[DRAFT] Interest Rate Swap Modeling

Stochastic Volatility with Regime-Switching

The Cox-Ingersoll-Ross (CIR) model with regime-switching captures the impact of shifting economic conditions on interest rate volatility:

\[ dr(t) = \alpha(X(t))(\beta(X(t)) - r(t))dt + \eta(X(t)) \sqrt{r(t)} dW(t), \]

where \( r(t) \) is the instantaneous rate, \( \alpha(X(t)) \) and \( \beta(X(t)) \) depend on regime \( X(t) \), \( \eta(X(t)) \) is the volatility factor, and \( W(t) \) is a Wiener process. The regime \( X(t) \) follows a continuous-time Markov process, governed by transition matrix \( Q \).

Interest Rate Swap Discounting and Forward Rates

The price \( P(t, T) \) of a zero-coupon bond maturing at \( T \) can be expressed in the regime-switching CIR framework as:

\[ P(t, T) = \exp \left( A(t, T, X(t)) - B(t, T) r(t) \right), \]

where \( A(t, T, X(t)) \) and \( B(t, T) \) are solutions to differential equations. The forward rate between dates \( T_i \) and \( T_{i+1} \) is:

\[ F(T_i, T_{i+1}) = \frac{1}{T_{i+1} - T_i} \left( \frac{P(t, T_i)}{P(t, T_{i+1})} - 1 \right). \]

Multivariate Markov-switching GARCH for Spread and Volatility Modeling

For spread and volatility modeling in IRS, a Multivariate Markov-switching GARCH (MSGARCH) model is used, particularly when IRS rates correlate with other rates such as LIBOR:

\[ \mathbf{r}_t = \mathbf{\mu}(X_t) + \mathbf{\Sigma}_{t, X_t}^{1/2} \mathbf{Z}_t, \]

where \( \mathbf{\mu}(X_t) \) is the regime-dependent mean, \( \mathbf{\Sigma}_{t, X_t} \) is the covariance matrix, and \( \mathbf{Z}_t \sim N(0, \mathbf{I}) \). The components of \( \mathbf{\Sigma}_{t, X_t} \) follow regime-specific GARCH processes:

\[ \sigma_{i, t}^2 = \alpha_{0, i}(X_t) + \alpha_{1, i}(X_t) \epsilon_{i, t-1}^2 + \beta_{1, i}(X_t) \sigma_{i, t-1}^2. \]

IRS Valuation Using the Hybrid Model

The present value \( PV_{\text{IRS}} \) of a swap with notional \( L \), fixed rate \( K \), and maturity periods \( T_1, T_2, \dots, T_N \) is:

\[ PV_{\text{IRS}} = L \sum_{i=1}^{N} P(t, T_i) \left( F(T_{i-1}, T_i) - K \right) \Delta T_i, \]

where \( P(t, T_i) \) is the discount factor, \( F(T_{i-1}, T_i) \) is the forward rate, and \( \Delta T_i \) is the year fraction for period \( i \).

Future Research Directions

Future research could explore machine learning integration to dynamically select regime parameters and high-frequency data analysis to capture intraday volatility in IRS markets, aligning valuation tools with rapid market shifts.

References

Jiling Cao, Teh Raihana Nazirah Roslan, and Wenjun Zhang. (2018). Pricing Variance Swaps in a Hybrid Model of Stochastic Volatility and Interest Rate with Regime-Switching. Methodology and Computing in Applied Probability, 20(4), 1359–1379. https://doi.org/10.1007/s11009-018-9624-5

John Hull and Alan White. (2014). OIS Discounting, Interest Rate Derivatives, and the Modeling of Stochastic Interest Rate Spreads. Journal of Investment Management. https://ssrn.com/abstract=2359610