# Itô Protocol

## Core Concept

A **Stochastic AMM** combines the usual automated market-making mechanisms with **stochastic modeling of price dynamics**, meaning the pool's pricing formula incorporates randomness typically simulating asset prices with a **Geometric Brownian Motion (GBM)**.

## Stochastic Pricing Model

Asset Prices will be modelled using a Geometric Brownian Motion (GBM) because of its simplicity and common use in finance. The GBM equation is:

$$
dS_t = \mu S_t \, dt + \sigma S_t \, dW_t
$$
Where:

- S_t = asset price at time t
- μ = drift (expected return)
- σ = volatility
- dW_t = Wiener process (random walk)

In Discrete terms of the Implementation resolves to:
$$
S_t = S_0 \exp\left( \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma \sqrt{t} Z\right)
$$
where Z is a standard normal random variable (which we will get from Chainlink Functions) Z-> `[0,1)`.

However, note that we are not simulating the entire path but sampling a price at the time of swap. So we can use:

$$
\text{EffectivePrice} = P_{\text{market}} \times \exp\left(\underbrace{-\frac{\sigma^2 \Delta t}{2}}_{\text{Convexity Adjustment}} + \underbrace{\sigma \sqrt{\Delta t} Z_0}_{\text{Random Shock}}\right)
$$

### Component Overview

1. **Drift Adjustment Term** (-σ²/2):
- Compensates for Jensen's Inequality in lognormal distributions
- Ensures E[S_t] = S₀e^{μt} (martingale property)
- Without this, prices would artificially drift upward
2. **Volatility Scaling** (σ√Δt):
- Annualized volatility scaled to time period
- Square root law comes from variance scaling in Brownian motion
- Ensures consistency across timeframes
3. **Random Shock** (Z):
- Standard normal variable (μ=0, σ=1)
- Captures unpredictable market movements

However, it's not perfect. Alternatives include:

- **Ornstein-Uhlenbeck (OU) Process**: For stablecoins (mean-reverting).
- **Jump-Diffusion Models**: To account for sudden market crashes.
---

## Liquidity

$$
Token B/Token A = (σ * currentRatio + (1 - σ) * oracleRatio)
$$
$$
Current Ratio = reserveA / reserveB
$$
$$
Oracle Ratio = 1 / Price
$$

---
## Volatility Estimation

We need the volatility σ. We can get this from historical price data. We can use Chainlink data feeds to compute the volatility over a recent period (e.g., last 30 days).

#### Why 30-day volatility?

- **Statistical Significance**: 30 days provides a more stable and reliable measure of volatility. 24h volatility can be extremely noisy and may overreact to short-term events (e.g., news, market manipulation).
- **Industry Standard**: In traditional finance, 30-day (or 1-month) volatility is widely used for options pricing and risk management.
- **Mean-Reverting Properties**: Volatility itself is mean-reverting. Using a longer time frame helps capture the "typical" volatility level rather than transient spikes.

However, for highly volatile assets or during market crises, shorter time frames (like 24h) might be more responsive. The choice depends on the asset and risk tolerance.

#### Why not use 24h?

- **Overfitting to Noise**: 24h volatility can be misleading. A single day of high volatility might not represent the asset's true risk profile.
- **Manipulation Risk**: Short-term volatility is easier to manipulate with large trades.
- **Inconsistency**: If we update the volatility too frequently (e.g., every block), it might lead to erratic fee changes and pricing.

### Fee Calculation

$$
\text{Fee} = \text{Base} + \sigma \times \text{VolMultiplier} + \frac{\text{TradeSize}}{\text{Reserves}} \times \text{DepthFactor}
$$

- Increases during high volatility
- Scales with trade size relative to pool depth
- Compensates LPs for increased risk.

### User Flow & Sequence Diagram

sequenceDiagram
    participant User
    participant SAMM
    participant DataFeed
    participant VRF
    participant LiquidityPool
    
    User->>SAMM: swap(1 ETH)
    activate SAMM
    SAMM->>DataFeed: Request volatility data
    activate DataFeed
    DataFeed-->>SAMM: Response (price, vol)
    deactivate DataFeed
    SAMM->>VRF: Request randomness
    activate VRF
    VRF-->>SAMM: Random number
    deactivate VRF
	SAMM->>SAMM: Compute Inverse CDF with Randomness
    SAMM->>SAMM: Calculate stochastic price
    SAMM->>LiquidityPool: Update Liquidity Pool
    SAMM->>User: Transfer output tokens
    deactivate SAMM

### Impermanent Loss Comparision


## References

- [Ito's Lemma Applied to Stock Trading](https://www.interactivebrokers.com/campus/ibkr-quant-news/itos-lemma-applied-to-stock-trading/)