Institutional Quantitative Fixed Income & Term Structure

Two-Factor Hull-White (G2++) Model Lab

Simulate John Hull, Alan White, Damiano Brigo, and Fabio Mercurio's Two-Additive-Factor Gaussian Model (G2++). Evaluate yield curve twisting and slope dynamics, analyze non-trivial forward rate correlation structures, and price European swaptions via 1D quadrature.

Launch Simulation
Calibrated Institutional Presets

Select an empirical market regime or configure custom Gaussian parameters below.

Model Parameters
Factor 1 (Fast Decay / Slope)
Factor 2 (Slow Decay / Trend)
Initial Market Yield Curve
Swaption Specification
5Y×5Y Payer Swaption
$0
0.0 bps
Forward Par Swap Rate
0.000%
Receiver: $0
2Y vs 10Y Forward Corr
0.0%
De-correlation Active
Instantaneous Short Vol
0.00%
0 bps / yr

Interpretation: In single-factor models, parallel shifts are the only possible yield curve deformation. Under G2++, contrasting shocks in factor 1 (short-rate slope) and factor 2 (long-rate trend) generate genuine curve twisting, bull steepeners, and inversions around intermediate pivot tenors.

Interpretation: The red dashed line demonstrates the 1-factor model failure where \(\text{Corr}(df(t, T_1), df(t, T_2)) \equiv 1.0\) for all tenors. G2++ models realistic de-correlation, where distant forward rates (e.g. 1Y vs 30Y) exhibit realistic lower correlation, crucial for pricing curve spread options and Bermudan swaptions.

Interpretation: 1,000 correlated Gaussian paths showing the 5th, 25th, 50th (median), 75th, and 95th percentile envelopes for the short rate \(r(t) = x(t) + y(t) + \varphi(t)\) over a 10-year horizon.

Swaption Value Sensitivity: \(\rho\) vs \(\sigma\)
5Y×5Y Payer Swaption Cost (in basis points of notional)
Corr \(\rho\) \ \(\sigma\) 0.60% 0.90% 1.20% 1.60% 2.00%
Forward Rate Correlation: \(a\) vs \(b\)
2Y vs 10Y Forward Rate Correlation \(\text{Corr}(df(t, 2), df(t, 10))\)
Reversion \(a\) \ \(b\) 0.01 0.02 0.04 0.08 0.15

Technical Mechanics: The Two-Additive-Factor Gaussian Model (G2++)

1. Short Rate Decomposition & Diffusion Dynamics

In the Two-Factor Hull-White (G2++) framework formalised by Hull & White (1994) and Brigo & Mercurio (2001, 2006), the instantaneous short rate \(r(t)\) is modeled as the sum of two correlated Gaussian state variables \(x(t)\) and \(y(t)\), shifted deterministically by \(\varphi(t)\):

\[ r(t) = x(t) + y(t) + \varphi(t) \] \[ dx(t) = -a x(t) dt + \sigma dW_1(t), \quad x(0) = 0 \] \[ dy(t) = -b y(t) dt + \eta dW_2(t), \quad y(0) = 0 \] \[ d\langle W_1, W_2 \rangle_t = \rho dt \]

Here, \(x(t)\) represents a fast-decaying factor (governed by mean-reversion speed \(a\)) that absorbs short-to-medium term yield curve slope changes, while \(y(t)\) represents a slower-decaying factor (governed by \(b\)) that captures long-term structural shifts. The parameter \(\rho \in [-1, 1]\) captures the contemporaneous correlation between the underlying Brownian innovations.

2. Exact Market Term Structure Calibration \(\varphi(t)\)

To ensure strict arbitrage-free pricing that matches the observed market zero-coupon bond prices \(P^M(0, T)\) and instantaneous forward rates \(f^M(0, T) = -\frac{\partial \ln P^M(0, T)}{\partial T}\), the deterministic function \(\varphi(t)\) is calibrated analytically:

\[ \varphi(t) = f^M(0, t) + \frac{\sigma^2}{2a^2}(1 - e^{-at})^2 + \frac{\eta^2}{2b^2}(1 - e^{-bt})^2 + \rho \frac{\sigma \eta}{ab}(1 - e^{-at})(1 - e^{-bt}) \]
3. Analytical Zero-Coupon Bond Pricing & Variance Function

