Mathematical Framework¶

The Aethel engine generates correlated, multi-path stochastic simulations of macroeconomic indicators and asset classes using three continuous-time stochastic processes.

1. Short Rates: Cox-Ingersoll-Ross (CIR) Model¶

Nominal short-term interest rates are modeled using the CIR square-root diffusion process. This model enforces mean reversion and prevents the nominal short rate from dropping below zero under standard configurations:

\[ dr_t = \theta_r (\mu_{r,t} - r_t) dt + \sigma_r \sqrt{r_t} dW_{1,t} \]

Where:

\(r_t\) is the nominal short rate at time \(t\).
\(\theta_r\) is the speed of mean reversion.
\(\mu_{r,t}\) is the time-varying rate target, representing central bank monetary intervention.
\(\sigma_r\) is the short-rate volatility parameter.
\(W_{1,t}\) is a standard Brownian motion.

Central Bank Policy Rule¶

The parameter \(\mu_{r,t}\) is not static. It dynamically adjusts according to a Taylor-like feedback rule based on the smoothed inflation rate:

\[ \mu_{r,t} = r^{\text{real}}_{\text{target}, t} + s_t + \gamma_t (s_t - \pi_{\text{target}, t}) \]

Where: * \(r^{\text{real}}_{\text{target}, t}\) is the structural real interest rate target. * \(s_t\) is the smoothed (exponentially moving-averaged) inflation rate. * \(\pi_{\text{target}, t}\) is the explicit target inflation rate. * \(\gamma_t\) is the policy response coefficient to inflation deviations.

2. Inflation: Ornstein-Uhlenbeck (OU) / Shifted-CIR Model¶

Consumer price index (CPI) dynamics are simulated with a mean-reverting process that incorporates central bank monetary policy feedback loops:

\[ dy_t = \theta_{\pi} (y_{\text{target},t} - y_t) dt + \sigma_{\pi} \sqrt{y_t} dW_{2,t} \]

Where the actual annualized inflation rate \(\pi_t\) is derived as:

\[ \pi_t = y_t + \pi_{\text{min}} \]

The feedback variable \(y_{\text{target},t}\) represents target-adjusting central bank behavior. It dynamically tightens or loosens based on the nominal interest rate gap to model economic drag:

\[ y_{\text{target},t} = (\mu_{\pi} - \pi_{\text{min}}) - \beta_{\text{drag}} (r_t - s_t) \]

Where:

\(\mu_{\pi}\) is the long-term structural inflation target.
\(\pi_{\text{min}}\) is the minimum possible inflation rate (deflation floor).
\(s_t\) is the smoothed inflation rate at time \(t\).
\(\beta_{\text{drag}}\) represents the drag coefficient of monetary intervention.
\(W_{2,t}\) is a standard Brownian motion.

3. Equities: Merton Jump-Diffusion Model¶

Stock returns are modeled utilizing a Merton Jump-Diffusion formulation. This structure combines log-return growth dynamics with asymmetric Poisson-driven market shocks (jumps) to represent systemic downturns:

\[ \frac{dS_t}{S_{t^-}} = (r_t + \eta_{\text{ERP},t}) dt + \sigma_S dW_{3,t} + d\left( \sum_{i=1}^{N_t} (Y_i - 1) \right) \]

Where:

\(\sigma_S\) is the continuous volatility of the stock process.
\(N_t\) is a Poisson process with jump intensity parameter \(\lambda_J\).
\(Y_i\) is the random jump size factor, where log-jump returns are normally distributed: \(\ln(Y_i) \sim \mathcal{N}(\mu_J, \sigma_J^2)\).
\(W_{3,t}\) is a standard Brownian motion.

Dynamic Equity Risk Premium¶

The Equity Risk Premium \(\eta_{\text{ERP},t}\) adjusts dynamically to reflect the macroeconomic regime, expanding or contracting based on the deviation of inflation from the policy target:

\[ \eta_{\text{ERP},t} = \eta_{\text{base}} + \eta_{\text{erp}} (s_t - \pi_{\text{target}, t}) \]

Discrete Time Step Log-Return Representation¶

For discrete-time path generation, the log-return over a time step \(dt\) is computed as:

\[ \ln\left(\frac{S_{t+dt}}{S_t}\right) = \left( r_t + \eta_{\text{ERP},t} - \lambda_J k_J - \frac{1}{2}\sigma_S^2 \right) dt + \sigma_S \sqrt{dt} Z_{3,t} + \sum_{i=1}^{N_{dt}} \ln(Y_i) \]

Where:

\(Z_{3,t} \sim \mathcal{N}(0, 1)\) is the standard normal shock for the continuous component.
\(N_{dt} \sim \text{Poisson}(\lambda_J dt)\) is the number of jump events occurring in the interval.
\(k_J\) is the expected relative jump size.

