Are Cryptocurrency Prices in Line with Fundamental Assets? ^†

Abstract

This paper presents the first rigorous empirical investigation into a fundamental question of cryptocurrency valuation: Are cryptocurrency prices in line with the prices of fundamental assets? To answer this, we analyze the nine largest cryptocurrencies by market capitalization—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Binance Coin (BNB), Ripple (XRP), Cardano (ADA), Litecoin (LTC), Tron (TRX), and the stablecoin DAI—against a suite of traditional benchmarks, including major fiat currencies (EUR, CAD, JPY), gold, and the S&P500 index. Our dataset spans from 1 January 2014 to 30 June 2025, with start dates varying for newer cryptocurrencies to ensure robust time series analysis. Guided by the asset pricing theory, we formulate a martingale test: if a cryptocurrency is priced in line with a fundamental numeraire asset, its price ratio relative to that numeraire must follow a martingale process. Our extensive empirical analysis reveals that the prices of major cryptocurrencies (BTC, ETH, SOL, BNB) consistently reject the martingale hypothesis when traditional assets (currencies, gold, equities) serve as the numeraire, indicating a decoupling from fundamental valuation anchors. Conversely, when Bitcoin or Ethereum itself is used as the numeraire, most smaller cryptocurrencies are priced in line with these crypto benchmarks, suggesting an internal valuation ecosystem that operates independently of traditional finance.

1. Introduction

Since the advent of Bitcoin in 2009, the cryptocurrency market has experienced exponential growth, with over 15,000 distinct cryptocurrencies now created and traded across more than 250 digital platforms. As of 1 July 2025, the aggregate market capitalization of all cryptocurrencies exceeds USD 3.25 trillion. Bitcoin remains the dominant cryptocurrency, with a market capitalization of USD 2.10 trillion, while Ethereum follows as a distant second at USD 291.524 billion (source: https://coinmarketcap.com, accessed on 1 July 2025). Given the rapid expansion of this asset class and growing investor interest, a critical question arises concerning whether cryptocurrencies are appropriately priced within financial markets. Specifically, this paper examines whether cryptocurrency prices align with those of fundamental assets.

This study constitutes a first attempt to empirically address this question. We selected the nine largest cryptocurrencies by market capitalization—Bitcoin, Ethereum, Solana, Binance Coin, Ripple, Cardano, Litecoin, Tron, and DAI—and compared their pricing to several traditional currencies, the price of gold, and the S&P500 index. Grounded in the asset pricing theory involving state price deflators, as presented by Back (2005), we formulate our testable hypothesis: if a cryptocurrency price aligns with a fundamental asset, then the ratio of the cryptocurrency price to the numeraire asset price must follow a martingale process. Three categories of numeraire assets are employed: (1) traditional currencies, which represent the established monetary systems that cryptocurrencies may compete with or complement; (2) gold, serving as a fundamental investment commodity and store of value; and (3) the S&P500 index, acting as a proxy for the broad equity market.

Prior to formal hypothesis testing, we documented key empirical properties of the selected cryptocurrencies. Summary statistics indicate that Bitcoin’s annualized return over the past 11 years is 89.13%, significantly exceeding the S&P500’s return of 12.40%. Concurrently, Bitcoin’s return volatility stands at 66.42%, approximately four times higher than the 17.37% volatility of the S&P500. Among the nine cryptocurrencies, Ethereum exhibits the highest annualized return (110.26%) and volatility (88.66%). In contrast, DAI shows the lowest return (0.61%) and volatility (11.07%) over the past eight years, consistent with its design objective to maintain a stable value of USD 1.00. Gold displays an annualized return and volatility of 9.93% and 14.09%, respectively, both lower than those of the S&P500. The three traditional currencies exhibit considerably lower return and volatility measures: the Euro shows −0.99% return with 7.78% volatility, the Canadian dollar yields 2.35% return with 7.18% volatility, and the Japanese Yen offers 3.09% return with 8.99% volatility. Notably, the Euro demonstrates the lowest return, while the Japanese Yen exhibits the highest volatility among the three currencies over the past 11 years.

Analysis of return correlations reveals that Bitcoin maintains positive correlations with most other cryptocurrencies, with coefficients generally exceeding 0.5. Most cryptocurrencies show negative correlations with traditional currencies such as the Canadian dollar and Japanese Yen. Conversely, cryptocurrencies generally demonstrate positive correlations with both gold and S&P500.

We further examine price ratios of cryptocurrencies relative to numeraire assets, revealing several noteworthy patterns. The Bitcoin-to-Euro ratio exhibits the highest mean and standard deviation among all ratios, with its standard deviation measuring 1.2441 times its mean value. A general pattern emerges wherein all cryptocurrency-to-Euro ratios (excluding DAI) demonstrate substantially higher volatility compared to ratios involving other benchmark assets. Additionally, most cryptocurrency-to-Euro ratios show strong positive correlations with each other, particularly the BTC/Euro and TRX/Euro pair, which shows a correlation coefficient of 0.9264. Finally, cryptocurrency ratios relative to both gold and S&P500 are predominantly positively correlated, with Bitcoin/Euro showing correlation coefficients of 0.9134 with gold/Euro and 0.9182 with S&P500/Euro.

The primary objective of this study was to test the hypothesis that if cryptocurrency prices align with fundamental assets, their price ratios relative to numeraire assets should follow martingale processes. We conducted two sets of tests: first using traditional currencies as numeraires, and second using gold and S&P500 as numeraires. Additionally, we employ Bitcoin and Ethereum as benchmark cryptocurrencies to examine whether other cryptocurrency prices align with these market leader cryptocurrencies. Our comprehensive empirical analysis indicates that several major cryptocurrencies—including Bitcoin, Ethereum, Solana, and Binance Coin—fail to satisfy the martingale property when traditional assets serve as numeraire assets. However, when Bitcoin or Ethereum functions as the numeraire, most smaller cryptocurrencies appear priced in alignment with these crypto benchmarks.

The academic and professional literature on cryptocurrency valuation remains in a nascent stage, with limited research directly addressing the core question of whether digital assets are priced in line with fundamental values. Existing studies tend to focus on specific aspects of cryptocurrency markets rather than providing comprehensive empirical tests of their fundamental alignment.

Adams et al. (2024) contribute to the theoretical understanding of cryptocurrency pricing by qualitatively examining the drivers of digital asset values. Their work suggests that cryptocurrency prices respond to factors similar to those affecting traditional assets, including supply–demand dynamics, market sentiment, and macroeconomic indicators. However, their analysis remains primarily conceptual without providing empirical validation of whether these similar drivers actually result in price alignment between crypto and traditional assets.

In the domain of derivative pricing, Cao and Celik (2021) made a valuable contribution by developing valuation frameworks for Bitcoin options and futures. Their research focuses specifically on derivative instruments rather than addressing the fundamental valuation of the underlying spot prices. This leaves the question of whether the spot prices themselves reflect fundamental values unanswered.

Several industry publications have offered introductory comparisons between cryptocurrency and traditional investments. The Cindicator Team (2024) provided an overview of potential benefits associated with crypto investments relative to traditional assets, while Taremi (2025) offered conceptual comparisons using accessible language for novice investors. These works serve primarily as educational resources rather than rigorous academic analyses.

Cryptopedia Staff (2025) advances the empirical discussion by examining co-movement patterns among different cryptocurrencies, documenting significant correlation structures within the digital asset ecosystem. This research provides important insights into inter-crypto relationships but does not extend the analysis to examine whether cryptocurrencies as a class align with traditional fundamental assets.

Despite these contributions, the literature reveals a significant gap: no existing study provides a comprehensive empirical test of whether cryptocurrency prices align with fundamental asset values using established financial theory. The current paper addresses this gap by applying the asset pricing framework from Back (2005) to test the martingale property of cryptocurrency price ratios relative to multiple traditional numeraire assets, thereby providing the first rigorous examination of whether cryptocurrencies are priced in line with fundamental assets.

The remainder of this paper proceeds as follows. Section 2 introduces the cryptocurrency market and provides detailed descriptions of the nine cryptocurrencies under study. Section 3 presents summary statistics and correlation analyses for both cryptocurrencies and fundamental assets. Section 4 outlines the empirical methodology and presents regression results. Section 5 concludes the paper.

2. Cryptocurrency Market and the Nine Most Actively Traded Cryptocurrencies

2.1. Cryptocurrency Market

Bitcoin (hereafter BTC), a cryptocurrency, was invented in 2008 by an unknown person or group of people using the name Satoshi Nakamoto, and launched in 2009 when its source code was released as open-source software. Bitcoins are created as rewards for a process known as mining. They can be exchanged for other currencies, products, and services. Bitcoin is a decentralized digital currency that can be transacted among users on the peer-to-peer bitcoin network without intermediaries. Transactions are verified by network nodes through cryptography and recorded in a public distributed ledger called a blockchain. More information on bitcoin can be found on the following website: https://en.wikipedia.org/wiki/Bitcoin#cite_note-174, accessed on 1 July 2025, recent financial downturn.

