RWA Financial Model

Building a Self-Financing Replicating Portfolio for Valuation of Tokenized Debt Obligations

Black-Scholes-Merton Model

The asset pricing theory that uses Wiener processes and stochastic differential equations (SDEs) to model financial asset returns is the Black-Scholes-Merton model.

Wiener Process and SDEs in Finance

• Diffusion processes, such as Brownian motion or the Wiener process, play a prominent role in the mathematical theory of finance and are key to deriving the Black-Scholes equation for pricing certain financial assets.

Black-Scholes Model

• The Black-Scholes model uses a Wiener process to model the price dynamics of the underlying asset. This leads to a partial differential equation whose solution provides the price of financial options.

GBM Implementation via the Euler-Maruyama Scheme

The Geometric Brownian Motion (GBM) we are using to model loan values is numerically implemented via the Euler-Maruyama scheme. This is a numerical approximation method for stochastic differential equations (SDEs).

The stochastic differential equation for GBM is:

$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t$

Where:

• $S_t$ is the loan value at time t
• $\mu$ is the drift (expected rate of return)
• $\sigma$ is the volatility
• $W_t$ is a standard Wiener process

The Euler-Maruyama scheme for this SDE is expressed as:

$S_{t+\Delta t}=S_t+\mu S_t\Delta t+\sigma S_t\sqrt{\Delta t}{Z}_t$

Where:

• $\Delta t$ is the time step
• $Z_t\sim N(0,1)$ is a standard normal random variable

In practice, for simulation, we use the logarithmic form to ensure positivity and improve numerical stability:

$S_{t+\Delta t}=S_t\exp \left((\mu -\frac{1}{2}{\sigma }^2)\Delta t+\sigma \sqrt{\Delta t}{Z}_t\right)$

This Euler-Maruyama scheme implementation for GBM is what we will use in our model to simulate loan value trajectories over time.

In the context of on-chain implementation, this precise formulation of the Euler-Maruyama scheme could be coded directly into a smart contract or, more likely, implemented in an off-chain oracle that then feeds the results to the smart contract for decision-making.

Key Characteristics of This Implementation:

1. Time discretization: We divide the total period into discrete steps $\Delta t$. Loan payments and interest accrual generally occur at discrete intervals. A discrete-time model directly reflects this reality.
2. Numerical approximation: It is a first-order approximation of the exact solution.
3. Convergence: Convergence is of strong order 0.5 and weak order 1.0.
4. Positivity preservation: The exponential form ensures that simulated values are always positive.

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Self-Financing Replicating Portfolio Modeling Using DeFi Loan Data

1. Introduction

This technical note describes an approach to modeling self-financing replicating portfolios using loan data from DeFi platforms. The objective is to develop a model that enables the creation of sophisticated investment strategies and risk management in the context of decentralized finance.

2. Input Data

The main data comes from the cleaned_loans.csv file, which contains the following relevant information:

• loan_id: Unique loan identifier
• base_interest_rate: Base interest rate (in %)
• loan_duration_days: Loan duration in days
• compounding_periods: Number of compounding periods
• normalized_cumulative_return: Normalized cumulative return

3. Stochastic Process Modeling

We will assume that each loan’s value follows a Geometric Brownian Motion (GBM):

$dS=\mu Sdt+\sigma SdW$

Where:

• $S$ is the loan value
• $\mu$ is the drift (estimated from base_interest_rate)
• $\sigma$ is the volatility (estimated from the variation in normalized_cumulative_return)
• $dW$ is a standard Wiener process

4. Parameter Estimation

4.1 Drift ($\mu$)

$\mu =\text{base\_interest\_rate}/100$

4.2 Volatility ($\sigma$)

Volatility will be estimated using the maximum likelihood method on the logarithmic returns derived from normalized_cumulative_return.

5. Portfolio Construction

We define a portfolio $\Pi$ as a linear combination of $n$ loans:

$\Pi ={\sum }_{i=1}^n{w}_i{S}_i$

Where $w_i$ are the weights of each loan in the portfolio.

6. Self-Financing Condition

For the portfolio to be self-financing, it must satisfy:

$d\Pi ={\sum }_{i=1}^n{w}_{i}d{S}_i$

7. Monte Carlo Simulation

We will implement a Monte Carlo simulation to estimate the future value of the portfolio:

1. Generate $M$ trajectories for each loan using the discretized GBM:

   $S_{i,t+1}=S_{i,t}\exp (({\mu }_i-\frac{1}{2}{\sigma }_i^2)\Delta t+{\sigma }_i\sqrt{\Delta t}{Z}_t)$

   Where $Z_t\sim N(0,1)$ and $\Delta t=\text{loan\_duration\_days}/365$

2. Calculate the portfolio value for each trajectory and time step.

8. Portfolio Optimization

We will use the Markowitz mean-variance optimization method to determine the optimal weights $w_i$:

${\max }_w{w}^T\mu -\frac{\lambda }{2}{w}^T\Sigma w$

Subject to: ${\sum }_{i=1}^n{w}_i=1$, $w_i\ge 0$

Where:

• $\mu$ is the vector of estimated drifts
• $\Sigma$ is the return covariance matrix
• $\lambda$ is the risk aversion parameter

9. Sensitivity Analysis

We will calculate the following “Greeks” for the portfolio:

• Delta: $\Delta =\frac{\partial \Pi }{\partial S}$
• Gamma: $\Gamma =\frac{{\partial }^2\Pi }{\partial S^2}$
• Theta: $\Theta =\frac{\partial \Pi }{\partial t}$

10. Backtesting

We will backtest the model using historical data:

1. Split the data into training and test sets.
2. Fit the model and optimize the portfolio using the training set.
3. Evaluate the performance of the optimized portfolio on the test set.
4. Calculate performance metrics such as the Sharpe ratio and maximum drawdown.

