From Financial Mathematics to DeFi: Agent‑Based Interest Rate Simulations and Borrowing Analysis

Introduction
The world of financial mathematics has long relied on elegant formulas and clean assumptions to model interest rates, discount cash flows, and assess risk. Classic theories such as the Black–Scholes framework, the Vasicek and Cox–Ingersoll–Ross (CIR) processes, and the Heath–Jarrow–Morton (HJM) model provide a solid foundation for pricing fixed‑income instruments. Yet the rise of decentralized finance (DeFi) challenges these paradigms. DeFi platforms operate on blockchains, use algorithmic governance, and expose new mechanisms for borrowing and lending, which are central topics in Exploring Borrowing Mechanics in DeFi Through Dynamic Interest Rate Agent Models.
To understand how interest rates evolve in such an ecosystem, one must combine the rigor of financial mathematics with the flexibility of agent‑based simulation, as explored in Agent‑Based Simulations for DeFi Interest Rate Dynamics and Borrowing Costs.

The evolution from fixed‑rate models to dynamic DeFi
Traditional finance treats interest rates as continuous‑time stochastic processes, a perspective that contrasts with the insights from Dynamic Interest Rate Modeling in DeFi with Agent‑Based Borrowing Mechanics. Market participants are often abstracted as a representative investor or a risk‑neutral pricing measure. DeFi, however, features thousands of distinct actors—individual lenders, institutional investors, arbitrage bots, and governance token holders—each with their own incentives. Rates in DeFi are often computed on a daily basis from supply‑demand dynamics within liquidity pools, and they adjust automatically in response to market changes. This dynamic behavior cannot be captured by a single stochastic differential equation; it requires a model that accounts for heterogeneous behavior, discrete events, and feedback loops.

Core concepts in financial mathematics for interest rates
Spot and forward rates form the backbone of any interest‑rate framework. The spot rate is the instantaneous rate observed today, while the forward rate represents the expected rate at a future date. Yield curves summarize the relationship between maturities and rates, providing a snapshot of market expectations. Discounting future cash flows uses the risk‑neutral measure, ensuring that the expected present value equals the current price. Classic models such as Vasicek and CIR describe the evolution of short rates, often assuming mean reversion and normal or square‑root diffusion. The HJM framework lifts the model to the level of forward rates, allowing for a richer term‑structure dynamics.

How DeFi disrupts traditional models
In DeFi, interest rates are not set by central banks or derived from market data; they emerge from the interaction of supply and demand in algorithmic liquidity pools. The rate can jump daily, reacting to large withdrawals, new deposits, or changes in collateral availability. Smart contracts enforce rules automatically, but they also introduce constraints such as fixed borrowing thresholds, liquidation thresholds, and interest‑rate caps. Because these constraints are hard‑coded, the dynamics of rates can exhibit non‑linear behavior that traditional continuous‑time models fail to capture.

Agent‑based modeling as a bridge
Agent‑based models (ABMs) simulate a system as a collection of autonomous agents, each following simple rules. The macro‑behavior of the system emerges from micro‑interactions. In the context of DeFi, an ABM can incorporate:

• Lender agents: decide how much to supply to a pool based on expected returns, risk tolerance, and governance incentives.
• Borrower agents: choose to borrow if collateral value exceeds a threshold and expected interest costs are acceptable.
• Protocol agents: represent smart contracts that enforce interest‑rate algorithms, liquidation rules, and governance decisions.

ABMs allow us to capture heterogeneity in behavior, the stochastic nature of blockchain events, and the path dependence introduced by protocol rules.

Building a dynamic interest rate simulation
Below is a step‑by‑step guide to construct a basic ABM for DeFi interest rates.

Step 1 – Define the state variables
Each day, the simulation tracks:

• Total liquidity (L_t) supplied to the pool.
• Total outstanding debt (D_t) borrowed from the pool.
• Current interest rate (r_t) applied to new loans.
• Collateral pool value (C_t) (in the protocol’s collateral token).
• Market price of the collateral (p_t).

These variables are updated daily.

Step 2 – Specify agent rules
Lender rules

• If the expected daily return (r_t) exceeds a personal hurdle rate (\theta), the lender supplies a fraction (\alpha) of their idle funds.
• Lenders monitor the volatility of (r_t); if volatility surpasses a threshold, they withdraw a fraction (\beta).

Borrower rules

• Borrower can request a loan if (C_t / D_t \geq \lambda), where (\lambda) is the required collateral ratio.
• Borrowers estimate the cost of borrowing as (r_t \times D_t). If this cost is below a personal threshold, they accept the loan, a behavior modeled in Dynamic Interest Rate Modeling in DeFi with Agent‑Based Borrowing Mechanics.

Protocol rule

• The protocol sets (r_t) based on the utilization ratio (u_t = D_t / L_t). A simple linear rule is: [ r_t = r_{\text{base}} + k \times (u_t - u_{\text{target}}) ] where (r_{\text{base}}) is the base rate, (k) is a slope, and (u_{\text{target}}) is the target utilization.

