Prepayment Model
\[ \lambda_{i,t} = \left(1+\exp{ \left(-5.1\left(f(x_{i,t}, \beta)-0.85\right)\right)}\right)^{-1}\]
where \(f(x_{i,t}, \beta) \) is a function in the explanatory variables \(x_{i,t}\) and the estimated prepayment model parameters \(\beta\).
Stochastic Interest Rate Model
\[ dr(t) = \left(\Theta(t)-\kappa r(t) \right) dt + \sigma(t) dW^{\mathbb{Q}}(t) \]
\[ \sigma(t) = \begin{cases} \sigma_0 \, , \quad t \in [t_0;t_1[ \\ \sigma_1 \, , \quad t \in [t_1;t_2[ \\ \vdots \\ \sigma_N \, , \quad t \in [t_{N};t_\infty[ \end{cases} \]
Assumptions
- Prepayment Assumptions:
- The scheduled redemption structure follows an annuity schedule.
- All cash flow calculations are performed on payment dates, after redemption and interest payments have been made.
- Prepayments can be made at par.
- Prepayments occur on scheduled payment dates, with no notice period required.
- Gain is calculated as the bond coupon minus the ten-year swap rate, with an additional spread of 40 basis points.
- The prepayment model depends on the pool factor, introducing path dependency into the pricing problem.
- The prepayment model is calibrated using Danish prepayment data from January 2016 to January 2024.
- The prepayment model is estimated using Quasi Maximum Likelihood estimation.
- Risk Calculations:
- Risk is quantified using basis point value (BPV) and convexity. BPV is the first-order sensitivity of the theoretical value to changes in the spot rate, while convexity represents the second-order sensitivity.
- OAS Determination:
- Option-Adjusted Spread (OAS) is defined as the constant spread added to the yield curve that equates the theoretical bond price to its market value.
- OAS affects only discounting; it does not influence prepayment behavior.
- OAS is determined using a naive search approach, which may be computationally intensive.
- Scenario Calculations:
- Scenarios are specified as constant spreads added to the yield curve.