BTC can be bought on digital currency exchanges and is often used as an investment vehicle. BTC prices have seen dramatic fluctuations since inception. The BTC price quoted in U.S. dollars rose from a few dimes to more than USD 30 and then went back down to a few dollars in 2011. In the latter part of 2012 and during the 2012–2013 Cypriot financial crisis, bitcoin prices reached a high of USD 266 on 10 April 2013 and then crashed to around USD 50. On 29 November 2013, BTC price reached a new high of USD 1242, only to return to USD 600 in August 2014. Subsequently, BTC price has seen tremendous volatilities although the general trend is impressively upward. As of 1 July 2025, BTC price is around USD 105,609.18, reaching a historical high. The supply of BTC is 19.85 million, and the market cap for bitcoin is USD 2.1 trillion, representing 64.62% of the entire crypto market cap of USD 3.25 trillion. It is evident that bitcoin prices exhibit extremely high volatility compared with gold price or the U.S. stock market indices.

With the introduction of BTC, many other cryptocurrencies quickly came to the market. These newcomers share the same fundamental features of a public ledger and decentralization but differ in many other ways. The development of the cryptocurrency market is influenced by various factors, such as supply and demand, technological innovation, regulatory developments, macroeconomic trends, and investor sentiment. Up to now, there are more than 15,000 different coins and tokens traded in the cryptocurrency market, each with its own unique features and use cases.

In 2018, the cryptocurrency market experienced a severe crash, losing more than 80 percent of its value from its peak in late 2017. The crash was attributed to various factors, such as regulatory crackdowns, hacking incidents, market manipulation, and investor fatigue. In 2019, the cryptocurrency market recovered a little, driven by the launch of new projects and platforms, such as Facebook’s Libra, later renamed Diem. In 2020, the cryptocurrency market witnessed a surge in adoption and innovation, fueled by the COVID-19 pandemic, raised interest, and the rise in decentralized finance (DeFi). BTC broke its previous all-time high of nearly USD 20,000 in December 2020 and continued to rally into 2021. In 2021, the cryptocurrency market reached new heights, with Bitcoin surpassing USD 60,000 and the total market cap exceeding USD 2 trillion. The market also saw the emergence of new trends and phenomena, such as non-fungible tokens (NFTs), meme coins, layer-2 solutions, and environmental concerns. In 2022, the cryptocurrency market faced a sharp correction, losing more than USD 300 billion in value in a week. The market was negatively impacted by rising interest, geopolitical tensions, and regulatory uncertainty. FTX, one of the largest players in the cryptocurrency industry, went bankrupt in November 2022. Its bankruptcy triggered a sell-off of FTX’s native token FTT, which lost more than 90 percent of its value in a week. It also caused a loss of confidence in the crypto sector, as many customers were unable to access their funds or withdraw their assets. The bankruptcy of FTX also raised questions about the regulation and oversight of the cryptocurrency industry, as well as the risks and challenges of operating a global and decentralized platform. The cryptocurrency market has also witnessed several major events recently years, such as the exit of Binance from Canada, the launch of Ordinals, and the transition of Ethereum to proof-of-stake (PoS).

2.2. The Nine Most Actively Traded Cryptocurrencies

2.2.1. Bitcoin (BTC)

BTC was launched in 2009 and has been dominating the entire cryptocurrency market ever since its inception. We provided many details of BTC in the previous section. Below, we turn to eight other significant cryptocurrencies, namely Ethereum, Solana, Binance, Ripple, Cardano, Litecoin, Tron, and DAI.

2.2.2. Ethereum (ETH)

Ethereum (hereafter ETH), launched in 2015, runs on the Ethereum network which is a decentralized platform that can run various applications, such as DeFi, smart contracts, and NFTs. This ETH coin can be used to pay for transactions on the Ethereum network, which are denominated in gwei, a unit of ETH coin. It can also be used to participate in various applications that run on Ethereum, such as lending, borrowing, trading, and gaming. As of 1 July 2025, the total supply of ETH is 120.71 million, and it is trading at USD 2416.41. The market cap is 291.524 billion, representing 8.95% of the crypto market.

2.2.3. Solana (SOL)

Solana (hereafter SOL) was rolled out in 2020 and is a third-generation blockchain that supports an array of DeFi solutions, including the development of decentralized applications (DApps) and smart contracts. Its founder is Anatoly Yakovenko who used to be a senior staff engineer manager at the American multinational corporation Qualcomm. SOL runs on an open-source platform that allows for better efficiency. Different from other blockchains, Solana uses a hybrid consensus algorithm that combines proof-of-history (PoH) with proof-of-stake (PoS), enabling the network to carry out up to 50,000 transactions per second. As of 1 July 2025, SOL has a total supply of 534.6 million and is trading at USD 145. Its market cap is 78 billion, representing 2.4% of the crypto market.

2.2.4. Binance (BNB)

Binance (hereafter BNB) is one of the world’s largest and most well-known cryptocurrency exchanges, offering a wide range of services for trading, investing, and managing digital assets. Founded in 2017 by Changpeng Zhao, Binance quickly grew to dominate the crypto market due to its low fees, extensive selection of cryptocurrencies, and innovative features. Binance has faced scrutiny from regulators worldwide (e.g., the U.S. SEC, CFTC) over compliance issues, leading to restrictions in some countries. In 2023, Changpeng Zhao stepped down as CEO after Binance settled with U.S. authorities for USD 4.3 billion over anti-money laundering violations.

BNB is the native cryptocurrency of BNB exchange and was initially launched as an ERC-20 token on Ethereum in 2017. BNB later migrated to Binance Chain. It plays a crucial role in the Binance ecosystem, offering utility, discounts, and governance features. BNB can be used to pay transaction and trading fees on the Binance exchange, as well as for other purposes such as participating in token sales, staking, and decentralized finance (DeFi) services. As of 1 July 2025, the total supply of BNB is 140.88 M, and it is trading at USD 646.32. The market cap is 91 billion, representing 2.8% of the crypto market.

2.2.5. Ripple (XRP)

Ripple (hereafter XRP), launched in 2012, is a cryptocurrency that runs on the XRP Ledger, which is a digital asset that aims to improve upon traditional banking systems by enabling real-time global payments and settlements. XRP coin can be used to transfer value between users on the XRP Ledger, as well as to participate in various applications that run on the XRP Ledger, such as DEX, stablecoins, and NFTs. XRP transactions typically settle in 3–5 s, making it one of the fastest digital assets for moving funds globally. The transaction fees on the XRP Ledger are minimal, often costing a fraction of a cent, which is advantageous for large-volume and micro-transactions alike. As of 1 July 2025, XRP has a total supply of 59 billion and is trading at USD 2.16. The market cap is 128 billion, representing 3.94% of the crypto market.

2.2.6. Cardano (ADA)

Cardano (hereafter ADA), launched in September 2017, runs on a decentralized open-source blockchain platform founded in 2015 by Charles Hoskinson, who is one of the co-founders of Ethereum. ADA aimed to deliver more advanced features than any protocol previously developed, focusing on sustainability, scalability, and interoperability through a rigorous scientific and peer-reviewed approach. ADA uses a unique PoS algorithm called Ouroboros, which is designed to be energy-efficient while maintaining high levels of security. ADA coin was created to serve as a secure and scalable medium of exchange and a way to stake in the ADA network. Staking means delegating ADA coins to a stake pool that validates transactions and earns rewards for securing the network. As of 1 July 2025, the total supply of ADA is 35.37 billion and is trading at USD 0.54. The market cap is 19.112 billion, representing 0.58% of the crypto market.

2.2.7. Litecoin (LTC)

Litecoin (hereafter LTC), launched in October 2011, is a decentralized, open-source cryptocurrency created by Charlie Lee, a former Google engineer. LTC coin was invented to address the developer’s concerns that Bitcoin was becoming too centrally controlled and to make it more difficult for large-scale mining firms to gain the upper hand in mining. It also aimed to provide faster, cheaper, and more secure transactions than Bitcoin. LTC coin can be used to transfer value between users on the Litecoin network, as well as to participate in various applications that run on Litecoin, such as DEXs, stablecoins, and NFTs. Litecoin often serves as a platform to test new technologies that can later be adopted by Bitcoin, due to their technical similarities. With faster transaction times and lower fees, Litecoin is practical for everyday transactions and micro-payments. As of 1 July 2025, the total supply of LTC is 76 million and is trading at USD 83.98. The market cap is 6.385 billion, representing 0.184% of the crypto market.

2.2.8. Tron (TRX)

Tron (hereafter TRX), launched in 2017, is the cryptocurrency that runs on the TRON network, which is a decentralized platform that aims to create a decentralized Internet where users can connect directly with content creators and pay them for their content without intermediaries. TRX coin was initially an Ethereum token but later migrated to its own blockchain in 2018. TRX coin can be used on the TRON network to pay for transactions, which are denominated in sun, a unit of TRX coin. It can also be used to participate in various applications that run on TRON, such as DEXs, gaming, gambling, and social media. As of 1 July 2025, the total supply of TRX is 94.79 billion and is trading at USD 0.2791 with a market cap of 26.46 billion, representing 0.814% of the crypto market.

2.2.9. DAI

DAI, launched in 2017, is a stablecoin that runs on the Ethereum network and attempts to peg at USD 1.00. Unlike centralized stablecoins, DAI coins are not backed by US dollars in a bank account. Instead, it is backed by collateral on the Maker platform. DAI coin can be used to transfer value between users on the Ethereum network, as well as to participate in various applications that run on Ethereum, such as DeFi, lending, borrowing, and trading. DAI coin can also be generated by users who deposit collateral on the Maker platform and pay a stability fee. As of 1 July 2025, the total supply of DAI is 5.36 billion and is trading at USD 0.9998. Its market cap is 5.36 billion, representing 0.165% of the crypto market.

