing approach.
Ai Jun Hou, Ning Wang, Cathy Y. H. Chen, Wolfgang Karl HĂ€rdle
Cryptocurrencies, especially Bitcoin (BTC), which comprise a new digital asset class, have drawn extraordinary worldwide attention. The characteristics of the cryptocurrency/BTC include a high level of speculation, extreme volatility and price discontinuity. We propose a pricing mechanism based on a stochastic volatility with a correlated jump (SVCJ) model and compare it to a flexible co-jump model by Bandi and RenĂČ (2016). The estimation results of both models confirm the impact of jumps and co-jumps on options obtained via simulation and an analysis of the implied volatility curve. We show that a sizeable proportion of price jumps are significantly and contemporaneously anti-correlated with jumps in volatility. Our study comprises pioneering research on pricing BTC options. We show how the proposed pricing mechanism underlines the importance of jumps in cryptocurrency markets.
Alla Petukhina, Simon Trimborn, Wolfgang Karl HĂ€rdle, Hermann Elendner
Cryptocurrencies (CCs) have risen rapidly in market capitalization over the past years. Despite striking volatility, their high average returns and low correlations have established CCs as alternative investment assets for portfolio and risk management. We investigate the benefits of adding CCs to well-diversified portfolios of conventional financial assets for different types of investors, including risk-averse, return-maximizing and diversification-seeking investors who may trade at different frequencies, namely, daily, weekly or monthly. We calculate out-of-sample performance and diversification benefits for the most popular portfolio-construction rules, including mean-variance optimization, risk-parity, and maximum-diversification strategies, as well as combined strategies. Our results demonstrate that CCs can improve the risk-return profile of portfolios, but their benefit depends on investor objectives. In particular, diversification strategies (maximizing the portfolio diversification index or equating risk contributions) draw appreciably on CCs and show, in line with spanning tests, CCs to be non-redundant extensions of the investment universe. However, when we introduce liquidity constraints via the LIBRO method to account for illiquidity of many CCs, out-of-sample performance drops considerably, while the diversification benefits persist. We conclude that the utility of CC investments strongly depends on investor characteristics.