This study presents the mathematical formulations of investor sentiment for investors in cryptocurrencies. We assume that bitcoin prices are driven by investor sentiment measured in terms of Google search volume and social media posts. The current generation of retail investors uses non-traditional methods such as social media posts and Google searches to obtain information so that an increase in posts and searches on ‘bitcoin,’ indicate positive or negative investor sentiment. Mathematical formulations describe investor sentiment separately for risk-averse, moderate risk, and risk-taking investors. Risk-averse investors are considered to be aberrant in their investment in cryptocurrencies as they are naturally resistant to high-risk investments such as cryptocurrencies. Only risk-taking investors capture the fullest extent of irrational exuberance that prevailed in the cryptocurrency markets. However, risk-takers with very high-risk tolerance, such as hedge funds, trade in investments with volatility to capitalize upon the highest market prices for cryptocurrencies. Their behavior is modeled in cryptocurrency futures and cryptocurrency call options, and cryptocurrency put options. The insight provided by this paper is that the history of cryptocurrency prices is stored in a Laplace transform so that investor sentiment is based on the trajectory of past prices for cryptocurrencies and cryptocurrency futures. For cryptocurrency options, the history of volatility of prices is embedded in the Laplace transform, with increasing volatility embedded in call option prices, and decreasing volatility embedded in put option prices.
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