3. Data

Our objective is to empirically examine whether cryptocurrency prices are in line with traditional assets, including currency prices, commodities, and stock markets. These traditional assets are referred to as benchmark assets. We use Euro/U.S.$, Canadian dollar/U.S.$, and Japanese Yen/U.S.$ to represent traditional currencies, and these three currencies were chosen so that different regions in the world are covered. Moreover, we use gold and S&P500 to represent commodities and overall stock markets, respectively. Daily data for the nine cryptocurrencies and the benchmark assets were retrieved from Refinitiv Workspace, powered by Reuters. It is important to note that cryptocurrencies are traded daily, entailing 365 observations in a year. However, traditional currencies, gold, and the S&P500 index are only traded on workdays. To synchronize the sample for our analysis, we exclude cryptocurrency trading prices for weekends and holidays.

As we know, Bitcoin was launched in 2009. Its prices before 2014 were either not publicly available or not accurate. As a result, the sample for Bitcoin starts on 1 January 2014. As discussed in the previous section, the other eight cryptocurrencies started at different times; hence their sample starting time is different. Specifically, samples for ETH start on 19 May 2017; SOL, on 2 September 2021; BNB, on 16 August 2021; XRP, on 29 March 2021; ADA, on 29 March 2021; LTC, on 4 December 2017; TRX, on 29 March 2021; and DAI, on 20 December 2017. Since Bitcoin has the longest sample period, we used its sample period to collect data for traditional assets. All samples end on 30 June 2025.

Table 1. Summary statistics and correlations for daily returns.

Panel A: Statistics for Daily Return (Annualized)
	Inception	Sample Period				Sample Size	Return	Std Dev	Min	Max	Skewness	Kurtosis	Normality Test
BTC	2009	1 January 2014 to 30 June 2025				2998	0.8913	0.6642	−66.0815	67.3366	0.0739	4.9017	***
ETH	2015	19 May 2017 to 30 June 2025				2116	1.1026	0.8866	−78.6182	95.5951	0.4950	5.5718	***
SOL	2020	2 September 2021 to 30 June 2025				997	0.7188	0.9897	−113.5443	97.6865	0.0897	5.6856	***
BNB	2017	16 August 2021 to 30 June 2025				1010	0.3275	0.5859	−53.6032	39.7575	−0.2368	4.1754	***
XRP	2012	29 March 2021 to 30 June 2025				1110	1.0918	0.9609	−72.0903	183.8659	2.9424	29.5473	***
ADA	2017	29 March 2021 to 30 June 2025				1110	0.2166	0.8622	−55.0388	97.5033	0.8793	6.0642	***
LTC	2011	4 December 2017 to 30 June 2025				1975	0.4998	0.9422	−81.0231	199.0374	1.6212	20.8162	***
TRX	2017	29 March 2021 to 30 June 2025				1110	0.9284	0.8616	−73.2308	228.8254	6.0158	94.0274	***
DAI	2017	20 December 2017 to 30 June 2025				1963	0.0061	0.1107	−26.3455	29.4213	1.1194	99.4652	***
CAD		1 January 2014 to 30 June 2025				2998	0.0235	0.0718	−4.8952	5.3953	−0.0165	1.4415	***
JPY		1 January 2014 to 30 June 2025				2998	0.0309	0.0899	−9.5365	8.1012	−0.3656	4.8735	***
EUR		1 January 2014 to 30 June 2025				2998	−0.0099	0.0778	−5.9763	7.7659	0.1615	2.3128	***
GOLD		1 January 2014 to 30 June 2025				2998	0.0993	0.1409	−14.4206	12.0916	−0.0699	2.6482	***
SP500		1 January 2014 to 30 June 2025				2998	0.1240	0.1737	−30.1996	23.9788	−0.3622	15.8681	***
Panel B: Correlations for Daily Return
	BTC	ETH	SOL	BNB	XRP	ADA	LTC	TRX	DAI	CAD	JPY	EUR	GOLD
BTC
ETH	0.7291
SOL	0.6951	0.7033
BNB	0.7060	0.7338	0.6182
XRP	0.5477	0.5420	0.5173	0.4981
ADA	0.6836	0.6917	0.6454	0.6383	0.6482
LTC	0.6802	0.7348	0.5861	0.6424	0.6119	0.6541
TRX	0.3933	0.4002	0.3204	0.4479	0.4468	0.3883	0.4156
DAI	−0.0650	−0.0590	0.0950	0.0910	0.0149	0.0243	−0.0547	−0.0274
CAD	−0.1081	−0.1943	−0.2671	−0.2603	−0.1783	−0.2584	−0.1620	−0.1329	0.0077
JPY	−0.0177	−0.0039	−0.0619	−0.0369	−0.0377	−0.0472	−0.0196	−0.0061	−0.0264	0.1558
EUR	0.0573	0.0956	0.1828	0.1737	0.1183	0.1765	0.0815	0.0689	0.0123	−0.4344	−0.4476
GOLD	0.0715	0.0910	0.0702	0.1020	0.0395	0.0705	0.0907	0.0512	−0.0008	−0.2882	−0.4593	0.3785
SP500	0.1883	0.2561	0.3690	0.3485	0.2569	0.3410	0.2367	0.1674	−0.0581	−0.4164	0.2281	0.0657	0.0187

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. *** indicates significance at 1%.

presents summary statistics and correlations for the returns of the nine cryptocurrencies and benchmark assets in Panels A and B, respectively. The annualized average return for BTC in the past 11 years is 89.13%, in contrast to the S&P500 annual return of 12.40%. The annualized return volatility for BTC is 66.42%, much higher than 17.37% for the S&P500. Note that ETH has the highest annualized return (110.26%) and volatility (88.66%) among the nine cryptocurrencies. On the contrary, DAI has the lowest return (0.61%) and volatility (11.07%) in the past 8 years among the nine cryptocurrencies. This may be a result of its attempt to maintain a value of USD 1.00. Gold’s annualized return and volatility are 9.93% and 14.09%, less than those of the S&P500. The three traditional currencies have much smaller returns and volatilities: −0.99% and 7.78% for Euro, 2.35% and 7.18% for Canadian dollar, and 3.09% and 8.99% for Japanese Yen. It is worth noting that Euro has the smallest return and Japanese Yen has the highest volatility in the past 11 years among the three traditional currencies. In addition, normality was strongly rejected for the nine cryptocurrencies.

Panel B presents correlations among the nine cryptocurrencies and other benchmark assets. In particular, BTC is highly positively correlated with six cryptocurrencies, with correlation coefficients being higher than 0.5. BTC has a slightly lower correlation with TRX at 0.3933. In contrast, BTC is not much correlated with DAI, with a correlation coefficient of only −0.0650. This is mostly because DAI is pegged with the U.S dollar at par.

Cryptocurrencies are negatively correlated with the Canadian dollar and Japanese YEN and slightly positively correlated with the Euro. The rationale is that they are acting as substitutes for traditional currencies. On the contrary, most cryptocurrencies are positively correlated with gold, while DAI’s correlation with Gold is −0.0008, and all coefficients are less than 0.11. Most cryptocurrencies have a slightly higher correlation with S&P500. In particular, the first eight cryptocurrencies have correlations between 0.2 and 0.37, while DAI’s correlation with S&P500 is −0.0581.

We now turn to the investigation of relationships between cryptocurrency prices and benchmark assets and present empirical patterns for these relationships.

3.1. Cryptocurrency Prices Relative to Traditional Currency

We first examine the relationship between cryptocurrency prices and traditional currencies. To do so, we compute the relative price ratio for each cryptocurrency using one of the three traditional currencies as the denominator; we then calculate the summary statistics and correlations for all relative price ratios. The reason that we investigate the price ratio of a cryptocurrency and a benchmark asset is presented in Section 4, where we discuss the well-known asset pricing theory and lay out the empirical methodology for our regression analysis. The main tests are based on the price ratios of cryptocurrencies and benchmark assets. Since the patterns of these summary statistics and correlations with respect to the three traditional currencies are similar, to save space, we only present results using the Euro as the denominator in

Table 2. Statistics and correlations for daily price ratio to euro.

