## Understanding the Price Impact Graph and Imbalance This lesson explains how price impact is calculated and visualized using a graph, focusing on the relationship between open interest imbalance and price adjustments. We'll explore the concepts of "same side" and "crossover" scenarios using a practical visualization built with the Desmos graphing calculator. ## Decoding the Desmos Visualization The visualization uses the Desmos graphing calculator to plot the relationship between market imbalance and the resulting price impact. * **Horizontal Axis (X-axis): Imbalance** * This axis represents the 'Imbalance' in the market, defined as `Long Open Interest - Short Open Interest`. * Units are measured in USD. * `x = 0`: Indicates that Long Open Interest equals Short Open Interest. * `x > 0`: Indicates that Long Open Interest is greater than Short Open Interest. * `x < 0`: Indicates that Short Open Interest is greater than Long Open Interest. * **Vertical Axis (Y-axis): Price Impact** * This axis represents the calculated 'Price Impact'. * Units are also measured in USD. * `y > 0`: Represents a positive price impact, generally indicating a favorable price adjustment for the protocol or liquidity pool. * `y < 0`: Represents a negative price impact, generally indicating an unfavorable price adjustment for the protocol or liquidity pool. * `y = 0`: Indicates no price impact arising from this specific imbalance mechanism (though other factors like a base spread might still apply). * **The Curve** * The curve plotted on the graph illustrates the calculated price impact (`y`) for any given level of imbalance (`x`). Its specific shape is dictated by the underlying mathematical formula governing price impact. * **Points** * Specific points plotted on the curve, represented as `(x, y)`, show the exact price impact (`y`) calculated for a specific imbalance value (`x`). ## Core Concepts: Imbalance and Price Impact Understanding the following concepts is crucial for interpreting the graph: 1. **Imbalance:** The fundamental input driving price impact, calculated as `Long Open Interest - Short Open Interest` (in USD). This value determines the position along the x-axis. 2. **Initial Imbalance (`x₀`):** This represents the imbalance state *before* a specific trade or market event occurs. In the Desmos tool, this is often controlled by a slider. 3. **Next Imbalance (`x₁`):** This represents the imbalance state *after* the trade or event has occurred. This is also typically controlled by a slider to simulate different scenarios. 4. **Price Impact (`p(x)`):** The output, shown on the y-axis. It's the price adjustment calculated based primarily on the imbalance level. The calculation often involves a formula that considers the initial imbalance (`x₀`) and parameters like a 'Price impact exponent' (`a`). For instance, a partially revealed formula might look like `p(x) = f_p(x)|x₀|^a - f_n(x)`, where `f_p` and `f_n` are functions related to positive/negative impact factors, and `a` (e.g., `a = 2`) influences the curvature and sensitivity of the impact. ## Same Side vs. Crossover Scenarios The relationship between the initial imbalance (`x₀`) and the next imbalance (`x₁`) defines two key scenarios: 1. **Same Side:** * A 'Same Side' scenario occurs when the *sign* of the imbalance does not change between the initial state (`x₀`) and the next state (`x₁`). It's crucial to note this refers to the sign of the imbalance (x-value), not necessarily the sign of the resulting price impact (y-value). * If `x₀ ≥ 0` (initial state has more or equal longs than shorts), a same-side transition requires `x₁ ≥ 0`. * If `x₀ ≤ 0` (initial state has more or equal shorts than longs), a same-side transition requires `x₁ ≤ 0`. * Essentially, if longs were dominant, they remain dominant (or balanced); if shorts were dominant, they remain dominant (or balanced). 2. **Crossover:** * A 'Crossover' scenario occurs when the *sign* of the imbalance *flips* between the initial state (`x₀`) and the next state (`x₁`). * If `x₀ > 0` (initially more longs), a crossover means `x₁ < 0` (ends with more shorts). * If `x₀ < 0` (initially more shorts), a crossover means `x₁ > 0` (ends with more longs). * This signifies a shift in market dominance between long and short positions. ## Illustrative Examples Let's walk through examples based on the graph: **Scenario 1: Starting with Positive Imbalance (`x₀ > 0`)** * Assume Initial Imbalance `x₀ = 2` USD (Longs exceed Shorts by 2 USD). The graph might show this starting point as `(2, 0)`. * **Same Side Example 1:** If a trade causes the Next Imbalance `x₁ = 1` USD. Since `x₁ > 0`, this is 'Same Side'. The graph might show the point `(1, 0.9)`, indicating a positive price impact of 0.9 USD. * **Same Side Example 2:** If `x₁ = 0.4` USD. Still 'Same Side' (`x₁ > 0`). The point could be `(0.4, 1.152)`. * **Same Side Example 3:** If `x₁ = 2.2` USD. Still 'Same Side' (`x₁ > 0`), even though the imbalance increased further from zero. This might lead to a *negative* price impact, e.g., `(2.2, -1.68)`. This highlights that "same side" refers only to the sign of `x`, not `y`. * **Crossover Example:** If a trade results in `x₁ = -0.3` USD. Since `x₁ < 0`, this is a 'Crossover'. The point could be `(-0.3, 1.02)`. **Scenario 2: Starting with Negative Imbalance (`x₀ < 0`)** * Assume Initial Imbalance `x₀ = -2` USD (Shorts exceed Longs by 2 USD). The corresponding point might be `(-2, 0)`. * **Same Side Example 1:** If the Next Imbalance `x₁ = -1.3` USD. Since `x₁ < 0`, this is 'Same Side'. The point could be `(-1.3, 0.693)`. * **Same Side Example 2:** If `x₁ = -0.2` USD. Still 'Same Side' (`x₁ < 0`). The point might be `(-0.2, 1.188)`. * **Same Side Example 3:** If `x₁ = -2.2` USD. Still 'Same Side' (`x₁ < 0`). This might result in a negative impact, e.g., `(-2.2, -1.68)`. * **Crossover Example:** If the Next Imbalance `x₁ = 0.3` USD. Since `x₁ > 0`, this is a 'Crossover'. The point could be `(0.3, 1.02)`. ## Key Characteristics of the Price Impact Function Observing the graph reveals important characteristics of how price impact behaves in this model: 1. **Penalization of Negative Impacts:** The shape of the curve often shows that negative price impacts (`y < 0`) can increase sharply as the imbalance moves further away from a balanced or initial state. This suggests the system is designed to heavily penalize large imbalances or changes that result in unfavorable price adjustments for the pool/protocol. 2. **Favoring Same Side Transitions:** The price impact calculation frequently results in more favorable outcomes (higher `y` values) for 'Same Side' transitions compared to 'Crossover' transitions, especially for changes of similar magnitude. * *Example:* Starting at `x₀ = 2`. A transition to `x₁ = 0.8` (Same Side) might yield a price impact `y = 1.008`. In contrast, a transition to `x₁ = -0.8` (Crossover), which is the same distance from `x=0` but flips the sign, might yield a much lower or even negative impact, like `y = -0.08`. This structure can incentivize trades that maintain the current direction of market imbalance rather than flipping it. 3. **Consistent Units:** Remember that all values discussed – Open Interest (Long and Short), Imbalance (`x`), and Price Impact (`y`) – are consistently represented in USD within this graphical context.