Abstract
Cryptocurrencies and Bitcoin, in particular, are prone to wild swings resulting in frequent jumps in prices, making them historically popular for traders to speculate. 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 introduces a bivariate jump-diffusion model to capture the dynamics of Bitcoin prices and the Bitcoin sentiment indicator, integrating trading volumes or Google search trends with Bitcoin price movements. We derive a closed-form solution for the Bitcoin price and the associated Black-Scholes equation for Bitcoin option valuation. The resulting partial differential equation for Bitcoin options is solved using an artificial neural network, and the model is validated with data from highly volatile stocks. We further test the model's robustness across a broad spectrum of parameters, comparing the results to those obtained through Monte Carlo simulations. Our findings demonstrate the model's practical significance in accurately predicting Bitcoin price movements and option values, providing a reliable tool for traders, analysts, and risk managers in the cryptocurrency market.