Panel A: Statistics for Daily Price Ratio to EUR
	Sample Period			Sample Size	Mean	Std Dev	Std Dev/Mean	Min	Max	Skewness	Kurtosis	Normality Test
BTC/1000	1 January 2014 to 30 June 2025			2998	19.5939	24.3771	1.2441	0.1508	101.5719	1.5040	1.5704	***
ETH/1000	19 May 2017 to 30 June 2025			2116	1.3340	1.1070	0.8298	0.0746	4.1423	0.5096	−0.9520	***
SOL	2 September 2021 to 30 June 2025			997	92.0545	63.2747	0.6874	7.8614	246.2654	0.2922	−1.1082	***
BNB	16 August 2021 to 30 June 2025			1010	389.4266	142.9090	0.3670	191.6729	705.0313	0.3759	−1.2173	***
XRP	29 March 2021 to 30 June 2025			1110	0.7929	0.6031	0.7606	0.2992	3.2179	1.9004	2.6250	***
ADA	29 March 2021 to 30 June 2025			1110	0.6756	0.4589	0.6792	0.2249	2.5352	1.5565	2.1025	***
LTC	4 December 2017 to 30 June 2025			1975	84.3472	44.1663	0.5236	20.3327	306.6040	1.6350	3.3894	***
TRX	29 March 2021 to 30 June 2025			1110	0.1039	0.0589	0.5667	0.0427	0.3977	1.4873	1.3237	***
DAI	20 December 2017 to 30 June 2025			1963	0.8992	0.0453	0.0504	0.7995	1.0421	0.1094	−0.1435	***
CAD	1 January 2014 to 30 June 2025			2998	1.1530	0.1311	0.1137	0.7738	1.4305	−1.0405	1.1312	***
JPY	1 January 2014 to 30 June 2025			2998	106.8174	19.7905	0.1853	72.7912	153.9340	0.6374	−0.5844	***
GOLD/1000	1 January 2014 to 30 June 2025			2998	1.4561	0.4744	0.3258	0.8764	3.0166	1.2161	1.0820	***
SP500/1000	1 January 2014 to 30 June 2025			2998	2.9797	1.1653	0.3911	1.2880	5.8959	0.6244	−0.6237	***
Panel B: Correlations for Daily Price Ratio to EUR
	BTC	ETH	SOL	BNB	XRP	ADA	LTC	TRX	DAI	CAD	JPY	GOLD
BTC
ETH	0.8314
SOL	0.8627	0.8523
BNB	0.9116	0.6978	0.9026
XRP	0.8150	0.3210	0.6060	0.6789
ADA	0.1902	0.5520	0.5511	0.3054	0.3245
LTC	0.2870	0.4272	0.5066	0.3413	0.3110	0.7977
TRX	0.9264	0.3280	0.6794	0.7825	0.8532	0.0224	0.0084
DAI	0.1969	0.2089	−0.3240	−0.1914	−0.1153	−0.5921	−0.4395	0.0259
CAD	0.3741	0.2511	−0.0684	0.1286	0.1499	−0.6491	−0.4979	0.3635	0.9103
JPY	0.6366	0.5905	0.0085	0.1635	0.0826	−0.7072	−0.2304	0.3904	0.7768	0.7788
GOLD	0.9134	0.6670	0.5495	0.7361	0.6752	−0.2758	−0.0585	0.8969	0.4153	0.5450	0.7398
SP500	0.9182	0.8477	0.7646	0.8385	0.6074	−0.1592	0.0703	0.8597	0.4502	0.5642	0.7776	0.9413

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. *** indicates significance at 1%.

.

As we know, BTC, ETH, gold, and S&P500 have high price levels compared to Euro. To facilitate comparison and discussion, we divided these price ratios by 1000. Panel A shows the summary statistics for price ratios. As expected, the BTC/Euro price ratio has the highest mean and standard deviation. To make a sensible comparison, we normalize the standard deviation by its mean for each price ratio. The normalized value for the BTC/Euro price ratio is 1.2441, indicating that the standard deviation of BTC/Euro is more than its average. The second highest normalized value is 0.8298, for ETH/Euro, suggesting that the standard deviation for ETH/Euro is 82.98% of its average. The general pattern is that all cryptocurrency prices/Euro (excluding DAI/Euro) have very high volatility compared to other benchmark asset prices/Euro while traditional currencies/Euro show very low fluctuations thanks to the low volatility for traditional currency prices. In addition to these empirical features, we also conducted normality tests for all price ratios, resulting in strong rejections for all price ratios.

Panel B presents correlations of price ratios among the nine cryptocurrencies and other benchmark assets. Specifically, the price ratios among the six cryptocurrencies, excluding ADA, LTC, and DAI, are highly positively correlated; most correlation coefficients are higher than 0.5. The highest correlation is between BTC/Euro and TRX/Euro: 0.9264. This positive correlation pattern among these six cryptocurrency price ratios is similar to the corresponding return correlations. As for DAI/Euro’s correlation with other cryptocurrency price ratios, it is indeterministic, ranging from −0.5921 to 0.2089. This indeterministic feature coincides with its return correlation pattern.

Now, we examine correlations between a cryptocurrency price/Euro and the CAD/Euro (or JPY/Euro). In contrast to the negative correlations for corresponding returns, the correlations for price ratios are mostly positive. The reason is that cryptocurrency prices are normalized with the Euro, and Euro prices move in directions opposite to those of cryptocurrencies.

As for other benchmark assets, gold and S&P500, their price ratios are mostly positively correlated with all cryptocurrency price ratios. BTC/Euro has the highest correlation coefficients, 0.9134 with gold/Euro and 0.9182 with S&P500/Euro.

3.2. Cryptocurrency Prices Relative to Gold Price

We now analyze the relationship between cryptocurrency prices and gold prices. We first compute relative price ratio for each cryptocurrency/gold, and then calculate the summary statistics and correlations for these relative price ratios. The results are presented in

Table 3. Summary statistics and correlations for daily price ratio to gold.

Panel A: Statistics for Daily Price Ratio to GOLD
	Sample Period			Sample Size	Mean	Std Dev	Std Dev/Mean	Min	Max	Skewness	Kurtosis	Normality Test
BTC	1 January 2014 to 30 June 2025			2998	10.5561	10.8245	1.0254	0.1446	40.1688	0.8957	−0.4632	***
ETH	19 May 2017 to 30 June 2025			2116	0.7489	0.5781	0.7720	0.0681	2.6182	0.7569	−0.0043	***
SOL * 1000	2 September 2021 to 30 June 2025			997	44.8942	30.1259	0.6710	4.6179	139.1468	0.5544	−0.4126	***
BNB * 1000	16 August 2021 to 30 June 2025			1010	193.9198	54.0547	0.2787	106.5954	355.7826	0.4444	−0.3485	***
XRP * 1000	29 March 2021 to 30 June 2025			1110	0.3908	0.2214	0.5665	0.1700	1.2234	1.3001	0.8713	***
ADA * 1000	29 March 2021 to 30 June 2025			1110	0.3779	0.3137	0.8302	0.1178	1.6635	1.7866	2.5746	***
LTC * 1000	4 December 2017 to 30 June 2025			1975	54.9343	36.2529	0.6599	18.5981	286.3372	2.0835	5.4167	***
TRX * 1000	29 March 2021 to 30 June 2025			1110	0.0508	0.0179	0.3531	0.0277	0.1581	1.1999	1.6308	***
DAI * 1000	20 December 2017 to 30 June 2025			1963	0.5756	0.1362	0.2366	0.2913	0.8516	0.1923	−0.5278	***
CAD * 1000	1 January 2014 to 30 June 2025			2998	0.8481	0.1986	0.2341	0.3958	1.3413	−0.0375	−0.8216	***
JPY * 1000	1 January 2014 to 30 June 2025			2998	77.1854	15.5458	0.2014	41.1325	117.0026	0.1638	−0.4842	***
EUR * 1000	1 January 2014 to 30 June 2025			2998	0.7495	0.2024	0.2701	0.3315	1.1410	−0.1970	−1.1277	***
SP500	1 January 2014 to 30 June 2025			2998	2.0099	0.2900	0.1443	1.3325	2.6635	−0.1576	−0.8307	***
Panel B: Correlations for Daily Price Ratio to GOLD
	BTC	ETH	SOL	BNB	XRP	ADA	LTC	TRX	DAI	CAD	JPY	EUR
BTC
ETH	0.8264
SOL	0.8642	0.8006
BNB	0.7701	0.7559	0.8979
XRP	0.7111	0.2757	0.5161	0.4588
ADA	0.3580	0.7527	0.6446	0.5518	0.4531
LTC	0.1378	0.2945	0.5340	0.5241	0.4238	0.8331
TRX	0.8068	0.0777	0.5264	0.4170	0.7387	0.0322	0.0085
DAI	−0.7531	−0.5125	−0.2955	−0.2501	−0.3735	0.3857	0.3242	−0.7204
CAD	−0.8199	−0.5999	−0.3810	−0.3262	−0.4407	0.2269	0.2497	−0.7197	0.9884
JPY	−0.7181	−0.3879	−0.4994	−0.4728	−0.6158	−0.2097	0.1418	−0.6090	0.8166	0.8993
EUR	−0.8187	−0.5228	−0.1762	−0.1548	−0.2458	0.5665	0.4329	−0.6506	0.9813	0.8693	0.7924
SP500	0.6517	0.6843	0.3924	0.3212	−0.1016	0.5479	0.2802	−0.2474	0.0466	−0.3569	−0.2709	−0.5151

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. *** indicates significance at 1%.

.

Since gold prices are much higher than most of the cryptocurrencies and traditional currencies, we multiplied them by 1000 to scale up all measurements for presentation purposes. Panel A of Table 3 shows the summary statistics for price ratios. Not surprisingly, the BTC/gold price ratio has the highest mean and standard deviation. As in the previous subsection, we normalize the standard deviation with its mean for each price ratio. The normalized value for the BTC/gold price ratio is 1.0254, the highest among all assets in consideration. This suggests that the standard deviation of BTC/gold is more than its average. The second highest normalized value is 0.8302 for ADA/gold, suggesting that the standard deviation for ADA/gold is more than 80% of its average. It is worth noting that, when gold price is the denominator, the lowest price ratio volatility does not belong to traditional currency; it goes to S&P500/gold. This observation can be explained as follows. Although traditional currency prices do not move much, gold prices do, which results in fluctuations in the price ratios. On the other hand, even though S&P500 and gold prices move a lot, relative movements between S&P500 and gold prices are less significant. Regarding normality tests, all price ratios show strong rejections.

