Neural Network for Valuing Bitcoin: Abstract and Introduction

13 May 2024

This paper is available on arxiv under CC 4.0 license.


(1) Edson Pindza, Tshwane University of Technology; Department of Mathematics and Statistics; 175 Nelson Mandela Drive OR Private Bag X680 and Pretoria 0001; South Africa [];

(2) Jules Clement Mba, University of Johannesburg; School of Economics, College of Business and Economics and P. O. Box 524, Auckland Park 2006; South Africa [];

(3) Sutene Mwambi, University of Johannesburg; School of Economics, College of Business and Economics and P. O. Box 524, Auckland Park 2006; South Africa [];

(4) Nneka Umeorah, Cardiff University; School of Mathematics; Cardiff CF24 4AG; United Kingdom [].


Cryptocurrencies and Bitcoin, in particular, are prone to wild swings resulting in frequent jumps in prices, making them historically popular for traders to speculate. A better understanding of these fluctuations can greatly benefit crypto investors by allowing them to make informed decisions. It is claimed in recent literature that Bitcoin price is influenced by sentiment about the Bitcoin system. Transaction, as well as the popularity, have shown positive evidence as potential drivers of Bitcoin price. This study considers a bivariate jump-diffusion model to describe Bitcoin price dynamics and the number of Google searches affecting the price, representing a sentiment indicator. We obtain a closed formula for the Bitcoin price and derive the Black-Scholes equation for Bitcoin options. We first solve the corresponding Bitcoin option partial differential equation for the pricing process by introducing artificial neural networks and incorporating multi-layer perceptron techniques. The prediction performance and the model validation using various high-volatile stocks were assessed.

Keywords: Jump-diffusion model · Cryptocurrencies · PDE · Bitcoin · Black-Scholes equation · Artificial neural network.

JEL: C15 C45 C53 G17.

1. Introduction

[44],[39],[8], [15] Bitcoin, a decentralized network-based digital currency and payment system, is a special type of cryptocurrency developed in 2009 [36] by a person or a group of persons known under the name of Satoshi Nakamoto. The soar in bitcoin appreciation has been accompanied by high uncertainty and volatility, which surrounds future prices, and this has attracted rapidly increasing research into this digital asset. Policymakers globally are concerned whether bitcoin is decentralized and unregulated and whether it could be a bubble [7] which threatens the stability of a given financial system. Regardless of the speculation, traders are still interested in the capacity of Bitcoin to generate more returns and the use of bitcoin derivatives as a risk management and diversification tool. Blockchain is the underlying technology that powers Bitcoin by recording transactions in a distributive manner, removing alteration and censorship. Following the success of Bitcoin and its growing community, many other alternatives to Bitcoin have emerged. There are more than 5000 tradable cryptocurrencies 1 with a total market capitalization of USD 248 billion at the time of this writing (see [54, 55] for further background on Bitcoin and its technology).

On the one hand, the cryptocurrency market is known to be highly volatile ([12, 23]) due to its sensibility to new information, whether fundamental or speculative [7] since it does not rely on the stabilizing policy of a central bank. On the other hand, the relative illiquidity of the market with no official market makers makes it fundamentally fragile to large trading volumes and market imperfections and thus more prone to large swings than other traded assets, see [43]. This concept results in frequent jumps of larger amplitude than what a continuous diffusion process can explain. Due to its local Markov property, i.e., the asset price changes only by a small amount in a short interval of time. When analyzing cryptocurrency data, it is interesting to consider processes that allow for random fluctuations that have more than a marginal effect on the cryptocurrency’s price. Such a stochastic process that enables us to incorporate this type of effect is the jump process. This process allows the random fluctuations of the asset price to have two components, one consisting of the usual increments of a Wiener process; the second provides for “large” jumps in the asset price from time to time. Shortly after the development of the Black–Scholes option valuation formula, Merton [35] developed a jump-diffusion model to account for the excess kurtosis and negative skewness observed in log-return data (see, [32]). This jump process is appended to the Black–Scholes geometric stochastic differential equation.

