The Black-Sholes-Merton model is a mathematical model for pricing financial options. The model is used to calculate the theoretical value of an option, which is the amount that the option holder would be paid if they exercised the option. The model takes into account factors such as the underlying asset’s price, volatility, and time to expiration. The model is named after its creators, Fischer Black, Myron Scholes, and Robert Merton.
The Black-Sholes-Merton model is the most widely used model for pricing options. It is considered to be the standard model in the industry. The model is used by traders, investors, and risk managers to price options and assess risks. The model is also used by financial analysts to value companies and make investment decisions.
One of the most important parameters in the Black-Sholes-Merton model is volatility which is a constant. In order to improve the Black-Sholes-Merton model, several advanced pricing methods have been developed that utilize stochastic volatilities.
Reference  proposed a radically different approach for pricing options. It replaced the volatility by volume,
The present paper proposes a new framework for describing the stock price dynamics. In the traditional geometric Brownian motion model and its variants, volatility plays a vital role. The modern studies of asset pricing expand around volatility, trying to improve the understanding of it and remove the gap between the theory and market data. Unlike this, we propose to replace volatility with trading volume in stock pricing models. This pricing strategy is based on two hypotheses: a price-volume relation with an idea borrowed from fluid flows and a white-noise hypothesis for the price rate of change (ROC) that is verified via statistic testing on actual market data. The new framework can be easily adopted to local volume and stochastic volume models for the option pricing problem, which will point out a new possible direction for this central problem in quantitative finance.
In short, the stock price dynamics is driven by the following Stochastic Differential Equation,
where v is the trading volume per time unit and V is the number of floating shares.
We find this proposal creative. However, we still don’t see how the skew and term structure would fit into this framework. Should we start talking about the term structure of volume and volume surface?
Let us know what you think in the forum or comments below.
 Ben Duan, Yutian Li, Dawei Lu, Yang Lu, Ran Zhang, Pricing Stocks with Trading Volumes, 2022, https://doi.org/10.48550/arXiv.2208.12067
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