Panel B in Table 3 presents correlations of price ratios among the nine cryptocurrencies and other benchmark assets. The price ratios among the eight cryptocurrencies, excluding DAI, are positively correlated; most correlation coefficients are higher than 0.5. The highest correlation is between BTC/gold and SOL/gold: 0.8642. This positive correlation pattern among these eight cryptocurrency price ratios is similar to the corresponding price ratio with respect to the Euro. As for DAI/gold’s correlation with other cryptocurrency price ratios, it is mostly negative, ranging from −0.7531 to 0.3857, which is consistent with the indeterministic feature of the price ratio with respect to the Euro.

Now, we turn to correlations between a cryptocurrency price/gold and traditional currency/gold. Most of the correlations are negative. The contributing factor to this negative relationship is mentioned earlier: cryptocurrencies are substitutes for traditional currencies. In contrast, DAI/gold is highly positively correlated with traditional currency/gold price ratios. The driving force behind this positive relationship is that gold prices move a lot and dominate the price ratios for the less active DAI and traditional currencies.

Lastly, we examine the S&P500/gold price ratio. It is negatively correlated with XRP/gold, TRX/gold, and the three traditional currency prices/gold. The driving forces behind this negative correlation are twofold: more price movements in gold and less price movements in currencies (either crypto or traditional). The S&P500/gold price ratio is highly positively correlated with the remaining seven cryptocurrencies. This price ratio’s positive correlation is inherited from the corresponding returns’ positive correlation because the effect of using gold price for normalization is less significant.

3.3. Cryptocurrency Prices Relative to Stock Market Value

We now discuss the relative value of a cryptocurrency to the stock market value. We use S&P500 to represent the U.S. stock market and compute the relative price ratio for each cryptocurrency/S&P500. We summarize the statistics and correlations in

Table 4. Summary statistics and correlations for daily price ratio to S&P500.

Panel A: Statistics for Daily Price Ratio to S&P500
	Sample Period			Sample Size	Mean	Std Dev	Std Dev/Mean	Min	Max	Skewness	Kurtosis	Normality Test
BTC	1 January 2014 to 30 June 2025			2998	4.8466	4.8721	1.0053	0.0884	19.0651	0.9700	−0.1199	***
ETH	19 May 2017 to 30 June 2025			2116	0.3343	0.2363	0.7068	0.0323	1.0235	0.4901	−0.5965	***
SOL * 1000	2 September 2021 to 30 June 2025			997	19.8373	12.4772	0.6290	2.1773	53.2707	0.2450	−1.0123	***
BNB * 1000	16 August 2021 to 30 June 2025			1010	86.7880	22.5054	0.2593	45.9079	138.0756	−0.0534	−1.1103	***
XRP * 1000	29 March 2021 to 30 June 2025			1110	0.1768	0.1078	0.6099	0.0766	0.5512	1.4182	0.8415	***
ADA * 1000	29 March 2021 to 30 June 2025			1110	0.1630	0.1237	0.7589	0.0539	0.6634	1.7220	2.3847	***
LTC * 1000	4 December 2017 to 30 June 2025			1975	25.2350	16.2923	0.6456	8.7152	134.2523	2.3482	7.0202	***
TRX * 1000	29 March 2021 to 30 June 2025			1110	0.0230	0.0094	0.4094	0.0118	0.0691	1.3967	1.1635	***
DAI * 1000	20 December 2017 to 30 June 2025			1963	0.2696	0.0693	0.2569	0.1611	0.4621	0.2987	−1.0976	***
CAD * 1000	1 January 2014 to 30 June 2025			2998	0.4364	0.1369	0.3138	0.2193	0.7799	0.1328	−1.2345	***
JPY * 1000	1 January 2014 to 30 June 2025			2998	39.5581	11.2050	0.2833	23.2089	63.7220	0.4276	−1.0557	***
EUR * 1000	1 January 2014 to 30 June 2025			2998	0.3895	0.1481	0.3802	0.1696	0.7764	0.5173	−0.5126	***
GOLD	1 January 2014 to 30 June 2025			2998	0.5087	0.0780	0.1534	0.3754	0.7505	0.6756	−0.3072	***
Panel B: Correlations for Daily Price Ratio to S&P500
	BTC	ETH	SOL	BNB	XRP	ADA	LTC	TRX	DAI	CAD	JPY	EUR
BTC
ETH	0.7798
SOL	0.8118	0.7486
BNB	0.7940	0.5859	0.8458
XRP	0.7617	0.1763	0.4992	0.5361
ADA	0.2183	0.7031	0.5687	0.3843	0.3547
LTC	0.0276	0.1815	0.3979	0.3207	0.3251	0.8052
TRX	0.8532	−0.0303	0.5032	0.5333	0.7935	−0.0544	−0.0723
DAI	−0.8545	−0.7639	−0.6967	−0.6322	−0.4808	0.1278	0.2577	−0.7523
CAD	−0.8827	−0.8106	−0.7782	−0.6708	−0.4952	−0.0992	0.1889	−0.6767	0.9895
JPY	−0.8288	−0.7264	−0.8253	−0.7293	−0.5754	−0.5348	0.0746	−0.4838	0.8670	0.9480
EUR	−0.8176	−0.7339	−0.6037	−0.5692	−0.3612	0.3690	0.3720	−0.7095	0.9833	0.9127	0.9044
GOLD	−0.5560	−0.5854	−0.1647	0.1174	0.3724	−0.4155	−0.1566	0.5415	0.3427	0.7126	0.7058	0.7973

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. *** indicates significance at 1%.

.

Since S&P500 values are much higher than most of the cryptocurrencies and traditional currencies, we multiplied them by 1000 to scale up all measurements. Panel A of Table 4 shows that the BTC/S&P500 price ratio has the highest mean and standard deviation. The standard deviation normalized by its mean is still the highest, 1.0053, suggesting that the standard deviation of BTC/S&P500 is more than its average. The second highest normalized value is 0.7589 for ADA/S&P500, indicating that the standard deviation is almost 80% of its average. Not surprisingly, when the S&P500 price is the denominator, the lowest price ratio volatility belongs to gold/S&P500, confirming the observation in the previous subsection when gold is the denominator. This observation can be explained using the same reasons presented in the previous subsection. Normality is strongly rejected by all price ratios.

Panel B in Table 4 shows that the price ratios among the seven cryptocurrencies (excluding TRX and DAI) are positively correlated. The highest correlation is 0.8458 between BNB/S&P500 and SOL/S&P500 and the lowest correlation is 0.0276 between LTC/S&P500 and BTC/S&P500. This positive correlation pattern among these seven cryptocurrency price ratios is similar to the corresponding price ratio with respect to gold. As for DAI/S&P500’s correlation with other cryptocurrency price ratios, it is indeterministic, ranging from −0.8545 to 0.2577.

Now, we turn to correlations between a cryptocurrency price/S&P500 and the traditional currency/S&P500. Most correlations are negative. Again, the contributing factor to this negative relationship is cryptocurrencies being substitute for traditional currencies. In contrast, DAI/S&P500 is highly positively correlated with traditional currency/S&P500 price ratios. The driving force behind this positive relationship is S&P500 prices moving a lot and dominating the price ratios for the less active DAI and traditional currencies.

Lastly, we examine the gold/S&P500 price ratio. It is negatively correlated with BTC/S&P500, ETH/S&P500, SOL/S&P500, ADA/S&P500, and LTC/S&P500. This negative correlation is inherited from the corresponding returns’ negative correlation. However, gold/S&P500 is positively correlated with the remaining cryptocurrency and traditional currency prices/S&P500. The driving force behind this positive is the lower price movements in currencies, either crypto or traditional.

3.4. Cryptocurrency Prices Relative to Bitcoin

Lastly, we analyze the relative value of a cryptocurrency to BTC. Since BTC has a much higher value than these of all other assets, we multiplied all assets by 1000. The summary statistics and correlations are presented in

Table 5. Summary statistics and correlations for daily price ratio to Bitcoin.

