5.7 Token Price Simulation

The key principle of the model is that it integrates different factors that could influence the token price, including entropy (unpredictability), demand-supply dynamics, yield curve (interest rate), and random fluctuations. It also applies the principle of viscosity to simulate the slowing down of price growth as it nears the upper limit. This combination of factors makes the model dynamic and responsive to different scenarios. Price modeling is carried out using stochastic differential equation commonly used in financial modeling (Cong, Li, and Wang 2021), known as the geometric Brownian motion (GBM). It describes the expected change in a quantity, here the price of the token, as the sum of a deterministic trend and a random component:

$$dP_t=P_t μ_t^P dt+ P_t σ_t^P dZ_t \ \ \ \ \ \ \ \ \tag{39}$$

The main parts of the equation are:

o : This represents the price of the token at time t.

o : This represents the expected return or "drift" of the token price, and can depend on a variety of factors.

o : This represents the standard deviation of the returns or "volatility" of the token price, expressing the level of uncertainty or risk associated with it.

o : This is an infinitesimal increment of time.

o : This represents a Wiener process (or Brownian motion), a type of random process where the changes in consecutive, infinitesimally small time increments are independent, normally distributed with a mean of 0 and a variance of .

The deterministic part of the equation says that, on average, the price of the token grows at a rate of per time unit.

The stochastic part adds a random fluctuation to this deterministic trend, which is proportional to the token price (which means the model assumes a constant relative risk). The term introduces randomness, making the model stochastic. This means that the price follows a random walk and, more specifically, a geometric Brownian motion. The approach currently utilized in our model provides a simple, mathematical description of the intricate and unpredictable behavior of financial market prices over time. It operates on the assumption that relative price changes are normally distributed and independent from each other. This renders the model straightforward and effective for many scenarios, making it a popular choice for various financial applications.

However, it is important to understand that all models, including ours, are approximations and simplifications of the realities of financial markets. Therefore, they have certain limitations. For instance, the assumption of normally distributed and independent price changes may not hold true in real-world markets, especially in situations of high volatility or significant market events.

In such cases, alternative models, such as the Jump Diffusion Model (Ramponi 2022) or Mean-Reverting Models (Wong and Lo 2009), might offer more accurate representations. The Jump Diffusion Model, for instance, can account for sudden, significant price changes ("jumps"), while Mean-Reverting Models assume that prices tend to revert to a long-term average, which may be more suitable if we believe that our token price will behave in such a manner.

While these alternative models are not incorporated into our current model, they could be considered in future iterations or adaptations, depending on the characteristics of the token and its market behavior. Thus, it's essential to continually evaluate the performance of our model and consider modifications as necessary to reflect the evolving market dynamics.