11. Implementation

The model will be implemented in Python, using the following libraries:

• pandas for data handling
• numpy for numerical calculations
• scipy for optimization
• statsmodels for parameter estimation
• matplotlib for visualization

12. Limitations and Future Considerations

• The model assumes normality and stationarity, which may not always hold in DeFi markets.
• The model could be extended to include jumps and heavy tails in the return distribution.
• Liquidity and transaction costs in DeFi should be considered for practical implementation.
• Accuracy vs. Simplicity: While the Euler-Maruyama method is simple, more sophisticated discretization schemes (e.g., Milstein method) can improve accuracy.

This technical note provides a framework for modeling and optimizing self-financing replicating portfolios using DeFi loan data, laying the groundwork for more sophisticated investment strategies and risk management in this emerging space. See also Datasets for input data and Tokenomics for the on-chain implementation.

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Model Results and On-Chain Actions

To determine specific on-chain buy or sell actions, we need to focus on the model elements that provide action signals and can be efficiently implemented in a blockchain environment. Here are the most relevant aspects:

1. Portfolio Optimization:
   • The Markowitz mean-variance optimization is crucial for determining the optimal weights of each asset in the portfolio.
   • These weights will translate directly into buy or sell orders to adjust current positions.
2. Sensitivity Analysis (Greeks):
   • Delta: Indicates how much the position in each asset should be adjusted to maintain the desired exposure.
   • Gamma: Helps anticipate when more frequent rebalancing will be needed.
   • Theta: Signals how the passage of time affects the portfolio value, which can trigger rebalancing actions.
3. Self-Financing Condition:
   • Ensures that buy/sell decisions are made without additional capital injection, only by reallocating existing assets.
4. Monte Carlo Simulation:
   • Provides short-term risk estimates that can be used to dynamically adjust positions.
5. Parameter Estimation:
   • The estimated drift and volatility are fundamental for calculating return and risk expectations, directly influencing trading decisions.
6. Backtesting:
   • Backtesting results can be used to adjust and fine-tune trading strategies before implementing them on-chain.

For on-chain implementation, the following would be needed:

1. A smart contract that:
   • Stores the model parameters (mu, sigma, optimal weights).
   • Implements the rebalancing logic based on deviations from optimal weights.
   • Has functions to execute buys/sells based on model calculations.
2. An oracle that:
   • Provides updated price and interest rate data.
   • Possibly executes complex calculations off-chain and sends results to the smart contract.
3. A governance mechanism to:
   • Adjust model parameters.
   • Approve significant changes to the strategy.
4. A monitoring system that:
   • Detects when portfolio deviations from the optimal exceed certain thresholds.
   • Triggers the rebalancing functions in the smart contract.
5. Integration with DeFi protocols:
   • To execute buy/sell transactions through DEXs or lending pools.

The typical flow would be:

1. The oracle updates market data.
2. The model recalculates optimal weights and risk metrics.
3. If deviations exceed thresholds, a rebalancing transaction is triggered.
4. The smart contract executes the necessary buys/sells through integrated DeFi protocols.

This approach enables dynamic on-chain portfolio management, based on the principles of the self-financing replicating portfolio model, adapted to the particularities of the blockchain and DeFi environment.

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Citations: [1] https://www.redalyc.org/pdf/114/11400805.pdf [2] https://riunet.upv.es/bitstream/handle/10251/107085/Cortes%3BBurgos%3BJornet - El rol del Movimiento Browniano o proceso estocastico de Wiener estandar e…pdf?isAllowed=y&sequence=1 [3] https://masteres.ugr.es/estadistica-aplicada/sites/master/moea/public/ficheros/tfmespecial2023/TFM_Gutierrez_Trillo_Beatriz.pdf [4] https://es.wikipedia.org/wiki/Proceso_de_Wiener [5] https://repositorio.unal.edu.co/bitstream/handle/unal/24219/9855-17647-1-PB.pdf?isAllowed=y&sequence=1 [6] http://www.scielo.org.mx/scielo.php?pid=S1665-53462019000300397&script=sci_arttext [7] https://openaccess.uoc.edu/bitstream/10609/56287/8/mlagunmeTFM0616memoria.pdf [8] https://www.researchgate.net/publication/260478833_Modelizacion_de_los_mercados_financieros_mediante_ecuaciones_diferenciales_estocasticas_Aplicacion_de_tecnicas_no_parametricas_al_caso_de_los_tipos_de_interes_a_corto_plazo_en_Espana