Panel A: Statistics for Daily Price Ratio to BTC
	Sample Period			Sample Size	Mean	Std Dev	Std Dev/Mean	Min	Max	Skewness	Kurtosis	Normality Test
ETH * 1000	19 May 2017 to 30 June 2025			2116	50.2622	21.8149	0.4340	16.3139	143.6704	0.3841	−0.2558	***
SOL * 1000	2 September 2021 to 30 June 2025			997	1.8646	0.8278	0.4440	0.5051	4.2354	0.3567	−0.3943	***
BNB * 1000	16 August 2021 to 30 June 2025			1010	9.5629	2.6931	0.2816	5.2331	18.1893	0.8973	0.2426	***
XRP * 1000	29 March 2021 to 30 June 2025			1110	0.0177	0.0057	0.3231	0.0070	0.0370	−0.0607	−0.5139	***
ADA * 1000	29 March 2021 to 30 June 2025			1110	0.0174	0.0115	0.6626	0.0048	0.0604	1.3381	1.4503	***
LTC * 1000	4 December 2017 to 30 June 2025			1975	5.3906	4.6262	0.8582	0.7883	22.6499	1.4924	1.4681	***
TRX * 1000	29 March 2021 to 30 June 2025			1110	0.0024	0.0005	0.2218	0.0011	0.0044	−0.1102	−0.8895	***
DAI * 1000	20 December 2017 to 30 June 2025			1963	0.0704	0.0657	0.9322	0.0090	0.3131	1.4458	1.7862	***
CAD * 1000	1 January 2014 to 30 June 2025			2998	0.9523	1.5198	1.5959	0.0124	6.7235	1.7355	1.9429	***
JPY * 1000	1 January 2014 to 30 June 2025			2998	87.0074	142.7532	1.6407	1.2930	660.0281	1.8412	2.3641	***
EUR * 1000	1 January 2014 to 30 June 2025			2998	0.8821	1.3936	1.5799	0.0098	6.6323	1.6079	1.3818	***
GOLD * 1000	1 January 2014 to 30 June 2025			2998	936.1583	1439.1116	1.5373	24.8950	6917.2996	1.6491	1.6509	***
SP500 * 1000	1 January 2014 to 30 June 2025			2998	1615.6461	2445.2141	1.5135	52.4518	11,315.1781	1.7258	1.9478	***
Panel B: Correlations for Daily Price Ratio to BTC
	ETH	SOL	BNB	XRP	ADA	LTC	TRX	DAI	CAD	JPY	EUR	GOLD
ETH
SOL	0.0187
BNB	0.6806	−0.2189
XRP	−0.0081	−0.2529	0.0713
ADA	0.5759	0.4171	0.3299	0.3631
LTC	0.0511	−0.2045	0.7009	0.4000	0.7040
TRX	−0.0870	−0.6878	0.2888	0.2600	−0.3355	−0.0214
DAI	−0.3206	−0.5690	0.8072	0.1322	0.1768	0.6776	0.5616
CAD	0.0360	−0.5939	0.8017	0.1263	0.1242	0.6573	0.5926	0.9993
JPY	0.0748	−0.6196	0.7668	0.0824	0.0297	0.6517	0.6444	0.9966	0.9971
EUR	0.0580	−0.5699	0.7989	0.1449	0.2270	0.6911	0.5217	0.9985	0.9879	0.9901
GOLD	0.0048	−0.6680	0.7735	0.1047	0.0343	0.6130	0.6348	0.9838	0.9921	0.9924	0.9971
SP500	0.0236	−0.5895	0.7872	0.0841	0.1145	0.6370	0.5818	0.9921	0.9965	0.9978	0.9921	0.9961

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. *** indicates significance at 1%.

.

Panel A shows that the JPY/BTC price ratio has the highest mean and standard deviation. The standard deviation normalized by its mean is still the highest, 1.6407, suggesting that the standard deviation of JPY/BTC is more than 1.6 times its average. The second highest normalized value is 1.5959 for CAD/BTC, not much smaller than that of JPY/BTC. The lowest normalized standard deviation goes to TRX/BTC, 0.2218, suggesting that the relative movement between TRX and BTC is the smallest. This observation can be explained by the fact that TRX and BTC move together a lot in the same direction, and the relative movement is not large. As in the previous subsection, we conducted normality tests, which were strongly rejected by all price ratios.

Panel B in Table 5 shows that the correlations of price ratios among all cryptocurrencies are indetermined with a very wide range. The highest correlation coefficient is 0.8072, between BNB/BTC and DAI/BTC, and the lowest is −0.6878, between TRX/BTC and SOL/BTC. There is no unique pattern for price ratios when BTC is used as the denominator. This observation naturally leads to the following conclusion: movements of cryptocurrencies relative to BTC are random.

We now turn to correlations between a cryptocurrency price/BTC and a traditional currency/BTC. Again, correlations are indeterministic and the range is wide. The highest correlation coefficient is 0.9993, between DAI/BTC and CAD/BTC, and the lowest is −0.6196, between JPY/BTC and SOL/BTC. This observation also suggests that the movements of cryptocurrencies and traditional currencies relative to BTC are quite random too.

Lastly, we examine the gold/BTC and S&P500/BTC price ratios. Both price ratios are positively correlated with cryptos/BTC, except SOL/BTC. The driving force for positive correlations is caused by one factor: big movements in BTC dominate the price ratios for the rest of the assets. The driving force behind the negative positive correlation for SOL/BTC is twofold: relatively less movements in gold and S&P500 and relatively more movements in SOL/BTC.

We have now thoroughly discussed the empirical features for returns of cryptocurrencies, traditional currencies, gold, and S&P500, as well as price ratios using traditional currency, gold, the stock market, and BTC as benchmark assets. In the next section, we perform a regression analysis to test whether cryptocurrencies are priced in line with existing assets, including traditional currencies, gold, and the stock market value.

4. Empirical Analysis

Our empirical analysis is based on the well-known asset pricing theory with the state price deflator. To facilitate discussion, we reframe the theory outlined by Back (2005) below.

Suppose that there are multiple assets in an economy and there is no dividend paid from any asset. Let us denote these assets as n = 0, 1, 2, …, N. The asset pricing theory with the state price deflator indicates that the price of any asset is determined as

$$S_t^n=E_{t+1}\left[{{\pi }_{t+1}S}_{t+1}^n\right], \mathrm{where} {\pi }_{t+1}>0 \mathrm{is} \mathrm{the} \mathrm{state} \mathrm{price} \mathrm{and} \mathrm{n} = 0, 1, 2, 3, \dots , \mathrm{N}.$$

(1)

We can choose any asset as the numeraire asset, say Asset 0. Since Asset 0 also satisfies the above equation, we have $S_t^0=E_{t+1}\left[{{\pi }_{t+1}S}_{t+1}^0\right]$. Restating the above equation, we have

$${1=E}_{t+1}\left[{\pi }_{t+1}{\displaystyle \frac{S_{t+1}^0}{S_t^0}}\right].$$

Since ${\pi }_{t+1}{\displaystyle \frac{S_{t+1}^0}{S_t^0}}$ > 0, we can define ${dQ}_{t+1}^0={\pi }_{t+1}{\displaystyle \frac{S_{t+1}^0}{S_t^0}}{dP}_{t+1}$ as the equivalent probability so that

$$1=\int {dQ}_{t+1}^0=\int {\pi }_{t+1}{\displaystyle \frac{S_{t+1}^0}{S_t^0}}{dP}_{t+1}.$$

With this equivalent probability, any other asset price relative to Asset 0’s price can be expressed as

$${\displaystyle \frac{S_t^n}{S_t^0}}=E_{t+1}^{Q^0}\left[{\displaystyle \frac{S_{t+1}^n}{S_{t+1}^0}}\right], \to \mathrm{for} \mathrm{n} = 1, 2, 3, \dots , \mathrm{N}.$$

(2)

The above result suggests that the price ratio of any asset relative to Asset 0 follows martingale with respect to this equivalent probability.

Since cryptocurrencies do not pay dividends, we can use Equation (2) to test whether their price ratios relative to benchmark assets follow martingale. We chose three types of assets as numeraire assets. The first type of numeraire assets is traditional currencies since cryptos are substitutes for these assets. The second type is either gold or S&P500, where gold is taken as an investment commodity, and S&P500 represents the overall stock market. We examine whether cryptocurrencies are priced in line with traditional currencies, gold, and the overall stock market. To be consistent with Equation (2), we use the ex-dividend index value for S&P500. The third type of numeraire asset is BTC or another cryptocurrency. The purpose is to check whether cryptocurrencies are priced in line with each other.

Equation (2) involves the equivalent martingale probability measure, which is unobserved. To filter out the information for this unobserved variable, we perform the following regression:

$${\displaystyle \frac{S_{t+1}^I}{S_{t+1}^{Numeraire}}}=a+b{\displaystyle \frac{S_t^I}{S_t^{Numeraire}}}+{\epsilon }_{t+1},$$

(3)

where a traditional asset is taken as the instrumental variable, and we assume that a traditional asset is properly priced in the financial market. In the above regression, the estimated residual ${\epsilon }_{t+1}$ will contain information for the unobserved state price deflator. We then use the estimated residual ${\epsilon }_{t+1}$ to proxy for the unobserved state price deflator and run the following regression to test whether the price of a cryptocurrency with respect to a benchmark asset follows martingale:

$${\displaystyle \frac{S_{t+1}^{crypto}}{S_{t+1}^{Numeraire}}}=\alpha +\beta {\displaystyle \frac{S_t^{crypto}}{S_t^{Numeraire}}}+\gamma {\epsilon }_{t+1}+{\eta }_{t+1},$$

(4)

The joint hypothesis test is that α = 0 and β = 1.

4.1. Cryptocurrency Prices vs. Traditional Currencies

We used traditional currencies as numeraire assets to test the martingale result for each cryptocurrency. We used gold, S&P500, or BTC as an instrumental variable separately. The regression results are presented in

Table 6. Martingale test with traditional currencies as numeraire assets.

