🔣Bonding Curve 101

The Bonding Curve is a powerful mathematical concept that has gained popularity in the world of decentralized finance (DeFi) and blockchain-based networks. The Liquidity-driven Curation Mechanism offers a secure, transparent, and traceable guarantee by tokenizing untradable and intangible virtual assets such as data, IPs, AI Agents, influence, and MEMEs.

☄️ Bonding Curve 101

Let R be the current reserve of the parent currency(say, $BOOM); Let P be the price of BLP Tokens(the share token of Bonding Pool) ; Let S be the current circulating supply of BLP Tokens. So we have the reserve ratio K:

$$K=\frac{R}{PS}$$

When a user buys an infinitesimal amount of BLP tokens dS (selling simply means dS < 0), we have:

$$PdS=dR=K(SdP+PdS)$$

After performing the integral, we have(C is arbitrary constant for a given K):

$$P={e^C}{S^{\frac{1}{K}-1}}$$

$$R=e^CKS^{\frac{1}{K}}$$

If the total supply of BLP token increases from S_0 to S, then the price increases from P_0 to P. The relationship between the two can be expressed as:

$$\frac {P} {P_0} = (\frac {S} {S_0})^{\frac {1}{K}-1}$$

If a user buys a total of N BLP tokens, bringing the total supply from S_0 to S_0+N , the total paid amount of parent currency A is:

$$A=\int^{S_0+N}_{S_0}{PdS}=\int{^{S_0+N}_{S_0}{P_0}(\frac{S}{S_0})^{\frac {1}{K}-1}}dS$$

then, we have:

$$A=R_0\bigg((1+\frac{N}{S_0})^{\frac{1}{K}}-1\bigg)$$

$$N=S_0\bigg((1+\frac{A}{R_0}）^{K}-1\bigg)$$

Let's simplify the bonding curve function mentioned in the previous section. Let

$$e^C=m$$

$$\frac{1}{K}-1=n$$

Thus, we have:

$$P=mS^{n}$$

$$R=mKS^{n+1}$$