Fair Pricing Considerations
Standard Zero-Liquidation Loan (Peer-to-Peer)
A Zero-Liquidation Loan (ZLL) can be seen as a swap in which the borrower:
Pledges x collateral tokens.
Receives y(x) loan tokens.
Is allowed to reclaim their previously pledged collateral if they pay the locked-in repayment amount prior to the loan's expiry. Essentially, this represents a call option to buy x collateral tokens from the pool for R = x * K, where K denotes the call option's strike.
In other words, taking out a ZLL as a borrower is equivalent to doing the following swap:
x * S -> y(x) + x * C(S, K, ΔT)where:
x is the number of collateral tokens pledged.
S is the price of the collateral.
y(x) is the loan amount.
C(...) is the value of the call option.
K is the strike price of the call option.
ΔT is the tenor of the loan.
In a fair ZLL, both sides of the swap have the same value such that neither the borrower nor the LP (Liquidity Provider) are better or worse off after the ZLL. We can find the fair strike K^* that makes the swap have zero value at the inception of the loan by solving the following minimization problem:
K^* = argmin(K) (x * S - y(x) - x * C(S, K, ΔT))^2For simplicity, we assume that ZLL borrowers hold a European-style option and use the well-known Black-Scholes model as an approximation to price the call. The Black-Scholes model is given by:
C(S, K, ΔT) = S * Φ(d1) - K * e^(-r * ΔT) * Φ(d2)where:
r is the risk-free rate.
Σ is the price volatility of the collateral.
Φ is the Gaussian cumulative distribution function.
The values of d1 and d2 are calculated as follows:
d1 = (ln(S/K) + (r + Σ^2/2)(ΔT)) / (Σ * sqrt(ΔT))
d2 = d1 - Σ * sqrt(ΔT)Based on the fair strike calculation, we can infer the corresponding fair APR:
APR = ((x * K) / y(x) - 1) * (1 / ΔT)This represents the fair Annual Percentage Rate (APR) for the Zero-Liquidation Loan.
Convertible Zero-Liquidation Loan (Peer-to-Pool)
A Zero-Liquidation Loan (ZLL) can be extended to include a conversion feature, which embeds another call option giving the lender the right to buy the collateral tokens at a pre-agreed price. The resulting swap can be expressed as follows:
S -> S * l + C(K1) - C(K2)where:
S is the price of the collateral.
l is the Loan-to-Value of the loan.
C(...) is the value of the call option.
K1 is the strike of the embedded call option, giving the borrower the option to reclaim their collateral.
K2 is the strike of the embedded call option, giving lenders the option to convert.
Here, K1 = S * l * (1 + α * ΔT), where α is the annualized interest rate to be paid by the borrower (if no default) and ΔT is the loan tenor.
To ensure the convertible ZLL is fair, we need to find a fair conversion strike K that makes the swap have zero value at the inception of the loan. This can be achieved by solving the following minimization problem:
K* = argmin(K2) (S * (1 - l) - C(K1) + C(K2))^2Assuming both the lender and borrower hold European-style calls, the Black-Scholes model can be used to price the European calls:
C(S, Ki, σi, ΔT) = S * Φ(d1) - Ki * e^(-r * ΔT) * Φ(d2)where:
r is the risk-free rate.
σi is the price volatility of the collateral.
Φ is the Gaussian cumulative distribution function.
The values of d1 and d2 are calculated as follows:
scssCopy coded1 = (ln(S/Ki) + (r + σi^2/2)(ΔT)) / (σi * sqrt(ΔT))
d2 = d1 - σi * sqrt(ΔT)One can apply different volatility levels σ1 and σ2 for C(K1) and C(K2) to account for the volatility skew in the underlying collateral token.
Example: One-Period Convertible ZLL
As an example, let's assume a DAO holds 100 yTokens in its treasury, with each token currently worth $0.5. The DAO is willing to post 100 yTokens ($50) as collateral to borrow $25 from lenders (50% LTV) for 1 year. The DAO is also willing to pay a coupon of 10% p.a. The implied volatilities are σ1 = 200% and σ2 = 100%, and the risk-free rate is r = 4%. The resulting fair strike is K = 0.7484, which represents approximately a 50% conversion premium. The repayment schedule for this convertible ZLL would be as follows:
1 year
$55
73.49 yTokens
$0.7484
Multi-Period Convertible Zero-Liquidation Loans
The concept of convertible ZLLs can be extended to allow for multi-period conversions. Let's consider an example where a DAO holds 1000 xTokens in its treasury, with each token currently worth $1. The DAO is willing to post 500 xTokens ($500) as collateral to borrow $250 from lenders (50% LTV) for 4 years. The DAO offers to repay lenders a coupon of $6.25 each year (10% p.a.). In this example, the implied volatilities are σ1 = 200% and σ2 = 100%, and the risk-free rate is r = 4%.
The repayment schedule for this multi-period convertible ZLL would be as follows:
1 year
$6.25
6.250 xTokens
$1.0
2 years
$6.25
4.167 xTokens
$1.5
3 years
$6.25
3.125 xTokens
$2.0
4 years
$256.25
102.500 xTokens
$2.5
In each period, lenders have the option to either exercise their conversion right to receive the specified conversion amount or receive the loan token due amount if the borrower repays. If the borrower defaults, the loan is considered to have defaulted, and the lenders will receive a recovery value consisting of the previously pledged collateral amount and any future conversion amounts that haven't been exercised. For example, if the borrower defaults in year 3, the lenders will have a recovery value of 500 xTokens (the previously specified collateral amount) plus the remaining future conversion amounts of 105.625 xTokens ($102.5 + $3.125).
It's important to note that the collateral amount and potential future conversions can be independently parameterized. This allows borrowers to provide additional assurance to lenders, ensuring they can recover at least the explicitly allotted collateral amount in case of default. However, it's also possible to set the collateral amount to zero, resulting in the recovery value being solely based on any remaining conversion amounts.
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