	Sample Period	α	β	F-Statistics for α = 0 and β = 1	α	β	F-Statistics for α = 0 and β = 1	α	β	F-Statistics for α = 0 and β = 1
Panel A: CAD as Numeraire Asset
		Panel A1: GOLD as Instrument Asset			Panel A2: S&P500 as Instrument Asset			Panel A3: BTC as Instrument Asset
BTC	1 January 2014 to 30 June 2025	16.2300	1.0006	2.4281 *	16.2466	1.0006	2.4319 *
ETH	19 May 2017 to 30 June 2025	3.9371	0.9972	3.9021 **	3.9074	0.9972	3.9343 **	4.3264	0.9969	4.6486 ***
SOL	2 September 2021 to 30 June 2025	0.3601	0.9951	2.3108 *	0.3548	0.9952	2.2369	0.3690	0.9950	2.5355 *
BNB	16 August 2021 to 30 June 2025	1.7990	0.9948	2.5879 *	1.8019	0.9948	2.6404 *	1.7588	0.9949	2.4295 *
XRP	29 March 2021 to 30 June 2025	0.0038	0.9958	1.9467	0.0037	0.9959	1.8763	0.0038	0.9958	1.9788
ADA	29 March 2021 to 30 June 2025	0.0028	0.9943	0.6947	0.0028	0.9943	0.6890	0.0028	0.9943	0.6980
LTC	4 December 2017 to 30 June 2025	0.7542	0.9896	1.8821	0.7558	0.9895	1.8987	0.7541	0.9896	1.9359
TRX	29 March 2021 to 30 June 2025	0.0007	0.9935	1.7059	0.0007	0.9935	1.6987	0.0007	0.9936	1.7316
DAI	20 December 2017 to 30 June 2025	0.0171	0.9775	5.5078 ***	0.0172	0.9773	5.6347 ***	0.0170	0.9775	5.5105 ***
Panel B: JPY as Numeraire Asset
		Panel B1: GOLD as Instrument Asset			Panel B2: S&P500 as Instrument Asset			Panel B3: BTC as Instrument Asset
BTC	1 January 2014 to 30 June 2025	0.1865	1.0004	2.4281 *	0.1866	1.0004	2.6877 *
ETH	19 May 2017 to 30 June 2025	0.0408	0.9971	3.9021 **	0.0395	0.9972	2.9098 *	0.0451	0.9967	3.4686 **
SOL	2 September 2021 to 30 June 2025	0.0035	0.9948	2.3108 *	0.0035	0.9948	1.7361	0.0036	0.9946	1.8793
BNB	16 August 2021 to 30 June 2025	0.0179	0.9944	2.5879 *	0.0180	0.9944	2.2235	0.0174	0.9946	2.0438
XRP	29 March 2021 to 30 June 2025	0.0000	0.9951	1.9467	0.0000	0.9951	2.3524 *	0.0000	0.9952	2.3968 *
ADA	29 March 2021 to 30 June 2025	0.0000	0.9950	0.6947	0.0000	0.9950	0.4937	0.0000	0.9950	0.4945
LTC	4 December 2017 to 30 June 2025	0.0075	0.9904	1.8821	0.0075	0.9904	1.8569	0.0075	0.9904	1.9117
TRX	29 March 2021 to 30 June 2025	0.0000	0.9923	1.7059	0.0000	0.9922	1.8016	0.0000	0.9924	1.8265
DAI	20 December 2017 to 30 June 2025	0.0000	0.9979	5.5078 ***	0.0000	0.9980	1.6778	0.0000	0.9979	1.6928
Panel C: Euro as Numeraire Asset
		Panel C1: GOLD as Instrument Asset			Panel C2: S&P500 as Instrument Asset			Panel C3: BTC as Instrument Asset
BTC	1 January 2014 to 30 June 2025	21.1107	1.0005	2.4880 *	21.1484	1.0005	2.4963 *
ETH	19 May 2017 to 30 June 2025	4.3677	0.9974	4.4261 **	4.4409	0.9974	4.7185 ***	4.6846	0.9972	5.4108 ***
SOL	2 September 2021 to 30 June 2025	0.4146	0.9956	1.9832	0.4079	0.9956	1.9425	0.4365	0.9953	2.3399 *
BNB	16 August 2021 to 30 June 2025	2.1057	0.9951	2.3478 *	2.0912	0.9952	2.4197 *	2.0458	0.9953	2.2301
XRP	29 March 2021 to 30 June 2025	0.0045	0.9960	1.6073	0.0044	0.9961	1.5666	0.0045	0.9960	1.6795
ADA	29 March 2021 to 30 June 2025	0.0036	0.9939	0.8113	0.0036	0.9939	0.8052	0.0036	0.9939	0.8174
LTC	4 December 2017 to 30 June 2025	0.9981	0.9881	2.5430 *	0.9964	0.9881	2.5318 *	0.9980	0.9881	2.5989 *
TRX	29 March 2021 to 30 June 2025	0.0009	0.9931	1.8182	0.0009	0.9931	1.8028	0.0009	0.9932	1.8723
DAI	20 December 2017 to 30 June 2025	0.0120	0.9867	6.0359 ***	0.0122	0.9865	6.2008 ***	0.0121	0.9865	6.1235 ***

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. ***, **, and * indicate significance at 1%, 5%, and 10%, respectively.

, where Panels A, B, and C correspond to CAD, JPY, and the Euro as numeraire assets.

Let us examine the first case where gold is the instrumental variable in Panels A1, B1, and C1, where CAD, JPY, and Euro are numeraire assets, respectively. It is easy to see that BTC, ETH, BNB, and DAI are rejected, while XRP, ADA, and TRX are not in the three subpanels. SOL is twice rejected, while LTC is rejected once. We can infer the following from these results: more than half of the cryptocurrencies in consideration are not in line with traditional currency prices. The rejection for BTC, ETH, SOL, and BNB may be due to these cryptos having very high return volatilities relative to these three traditional currencies. The reason for DAI’s rejection may be because it is pegged to a U.S. dollar at par.

Similar observations can be made for the second case, where S&P500 is the instrumental variable in Panels A2, B2, and C2 and CAD, JPY, and Euro are numeraire assets, respectively. We also see that BTC and ETH are rejected, while SOL, ADA, and TRX are not in the three subpanels. XRP and LTC are rejected once, while BNB and DAI are rejected twice. The explanations for the complete rejections for BTC and ETH are the same as those explained above.

When BTC is the instrumental variable, we observe less rejections. In particular, we only see ETH as the only crypto that is rejected, and we see cryptos that ADA and TRX are not rejected in Panels A3, B3, and C3. To put it differently, if we assume that BTC is correctly priced by the market, other cryptocurrency prices are more in line with the traditional currencies.

Overall, the empirical evidence suggests that, among the nine largest cryptocurrencies in the market, BTC, ETH, BNB, and DAI are not in line with traditional currency prices, while smaller cryptocurrencies like ADA and TRX are priced consistently with traditional currencies.

4.2. Cryptocurrency Prices vs. Gold or Stock Market Value

Now, we examine the situation where gold or S&P500 is taken as a numeraire asset and traditional currencies CAD, JPY, and Euro are used as instrumental variables separately. The regression results are presented in

Table 7. Martingale test with gold or S&P500 as numeraire asset.

	Sample Period	α	β	F-Statistics for α = 0 and β = 1	α	β	F-Statistics for α = 0 and β = 1	α	β	F-Statistics for α = 0 and β = 1
Panel A: GOLD as Numeraire Asset
		Panel A1: CAD as Instrument Asset			Panel A2: JPY as Instrument Asset			Panel A3: Euro as Instrument Asset
BTC	1 January 2014 to 30 June 2025	0.01902	0.99920	3.99183 **	0.01902	0.99920	3.99242 **	0.01902	0.99920	3.99182 **
ETH	19 May 2017 to 30 June 2025	0.00277	0.99670	2.63307 *	0.00277	0.99669	2.64102 *	0.00277	0.99670	2.63305 *
SOL	2 September 2021 to 30 June 2025	0.00027	0.99314	2.34773 *	0.00027	0.99318	2.30983 *	0.00027	0.99314	2.34846 *
BNB	16 August 2021 to 30 June 2025	0.00197	0.98970	2.67064 *	0.00199	0.98958	2.75350 *	0.00197	0.98970	2.67073 *
XRP	29 March 2021 to 30 June 2025	0.00000	0.98988	3.51676 **	0.00000	0.99116	2.68828 *	0.00000	0.98989	3.49489 **
ADA	29 March 2021 to 30 June 2025	0.00000	0.99550	0.33305	0.00000	0.99506	0.43736	0.00000	0.99551	0.33378
LTC	4 December 2017 to 30 June 2025	0.00067	0.98738	4.11684 **	0.00066	0.98746	4.09545 **	0.00067	0.98738	4.11730 **
TRX	29 March 2021 to 30 June 2025	0.00000	0.98329	3.17735 **	0.00000	0.98166	2.85415 *	0.00000	0.98418	3.37762 **
DAI	20 December 2017 to 30 June 2025	0.00000	0.99842	2.06152	0.00000	0.99895	1.94225	0.00000	0.99883	1.63437
Panel B: S&P 500 as Numeraire Asset
		Panel B1: CAD as Instrument Asset			Panel B2: JPY as Instrument Asset			Panel B3: Euro as Instrument Asset
BTC	1 January 2014 to 30 June 2025	0.006634	0.999791	2.939014 *	0.006635	0.999791	2.939350 *	0.00663	0.99979	2.93901 *
ETH	19 May 2017 to 30 June 2025	0.001274	0.996659	2.995580 *	0.001276	0.996652	3.004558 *	0.00127	0.99666	2.99556 *
SOL	2 September 2021 to 30 June 2025	0.000116	0.993764	2.848474 *	0.000113	0.993891	2.678177 *	0.00012	0.99376	2.84969 *
BNB	16 August 2021 to 30 June 2025	0.000827	0.990642	3.287564 **	0.000829	0.990620	3.325384 **	0.00083	0.99064	3.28780 **
XRP	29 March 2021 to 30 June 2025	0.000001	0.992856	2.858636 *	0.000001	0.993823	2.234200	0.00000	0.99288	2.85198 *
ADA	29 March 2021 to 30 June 2025	0.000001	0.995105	0.380370	0.000001	0.994662	0.489788	0.00000	0.99512	0.37962
LTC	4 December 2017 to 30 June 2025	0.000318	0.986916	4.502518 **	0.000317	0.986945	4.538337 **	0.00032	0.98692	4.50310 **
TRX	29 March 2021 to 30 June 2025	0.000000	0.989713	2.353574 *	0.000000	0.988746	2.193138	0.00000	0.98995	2.43125 *
DAI	20 December 2017 to 30 June 2025	0.000001	0.997289	1.289955	0.000000	0.998119	0.611742	0.00000	0.99733	1.21864

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. ** and * indicate significance at 5% and 10%, respectively.