Several studies have used the daily bitcoin data to document the impact of jump as a crucial feature of the cryptocurrency dynamics. Chaim and Fry [6] observed that jumps associated with bitcoin volatility are permanent, whereas the jumps to mean returns are said to have contemporaneous effects. The latter equally capture large price bubbles, mainly negative, and are often associated with hacks and unsuccessful fork attempts in the cryptocurrency markets. Hillard and Ngo [17] further investigated the characteristics of bitcoin prices and derived a model which incorporates both jumps and stochastic convenience yield. The result was tested with data obtained from the Deribit exchange, and they observed that modelling bitcoin prices as a jump diffusion model outperformed the classical Black-Scholes models. Philippas et al. [41] observed that during periods of high uncertainty, some informative signals, which are proxied by Google search volume and Twitter tweets, have a partial influence on Bitcoin prices and price jumps.

In recent years, the focus has been on identifying the main drivers of Bitcoin price evolution in time. Many researchers have assigned the high volatility in Bitcoin prices to the sentiment and popularity of the Bitcoin system. Though these are not directly observable, they may be considered as indicators from the transaction volumes or the number of Google searches or Wikipedia requests about the topic, see for example, Kristoufek [27, 28], Kim et al. [26] and [4]. Authors in [4] use a Bitcoin sentiment measure from and develop a discrete-time model to show that excessive confidence in the system may boost a bubble in the Bitcoin system. These sentiment-based data were collected through Natural Language Processing techniques to identify a string of words conveying positive, neutral or negative sentiment on Bitcoin. Furthermore, the authors in [9] introduce a bivariate model in continuous time to describe both the dynamics of the Bitcoin sentiment indicator and the corresponding Bitcoin price. By fitting their bivariate model to market data, they consider both the volume and the number of Google searches as proxies for the sentiment factor.

While traditional financial theory relies on assumptions of market efficiency, normally distributed returns, and no-arbitrage, emerging research suggests cryptocurrency markets exhibit different characteristics. These markets appear prone to inefficiencies, fat-tailed non-normal distributions, and frequent arbitrage opportunities according to Kabaˇsinskas and Sutien˙e [22]. For example, the extreme volatility and relative illiquidity of cryptocurrencies can lead to dislocations between markets where arbitrageurs can profit. Additionally, the return distributions tend to exhibit higher kurtosis and skewness than normal distributions. However, despite violating some traditional assumptions, jump-diffusion models can still provide a useful starting point for modelling cryptocurrency dynamics. The jumps can account for extreme price fluctuations beyond what continuous diffusion alone predicts. While the model may require modifications over time as more data becomes available, it captures key features like volatility clustering and significant outliers. The sentiment indicator variable also represents an initial attempt to incorporate a behavioural factor affecting prices.

Our technique incorporates the one similar to Cretarola et al. [9], who developed a continuous bivariate model that described the bitcoin price dynamics as one factor, and a sentiment indicator as the second factor. We further added a jump-diffusion component to the SDE with the aim of capturing the occurrence of rare or extreme events in the bitcoin price return. Furthermore, we introduce the artificial neural network and propose a trial solution that solves the associated BlackScholes partial differential equation (PDE) for the bitcoin call options with European features. This concept is equally different from the univariate jump-diffusion model used, for example, by Chen and Huang (2021). We further implemented the number of Google searches as a Bitcoin sentiment indicator in this paper. This choice is due to their unique transparency in contrast to other social media-driven measures, and they have the tendency to gauge behaviour instead of searching for it. Therefore, using search-based data as sentiment indices has the potential to reveal the underlying beliefs of populations directly.

In this paper, we develop an initial modelling framework using a bivariate jump-diffusion model and sentiment indicator. This provides a foundation for pricing and derivatives valuation in cryptocurrency markets. We acknowledge the model’s limitations per the evolving understanding of these new markets. As future research expands knowledge of distributional properties, market microstructure issues, and other intricacies, the modelling approach can be enhanced. Nonetheless, our proposed model offers an initial step toward financial engineering in the cryptocurrency space. To this effect, the significant contributions of this paper are highlighted as follows:

• We present a bivariate jump-diffusion model to describe the dynamics of the Bitcoin sentiment indicator, which consists of volumes or Google searches and the corresponding Bitcoin price.

• We derive a closed-form formula for the Bitcoin price and the corresponding Black-Scholes equation for Bitcoin options valuation.

• The corresponding bitcoin option PDE is solved using the artificial neural network, and the proposed model was validated using data from highly volatile stocks.

The rest of the paper is organized as follows: Section 2 introduces the methodology and highlights the strengths and the limitations of the proposed model. Section 3 describes the concept of the artificial neural network, as well as its applications in solving the option pricing differential equations, Section 4 discusses the numerical implementation findings, and the last section concludes the work.