Bonding curves, simply

At first glance, bonding curves sound like some sort of fancy term for another abstract thing in computation, but bonding curves turn out to be a quite interesting thing in finance and mathematics.

Bonding curves simply put, are a mathematical formula for the calculation of an asset's value, in relation to the supply of the said asset.

Bonding curves are another mathematical model for asset valuation and pricing, in proportion to the asset supply and liquidity available for such assets. For better illustration, think of it as a line (or curve) on a graph that shows how the price of a token changes as more tokens are created or sold.

Say we have a token (LCD), and we want to use a bonding curve to determine the value of 1 LCD, we can derive the price using the mathematical formula being used in the specific bonding curve.

There are mainly two types of bonding curves;

• Linear Bonding Curves
• Exponential bonding curves

Linear Bonding Curves

A linear bonding curve increases the price of an asset in a straight line as more units of the assets are created. This means the price increases at a constant rate for each additional asset, or the asset price is directly proportional to asset supply (P∝S).

The formula for a linear bonding curve is:

P = m⋅S+b
where;

P = price of the asset
m = slope of the curve (rate of price increase per asset)
S = asset supply (number of existing assets)
b = initial price (when supply = 0)

A sample graph showing a linear bonding curve. The straight slope shows a direct proportion of asset price to asset supply

How it Works

Initial Price (b): This is the starting price of the asset when no assets have been issued yet.
Slope (m): This determines how much the price increases with each new asset issued.
Supply (S): As more assets are issued, the supply increases, and the price is recalculated.

Example:

Initial price (b) = $1
Slope (m) = $0.10 per asset

When no assets are issued:
P=0.10⋅0+1=1

When 1 asset is issued:
P=0.10⋅1+1=1.10

When 2 assets are issued:
P=0.10⋅2+1=1.20

Exponential Bonding Curves

An exponential bonding curve increases the price of an asset exponentially as more assets are issued. This means the price starts low but increases rapidly with each additional asset, or the asset value is exponentially proportional to the asset supply (P∝e^S)

The formula for an exponential bonding curve is:

P = a⋅e^(b⋅S)
where:

P: Price of the asset
S: Supply of the asset (number of existing assets)
a: Initial price (starting multiplier)
b: Exponential growth rate

A sample graph showing an exponential bonding curve. The slope advances slowly, and then increases sharply, showing an exponential increase with further supply addition.

How it Works

• Initial Price (a): This is the starting multiplier for the price.
• Exponential Growth Rate (b): This determines how quickly the price increases as more tokens are issued.
• Supply (S): As more tokens are issued, the supply increases, and the price is recalculated using the exponential formula.

Example:
Initial price (a) = $1
Growth rate (b) = 0.1

When no tokens are issued:
P=1⋅e^(0.1⋅0) =1
When 1 token is issued:
P=1⋅e^(0.1⋅1) ≈ 1.105
When 2 tokens are issued:
P=1⋅e^(0.1⋅2) ≈ 1.221

Conclusion

Bonding curves provide a systematic way to manage the price and supply of assets. They ensure that early buyers are rewarded and that the asset price reflects demand.

Bonding curves can be applied to DeFi, and used to reward early participants of token sales, or to maintain the price of decentralised assets while reflecting the demand. By understanding these curves, projects can create sustainable and fair economic models for their tokens.