Step 3 – Implement the market‑clearing condition
The simulation updates (L_t) and (D_t) by adding new deposits from lenders and new borrowings from borrowers. If (D_t > L_t), the protocol can issue a debt token or a short‑term bond to cover the shortfall. Conversely, if (L_t > D_t), excess liquidity can be held as reserves or redistributed to lenders.

Step 4 – Run the simulation
Iterate daily for a chosen horizon (e.g., 180 days). Record (r_t), (L_t), (D_t), and the number of active lenders and borrowers each day. Visualize the time series to observe cycles, spikes, and trends.

Borrowing mechanics in DeFi
DeFi borrowing introduces several distinctive features.

Collateralization and liquidation
Borrowers deposit collateral in a collateral token (often a stablecoin or a volatile asset). The protocol enforces a minimum collateral ratio. If the collateral value drops below this ratio, the protocol initiates liquidation: the collateral is sold at market price, and the debt is covered. The liquidation penalty incentivizes borrowers to maintain healthy collateral levels.

Variable vs stable borrowing rates
Some DeFi protocols offer a stable borrowing rate, fixed for a given period. Others use a variable rate that updates frequently, usually tied to a utilization‑based algorithm. The choice affects borrower behavior: stable rates provide predictability, while variable rates align incentives with liquidity provision. Variable versus stable borrowing rates are discussed in detail in Exploring Borrowing Mechanics in DeFi Through Dynamic Interest Rate Agent Models.

Flash loans and leverage
Flash loans allow a borrower to take a loan without collateral, provided the loan is repaid within the same transaction. Flash loan operators can arbitrage price differences across markets, impact interest rates, or trigger liquidation events. Their presence introduces a source of rapid, high‑volume shifts in (L_t) and (D_t).

Case study: Simulating a lending platform
Consider a simplified protocol with the following parameters:

• (r_{\text{base}} = 1%) per day
• (k = 5%) per day per utilization unit
• (u_{\text{target}} = 75%)
• Lender hurdle rate (\theta = 1.5%)
• Borrower collateral ratio (\lambda = 1.5)

The simulation runs for 180 days with initial liquidity of 1,000,000 units and initial debt of 500,000 units.

Result interpretation
The interest rate (r_t) fluctuates between 0.5% and 3% daily. Periods of high utilization (>80%) trigger higher rates, attracting new lenders. Conversely, a sudden influx of deposits reduces utilization, lowering rates and prompting borrowers to refinance.

Sensitivity analysis
Varying the slope (k) amplifies the feedback loop: a steeper slope produces sharper spikes in (r_t), leading to more aggressive borrowing and lending cycles. Adjusting the collateral ratio (\lambda) changes the volatility of debt: tighter collateral reduces borrowing volume, stabilizing rates.

Insights from agent‑based simulations

• Risk distribution: The ABM reveals that a small fraction of large lenders can dominate liquidity, creating concentration risk.
• Systemic stability: Sudden price drops in collateral can trigger simultaneous liquidations, causing a cascading drop in rates.
• Policy implications: Adjusting the target utilization (u_{\text{target}}) or implementing rate caps can dampen extreme volatility.

Integrating with traditional financial analytics
ABMs complement traditional analytics by providing a sandbox to test scenarios that would be costly or impossible in the real world.

• Backtesting: Simulated data can be compared against historical DeFi data to validate the model.
• Stress testing: Introduce extreme events such as a 30% collateral price crash to observe protocol resilience.
• Regulatory compliance: Stress tests satisfy supervisory requirements for capital adequacy and liquidity.

Practical steps for developers

1. Choose a programming environment
   Python with libraries like Mesa (for ABM) and Pandas (for data handling) is popular. Solidity is needed for on‑chain logic, but simulation can run off‑chain.
2. Define agent classes
   Implement lender, borrower, and protocol agents as separate Python classes.
3. Connect to blockchain data
   Use APIs such as Infura or Alchemy to fetch real‑time prices, transaction volumes, and pool balances.
4. Deploy on testnets
   Deploy smart contracts on Ropsten or Goerli to validate that on‑chain logic matches the simulation assumptions.
5. Monitor and iterate
   After each simulation run, analyze key metrics (utilization, rate volatility, liquidation frequency) and adjust parameters accordingly.

Conclusion
Transitioning from the tidy equations of classical financial mathematics to the chaotic, agent‑driven world of DeFi requires a hybrid approach. Agent‑based interest‑rate simulations bring the granularity of individual behavior into the modeling of dynamic borrowing markets. By carefully defining state variables, agent rules, and protocol mechanics, one can capture the emergent properties of DeFi lending platforms—rate cycles, liquidity shocks, and collateral dynamics. These simulations not only enhance our theoretical understanding but also provide actionable insights for developers, risk managers, and regulators navigating the rapidly evolving landscape of decentralized finance.