# 🛠 AlphaX Arbitrageurs Docs 💰

> AlphaX Contract Interfaces: https://hackmd.io/RfH7bNirRMCP8y9rIN066g

This doc will go into details of how to effectively arbitrage on AlphaX, including some PoCs.

## AlphaX Recap

To recap, AlphaX has 2 main types of tokens: **Long Strike Tokens** & **Short Strike Tokens**.

Long Strike Tokens come in the form **ETH-X**, e.g., ETH-3000, ETH-2500.
Short Strike Tokens come in the form of **Y-ETH**, e.g., 6450-ETH, 7100-ETH.

The price of ETH-X should be pegged to (ETH price - X).
The price of Y-ETH should be pegged to (Y - ETH price).
This is regarded as the **Oracle Price $P_O$**.

However, the actual ETH-X & Y-ETH token price might be trading on DEXes at another price. This price is the **Market Price $P_M$**.

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> There are multiple ways to arbitrage the gap between $P_M$ and $P_O$. Here, we present one way.

Two main arbitrage cases:

## Case 1. Market Price too low ($P_M < P_O$)

> TL;DR Buy XToken from the market.

How much XTokens to buy depends on trading pool reserves, market price, and oracle price.

Here are some calculations:

Let
$R_b$ = trading pool's base reserve (obtained from `pair.getReserves()`)
$R_q$ = trading pool's quote reserve (obtained from `pair.getReserves()`)
$f$ = trading pool's swap fee (0.3% = 0.003 in this case).

Let $x$ be the optimal amount of XTokens to buy to bring $P_M$ up to $P_O$. We want to solve for $x$.

After the swap, the new reserves become:
\begin{align*}
R'_b &= R_b + x \\
R'_q &= \frac{R_b \cdot R_q}{R_b + (1-f) \cdot x}
\end{align*}

The new market price becomes:
\begin{align*}
P_O = P'_M = \frac{R'_q}{R'_b} = \frac{R_b \cdot R_q}{(R_b + (1-f) \cdot x)\cdot(R_b+x)}
\end{align*}

Rearranging the equation gives a solvable quadratic equation:

\begin{align*}
P_O \cdot (1-f) \cdot x^2 + (P_O\cdot(2-f)\cdot R_b)\cdot x + P_O\cdot R_b^2 - R_b \cdot R_q = 0
\end{align*}

**Calculation Pseudocode:**

def solve_buy_amt(pair_addr, oracle_price):
    pair = interface.IUniswapV2Pair(pair_addr)
    f = 0.003 # 0.3% swap fee
    p = oracle_price 

    r_0, r_1, _ = pair.getReserves()
    r_b, r_q = (r_0, r_1) if pair.token1() == USDC else (r_1, r_0) # check ordering

    # quadratic equation: ax^2 + bx + c = 0
    a = p * (1-f)
    b = p * (2-f) * r_b
    c = p * r_b * r_b - r_b * r_q

    # solve x
    x = int((-b + sqrt(b*b - 4*a*c))/(2*a))
    
    return x

> NOTE: You can swap directly in the trading pool using USDC.e.

## Case 2. Market Price too high ($P_M > P_O$)

> TL;DR Sell XToken to the market. (Mint XToken pair if necessary).

How much XTokens to sell depends on trading pool reserves, market price, and oracle price.

Here are some calculations:

Let
$R_b$ = trading pool's base reserve (obtained from `pair.getReserves()`)
$R_q$ = trading pool's quote reserve (obtained from `pair.getReserves()`)
$f$ = trading pool's swap fee (0.3% = 0.003 in this case).

Let $x$ be the optimal amount of XTokens to sell to bring $P_M$ down to $P_O$. We want to solve for $x$.

After the swap, the new reserves become:
\begin{align*}
R'_b &= R_b - x \\
R'_q &= \frac{(1-f)\cdot R_b \cdot R_q + f\cdot R_q \cdot x}{(1-f)\cdot (R_b - x)}
\end{align*}

The new market price becomes:
\begin{align*}
P_O = P'_M = \frac{(1-f)\cdot R_b \cdot R_q + f\cdot R_q \cdot x}{(1-f)\cdot (R_b - x) \cdot(R_b -x)}
\end{align*}

Rearranging the equation gives a solvable quadratic equation:

\begin{align*}
P_O \cdot (1-f) \cdot x^2 - (2\cdot P_O \cdot(1-f)\cdot R_b + f \cdot R_q) \cdot x + (P_O \cdot (1-f)\cdot R_b^2 - (1-f)\cdot R_b \cdot R_q) = 0
\end{align*}

**Calculation Pseudocode:**

def solve_sell_amt(pair_addr, oracle_price):
    pair = interface.IUniswapV2Pair(pair_addr)
    f = 0.003 # 0.3% swap fee
    p = oracle_price 

    r_0, r_1, _ = pair.getReserves()
    r_b, r_q = (r_0, r_1) if pair.token1() == USDC else (r_1, r_0) # check ordering

    # quadratic equation: ax^2 + bx + c = 0
    a = p * (1-f)
    b = -(2 * p * (1-f) * r_b + f * r_q)
    c = p * (1-f) * r_b * r_b - (1-f) * r_b * r_q

    # solve x
    x = int((-b + sqrt(b*b - 4*a*c))/(2*a))
    
    return x

> NOTE: You can swap directly in the trading pool using XTokens. If you do not have sufficient XTokens, you can `mint` XTokens using the interfaces found here: https://hackmd.io/RfH7bNirRMCP8y9rIN066g