The expected relative jump size \(k_J\) is derived analytically as:

\[ k_J = \mathbb{E}[Y_i - 1] = \exp\left(\mu_J + \frac{1}{2}\sigma_J^2\right) - 1 \]

4. Stochastic Correlation Matrix¶

The three standard Brownian motions (\(W_{1,t}\), \(W_{2,t}\), and \(W_{3,t}\)) are correlated using a Cholesky decomposition of the symmetric correlation matrix:

\[ \Sigma = \begin{pmatrix} 1 & \rho_{12} & \rho_{13} \\ \rho_{12} & 1 & \rho_{23} \\ \rho_{13} & \rho_{23} & 1 \end{pmatrix} \]

Where:

\(\rho_{12}\) represents the short rate and inflation correlation.
\(\rho_{13}\) represents the short rate and equity return correlation.
\(\rho_{23}\) represents the inflation and equity return correlation.

5. Term Structure and Yield Curve Derivation¶

The Aethel engine generates entire term structures for nominal and real risk-free rates across arbitrary maturities (tenors) on-the-fly. This is done analytically to preserve memory capacity across high scenario counts.

Nominal Yield Curve (Analytical CIR Formulation)¶

Under the Cox-Ingersoll-Ross framework, the price of a nominal zero-coupon bond maturing in \(\tau\) years, \(P(t, \tau)\), can be evaluated analytically. The nominal yield \(Y(t, \tau)\) for tenor \(\tau\) at time \(t\) is defined as:

\[ Y(t, \tau) = -\frac{\ln P(t, \tau)}{\tau} = \frac{B(\tau)}{\tau} r_t - \left( \frac{2\theta_r \mu_{r,t}}{\sigma_r^2} \right) \frac{\ln A_{\text{base}}(\tau)}{\tau} \]

Where the components \(h\), \(A_{\text{base}}(\tau)\), and \(B(\tau)\) are derived as:

\[ h = \sqrt{\theta_r^2 + 2\sigma_r^2} \]

\[ A_{\text{base}}(\tau) = \frac{2 h \exp\left( \frac{(\theta_r + h)\tau}{2} \right)}{(\theta_r + h)(\exp(h\tau) - 1) + 2h} \]

\[ B(\tau) = \frac{2(\exp(h\tau) - 1)}{(\theta_r + h)(\exp(h\tau) - 1) + 2h} \]

Here, the time-varying short-rate target \(\mu_{r,t}\) reflects path-dependent monetary policy shifts, yielding a dynamic nominal yield curve term structure.

Real Yield Curve (Stochastic Fisher Relation)¶

Real yield curves are constructed by adjusting the analytical nominal term structure for expected inflation and an inflation risk premium over the holding period:

\[ Y_{\text{real}}(t, \tau) = Y(t, \tau) - \mathbb{E}_t[\bar{\pi}_{t, t+\tau}] - \operatorname{IRP}(\tau) \]

The calculation of the real yield curve relies on the following two core term-structure components:

Expected Average Inflation¶

Expected average inflation over the tenor horizon \(\tau\) is derived from the mean-reverting properties of the underlying inflation process:

\[ \mathbb{E}_t[\bar{\pi}_{t, t+\tau}] = \mu_{\pi, t} + (\pi_t - \mu_{\pi, t}) \Lambda(\tau) \]

Where \(\mu_{\pi, t} = y_{\text{target}, t} + \pi_{\text{min}}\) is the local central bank inflation target and \(\pi_t\) is the current spot inflation rate. The average reversion coefficient \(\Lambda(\tau)\) over the interval \(\tau\) is defined as:

\[ \Lambda(\tau) = \begin{cases} \frac{1 - \exp(-\theta_{\pi} \tau)}{\theta_{\pi} \tau} & \text{if } \theta_{\pi} \tau > 10^{-4} \\ 1 - \frac{1}{2}\theta_{\pi} \tau + \frac{1}{6}(\theta_{\pi} \tau)^2 & \text{if } \theta_{\pi} \tau \le 10^{-4} \end{cases} \]

(Note: The second branch uses a Taylor series expansion to maintain numerical stability and avoid division-by-zero errors when handling very short tenors.)

Inflation Risk Premium¶

The inflation risk premium compensates investors for the term-structure uncertainty of inflation, parameterized using a decay rate:

\[ \operatorname{IRP}(\tau) = \lambda_{\text{IRP}} \sigma_{\pi} \left( 1 - \exp(-\kappa_{\text{IRP}} \tau) \right) \]

Where:

\(\lambda_{\text{IRP}}\) is the constant market price of inflation risk.
\(\kappa_{\text{IRP}}\) is the risk premium speed of adjustment parameter.
\(\sigma_{\pi}\) is the continuous inflation volatility parameter.