Fair Interest Rates Are Impossible for Lending Pools: Results from Options Pricing

Abstract

Cryptocurrency lending pools are services that allow lenders to pool together assets in one cryptocurrency and loan it out to borrowers who provide collateral worth more (than the loan) in a separate cryptocurrency. Borrowers can repay their loans to reclaim their collateral unless their loan was liquidated, which happens when the value of the collateral dips significantly. Interest rates for these pools are currently set via supply and demand heuristics, which have several downsides, including inefficiency, inflexibility, and being vulnerable to manipulation. Here, we reduce lending pools to options, and then use ideas from options pricing to search for fair interest rates for lending pools. In a simplified model where the loans have a fixed duration and can only be repaid at the end of the term, we obtain analytical pricing results. We then consider a more realistic model, where loans can be repaid dynamically and without expiry. Our main theoretical contribution is to show that fair interest rates do not exist in this setting. We then show that impossibility results generalize even to models of lending pools which have no obvious reduction to options. To address these negative results, we introduce a model of lending pools with fixed fees, and model the ability of borrowers to top-up their loans to reduce the risk of liquidation. As a proof of concept, we use simulations to show how our model's predicted interest rates compare to interest rates in practice.

Figures (4)

• Figure 1: These graphs show how the duration and buyer's expected value decrease as the interest rate increases. They were generated from simulations; see Section 6 for more details on the simulations. Here, we simulated with $S_0 = 100, c = 1.7, c_0 = 1.2,$ and $r = 0.05$. The dotted line at $\alpha = 0.05$ indicates the smallest $\alpha$ which gives the lender an expected utility higher than the risk-free rate. Note that the value quickly converges to $S_0(1-1/c) \approx 41.17$, as the buyer's optimal strategy converges to immediately exercising the loan. The only fair interest rates are those which yield a buyer value of $S_0(1-1/c)$.
• Figure 2: These graphs show how buyer's expected value changes as different simulation parameters change. The base parameters for the simulation are $r = 0.03746$, $c = 1/0.805$, $c_0 = 1/0.83$, $S_0 = 100$, $\sigma = 0.46$, $\alpha = 0.0283$, $\delta = 0.005$, $\beta = 0.5$, and a monitor frequency of 10 times daily. The horizontal dashed line indicates the fair option value of $S_0(1-1/c) = 19.5$. The error bars represent the standard error of our test samples.
• Figure 3: This graph compares our model's interest rates to the real interest rates in Aave's lending pool over the past year. The solid lines refer to the interest rates, and the dashed lines display the market conditions during each month.
• Figure 4: This graph shows the relative impact of the risk-free rate and volatility. The dashed lines are the results of linear regressions.

Theorems & Definitions (18)

• Theorem 1
• proof
• Theorem 2: Adapted from Haug_2007
• Theorem 3
• proof
• Theorem 4
• proof
• Lemma 1
• proof
• Theorem 5