Arbitrage Trading Formula Derivations

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How much to trade formula derivation

Let’s define some variables, all of which are known quantities.

• g is the gas fees in ETH
• p_e is the price of ETH in USD
• p_t is the price of TC in USD
• q_e is how much ETH you have in your holdings (as a float: so 10.0 ETH, not 10000000000000000000 wei)
• q_t is how much TC you have in your holdings (as a float: so 100.0 TC, not 1000000000000)
• x_d, y_d, and k_d are the values from the DEX (respectively: ether liquidity, token liqudiity, and k); we’ll assume that these values are the floating-point values without all the extra decimals (so 1.5 for y_t instead of 15000000000). Likewise, we’ll assume that k_d = x_d * y_d, meaning that it too is scaled down without all those decimals.
• f is the percentage (out of 1.0) obtained after the DEX fees are removed – if f_n is the fee numerator (say, 3) and f_d is the fee denominator (say, 1000), then f = 1 − f_n/f_d. In this example, with f_n = 3 and f_d = 1000, f = 0.997. Note that this fee applies to both ETH and TC transactions.

The variable subscripts here all follow these rules:

• A subscript of t means it represents a value related to the TC (price, quantity, etc.)
• A subscript of e means it represents a value related to the ETH (price, quantity, etc.)
• A subscript of d means it represents a value of the DEX (x, y, k, etc.)

Your current holdings in USD are: h_before = q_e * p_e + q_t * p_t

You have two options as to how to interact with a DEX: you could trade in either ETH or TC. We’ll define those amounts as δ_e and δ_t, depending on which we are trading in.

If you are trading δ_t TC to the DEX (note that δ_t must be positive):

• q_t is decreased by δ_t; thus the amount of TC after, which we will represent by ω_t, is:
  • ω_t = q_t − δ_t
• The value of the DEX’s x_d after the transaction we’ll represent by x′_d = k_d/(y_d+δ_t)
• The amount of ETH received: r_e = f * (x_d−x′_d)
  • This amount is added to q_e
  • As TC was traded to the DEX, r_e will be positive, indicating a receipt of ETH, as expected
• The amount of ETH after (ω_e) is thus:
  • ω_e = q_e + r_e
  • ω_e = q_e + f * (x_d−x′_d)
  • ω_e = q_e + f * x_d − f * k_d/(y_d+δ_t)
• However, we have to deduct the gas fees of g, so the amount of ETH after is then:
  • ω_e = q_e + f * x_d − f * k_d/(y_d+δ_t) − g
• We can then figure out our holdings after the transaction:
  • h_after = ω_e * p_e + ω_t * p_t
  • h_after = (q_e+f*x_d−f*k_d/(y_d+δ_t)−g) * p_e + (q_t−δ_t) * p_t
  • We’ll separate out the gas fees to make it simpler a bit later on:
  • h_after = (q_e+f*x_d−f*k_d/(y_d+δ_t)) * p_e + (q_t−δ_t) * p_t − g * p_e
• We want to maximize this formula. The only variable that we can change is how much TC we are trading in (δ_t)
  • To do this, we are going to have to take the derivative of that function
  • Before we do, let’s put it into a reduced form: y = a + b/(c+x) + d * x = a + b * (x+c)⁻¹ + d * x
    • If we massage the above formula into the y = a + b/(c+x) + d * x form, we can derive what x, a, b, c, and d equal:
      • x, the variable we are solving for, to x = δ_t
      • a = q_e * p_e − g * p_e + q_t * p_t + p_e * f * x_d
      • b =  − f * k_d * p_e
      • c = y_d
      • d =  − p_t
  • We take the derivative of that:
    • y′ =  − b * (x+c)⁻² + d
  • To find the maxima, we set y′ = 0 and solve for x:
    • y′ = 0 =  − b * (x+c)⁻² + d
    • 0 =  − b * (x+c)⁻² + d
    •  − d =  − b * (x+c)⁻²
    • d = b * (x+c)⁻²
    • d = b/(x+c)²
    • d * (x+c)² = b
    • (x+c)² = b/d
    • x + c= √ (b/d)
    • x =  − c± √ (b/d)
  • Recall that:
    • x = δ_t
    • b =  − f * k_d * p_e
    • c = y_d
    • d =  − p_t
  • So we can rewrite that formula as:
    • δ_t =  − y_d± √ (f*k_d*p_e/p_t)
      • That formula does not render well in HTML, but the entire parenthetical is what we are taking the square root of
    • So once we know the DEX values (x_d, y_d, k_d, and the fees f) and the current prices (p_e and p_t), we can compute the points for the maxima and minima
    • Note that these are not guaranteed to make a profit! But if they do make a profit, one of them will make the most and/or least profit.
    • If we are trading in ETH, the formula is similar: δ_e =  − x_d± √ (f*k_d*p_t/p_e)
    • Because the variables in the parenthetical can never be negative, the square root will always return real values
  • We can simplify these equations somewhat. Although there is a ± (plus-minus), subtracting that value would make the overall δ_t or δ_e value negative, which doesn’t make sense in this case – what we derived is how much we are trading in to the DEX, not how much we are getting out of the DEX. So we will just replace the ± with a regular plus symbol.
    • If we are trading in TC: δ_t =  − y_d+ √ (f*k_d*p_e/p_t)
    • If we are trading in ETH: δ_e =  − x_d+ √ (f*k_d*p_t/p_e)
    • As above, those formulas do not render well in HTML, but the entire parenthetical is what we are taking the square root of