, where Panels A and B correspond to CAD, JPY, and Euro as instrumental variables.

We examine the first case in Panels A1 and B1, where CAD is the instrumental variable and gold or S&P500 is the numeraire asset, respectively. Panel A1 and B1 show that, except ADA and DAI, all cryptos reject the joint hypothesis. That is, if gold or S&P500 is priced correctly in the market, BTC, ETH, SOL, BNB, XRP, LTC, and TRX are not priced in line with fundamental assets, gold and the overall stock market S&P500. The rejection for these seven largest cryptos can be attributed to the fact that these cryptos have very high prices and very high return volatilities relative to the fundamental assets. It is interesting to note that DAI’s price seems to be consistent with gold and S&P500 although it is pegged with the U.S. dollar at par. This observation is in sharp contrast with the empirical evidence presented in the previous section, where traditional currencies are taken as numeraire assets.

Let us analyze the second case in Panels A2 and B2, where JPY is the instrumental variable and gold or S&P500 is taken as the numeraire asset. The pattern in Panel A2 is identical to that in Panel A1. In other words, except ADA and DAI, the seven largest cryptos are rejected in Panel A2, suggesting that BTC, ETH, SOL, BNB, XRP, LTC, and TRX are not priced correctly relative to gold price. When S&P500 is used as the numeraire asset, the rejection level is lower. Only five cryptos, including BTC, ETH, SOL, BNB, and LTC reject the joint hypothesis test, while XRP, ADA, TRX, and DAI do not. The reason for the five rejections is the same as that offered for Panels A1 and B1.

Lastly, we investigate the third case in Panels A3 and B3, where Euro is the instrumental variable and gold or S&P500 is as taken as the numeraire asset. The pattern in Panels A3 and B3 is identical to that in Panels A1 and B1, where CAD is the instrumental variable and gold or S&P500 is the numeraire asset.

With the empirical observations presented in Table 7, we conclude that, among the nine largest cryptocurrencies in the market, at least BTC, ETH, SOL, BNB, and LTC are not in line with gold and the overall market index S&P500.

4.3. Cryptocurrency Prices Among Themselves

In this subsection, we take BTC and other cryptos as the numeraire assets to test the martingale result for each cryptocurrency. Again, traditional currencies CAD, JPY, and Euro are used as instrumental variables. Since the results using other cryptos are similar, to save space, we only present the results when ETH is used as the numeraire asset. The regression results are presented in

Table 8. Martingale test with BTC or another cryptocurrency as numeraire asset.

	Sample Period	α	β	F-Statistics for α = 0 and β = 1	α	β	F-Statistics for α = 0 and β = 1	α	β	F-Statistics for α = 0 and β = 1
Panel A: BTC as Numeraire Asset
		Panel A1: CAD as Instrument Asset			Panel A2: JPY as Instrument Asset			Panel A3: Euro as Instrument Asset
ETH	19 May 2017 to 30 June 2025	0.00027	0.99415	1.82815	0.00026	0.99437	1.65300	0.00027	0.99415	1.82845
SOL	2 September 2021 to 30 June 2025	0.00001	0.99146	3.02109 **	0.00001	0.99270	1.71652	0.00001	0.99145	3.02795 **
BNB	16 August 2021 to 30 June 2025	0.00003	0.99614	0.43517	0.00003	0.99666	0.30665	0.00003	0.99614	0.43573
XRP	29 March 2021 to 30 June 2025	0.00000	0.98617	4.01418 **	0.00000	0.98629	3.92490 **	0.00000	0.98609	4.08143 **
ADA	29 March 2021 to 30 June 2025	0.00000	0.99715	0.17378	0.00000	0.99705	0.18880	0.00000	0.99716	0.17290
LTC	4 December 2017 to 30 June 2025	0.00002	0.99644	1.53068	0.00001	0.99703	1.04796	0.00002	0.99644	1.53177
TRX	29 March 2021 to 30 June 2025	0.00000	0.97236	3.80475 **	0.00000	0.97252	3.82524 **	0.00000	0.97227	3.82094 **
DAI	20 December 2017 to 30 June 2025	0.00000	0.99804	0.44127	0.00000	0.99809	0.39684	0.00000	0.99784	0.57558
Panel B: ETH as Numeraire Asset
		Panel B1: CAD as Instrument Asset			Panel B2: JPY as Instrument Asset			Panel B3: Euro as Instrument Asset
BTC	1 January 2014 to 30 June 2025	0.08960	0.99695	2.82266 *	0.08978	0.99694	2.83699 *	0.08960	0.99695	2.82261 *
SOL	2 September 2021 to 30 June 2025	0.00014	0.99681	1.56458	0.00013	0.99712	1.25833	0.00014	0.99681	1.56537
BNB	16 August 2021 to 30 June 2025	0.00101	0.99505	1.34615	0.00093	0.99549	1.15102	0.00101	0.99505	1.34666
XRP	29 March 2021 to 30 June 2025	0.00000	0.99834	0.90951	0.00000	0.99863	0.82405	0.00000	0.99828	0.94354
ADA	29 March 2021 to 30 June 2025	0.00000	0.99317	0.89624	0.00000	0.99316	0.90007	0.00000	0.99313	0.91691
LTC	4 December 2017 to 30 June 2025	0.00033	0.99694	1.44974	0.00021	0.99780	0.54434	0.00033	0.99694	1.44937
TRX	29 March 2021 to 30 June 2025	0.00000	0.99529	1.43133	0.00000	0.99535	1.40526	0.00000	0.99528	1.43510
DAI	20 December 2017 to 30 June 2025	0.00001	0.99734	0.72654	0.00001	0.99724	0.77033	0.00001	0.99669	1.12316

Notes: Time series for cryptocurrencies, traditional currencies, gold price, and S&P500 were downloaded from Refinitiv Workspace, powered by Reuters. ** and * indicate significance at 5% and 10%, respectively.

, where Panels A and B correspond to CAD, JPY, and Euro as instrumental variables.

We examine the results in Panel A where BTC is the numeraire asset and CAD, JPY, and Euro are instrumental variables. In this case, there are eight cryptos remaining. Only SOL, XRP, and TRX reject the hypothesis, and the other five cryptos, ETH, BNB, ADA, LTC, and DAI, do not. The rejection level is less than those of cases presented in the two previous subsections. This suggests that, when BTC is used as the numeraire asset, more than a half of the cryptos are priced in line with BTC.

We further examine results in Panel B where ETH is taken as the numeraire asset and CAD, JPY, and EURO are instrumental variables. Among the eight cryptos, only BTC rejects the hypothesis, and the other seven cryptos, SOL, BNB, XRP, ADA, LTC, TRX, and DAI, do not. This shows that, when ETH is used as the numeraire asset, only BTC price is off, and the others are priced properly relative to ETH.

In summary, the empirical evidence indicates that, among the nine largest cryptocurrencies in the market, their prices are priced in line with each other, except BTC. The inconsistency for BTC is mostly due to BTC’s high price level and high return volatility.

5. Conclusions

This study presents the first comprehensive empirical investigation into a fundamental question of cryptocurrency valuation: whether cryptocurrency prices align with those of fundamental assets. To address this question, we analyzed the nine largest cryptocurrencies by market capitalization—Bitcoin (BTC), Ethereum (ETH), Solana (SOL), Binance Coin (BNB), Ripple (XRP), Cardano (ADA), Litecoin (LTC), Tron (TRX), and the stablecoin DAI—comparing their pricing dynamics against several traditional currencies, gold, and the S&P500 index.

Guided by the asset pricing theory involving state price deflators, as established by Back (2005), we formulated and tested the hypothesis that if a cryptocurrency is priced in line with a fundamental asset, the ratio of their prices must follow a martingale process. Our empirical design incorporated three distinct categories of numeraire assets: (1) traditional currencies, which cryptocurrencies compete against; (2) gold, serving as a fundamental investment commodity and value storage; and (3) the S&P500 index, proxying the broad equity market. Additionally, we used Bitcoin or Ethereum as primary cryptos to examine whether other cryptocurrency prices are in line with them.

The results from our extensive empirical analysis reveal that when traditional assets serve as numeraire, the prices of major cryptocurrencies—including Bitcoin, Ethereum, Solana, and Binance Coin—consistently reject the martingale hypothesis. This suggests that these cryptocurrencies are not priced in alignment with fundamental traditional assets. In contrast, when Bitcoin or Ethereum is employed as the numeraire asset, the majority of smaller cryptocurrencies are priced in line with them.