Conditional on information \(\mathcal{F}_t\), the price of a zero-coupon bond maturing at \(T > t\) is given in closed form:

\[ P(t, T, x(t), y(t)) = \frac{P^M(0, T)}{P^M(0, t)} \exp\left( -\frac{1}{2}\big[V(t, T) - V(0, T) + V(0, t)\big] - \frac{1 - e^{-a(T-t)}}{a} x(t) - \frac{1 - e^{-b(T-t)}}{b} y(t) \right) \] where the deterministic variance integral \(V(t, T)\) is: \[ \begin{aligned} V(t, T) &= \frac{\sigma^2}{a^2}\left[ (T-t) + \frac{2}{a}e^{-a(T-t)} - \frac{1}{2a}e^{-2a(T-t)} - \frac{3}{2a} \right] \\ &+ \frac{\eta^2}{b^2}\left[ (T-t) + \frac{2}{b}e^{-b(T-t)} - \frac{1}{2b}e^{-2b(T-t)} - \frac{3}{2b} \right] \\ &+ 2\rho \frac{\sigma \eta}{ab}\left[ (T-t) + \frac{e^{-a(T-t)}-1}{a} + \frac{e^{-b(T-t)}-1}{b} - \frac{e^{-(a+b)(T-t)}-1}{a+b} \right] \end{aligned} \]
4. Overcoming the Single-Factor Correlation Failure

A notorious structural flaw of one-factor short-rate models (such as Vasicek, 1-factor Hull-White, and Black-Karasinski) is that the instantaneous correlation between any two forward rates \(f(t, T_1)\) and \(f(t, T_2)\) is identically \(1.0\). Under G2++, the forward rate correlation is:

\[ \text{Corr}\big(df(t, T_1), df(t, T_2)\big) = \frac{\sigma^2 e^{-a(T_1+T_2-2t)} + \eta^2 e^{-b(T_1+T_2-2t)} + \rho \sigma \eta \left(e^{-a(T_1-t)}e^{-b(T_2-t)} + e^{-b(T_1-t)}e^{-a(T_2-t)}\right)}{\sqrt{\left[\sigma^2 e^{-2a(T_1-t)} + \eta^2 e^{-2b(T_1-t)} + 2\rho\sigma\eta e^{-(a+b)(T_1-t)}\right] \cdot \left[\sigma^2 e^{-2a(T_2-t)} + \eta^2 e^{-2b(T_2-t)} + 2\rho\sigma\eta e^{-(a+b)(T_2-t)}\right]}} \]

When \(a \ne b\), this correlation decays smoothly as the tenor gap \(|T_1 - T_2|\) widens. This realism is mandatory when pricing curve-spread instruments (e.g. CMS 10Y-2Y steepeners) or multi-exercise Bermudan swaptions.

5. European Swaption Valuation via Brigo-Mercurio 1D Quadrature

Because bond prices depend on a two-dimensional state \((x, y)\), Jamshidian's (1989) one-dimensional trick cannot be applied directly. However, Brigo and Mercurio (2001) demonstrated that by conditioning on \(x(T_0)\), the coupon bond price is strictly decreasing in \(y\). A unique root \(y^*(x)\) solves \(\sum_{i=1}^n c_i P(T_0, T_i, x, y^*) = 1\), reducing the 2D pricing integral into an ultra-fast 1D Gaussian quadrature:

\[ \mathbf{Swaption}(0) = P^M(0, T_0) \int_{-\infty}^{\infty} \frac{e^{-\frac{x^2}{2\sigma_x^2}}}{\sqrt{2\pi}\sigma_x} \left[ \Phi\left(\frac{y^*(x) - \mu_y(x)}{\sigma_{y|x}}\right) - \sum_{i=1}^n c_i A(T_0, T_i) e^{-B_a(T_0, T_i)x - B_b(T_0, T_i)\mu_y(x) + \frac{1}{2}B_b^2 \sigma_{y|x}^2} \Phi\left(\frac{y^*(x) - \mu_y(x) + B_b \sigma_{y|x}^2}{\sigma_{y|x}}\right) \right] dx \]

Two-Factor Gaussian (G2++) Mastery Quiz

1. What is the fundamental theoretical deficiency of 1-factor short-rate models that G2++ resolves?

2. How does G2++ ensure that simulated bond prices match the initial market yield curve \(P^M(0, T)\) exactly?

3. What is the typical sign and range of the Brownian correlation \(\rho\) in empirical yield curve calibrations?

4. Why can European swaptions in G2++ be priced via 1D integration rather than full 2D Monte Carlo?

Frequently Asked Questions

The Two-Factor Hull-White model, also known as the Two-Additive-Factor Gaussian Model (G2++), represents the instantaneous short rate as the sum of two correlated Gaussian diffusion processes \(x(t)\) and \(y(t)\) plus a time-dependent deterministic shift \(\varphi(t)\): \(r(t) = x(t) + y(t) + \varphi(t)\). Developed by John Hull, Alan White, Damiano Brigo, and Fabio Mercurio, it allows interest rate term structures to experience both shift (level) and twist (slope/curvature) changes.

Single-factor models suffer from a fundamental theoretical flaw: all forward rates across all maturities are 100% correlated. Consequently, the yield curve can only shift monotonically up or down. The Two-Factor Hull-White (G2++) model breaks this degeneracy by introducing two distinct mean-reversion speeds (\(a \ne b\)) and correlation parameter \(\rho\), allowing short-term and long-term interest rates to de-correlate, enabling realistic yield curve twisting, steepening, flattening, and accurate pricing of correlation-dependent derivatives.

G2++ achieves exact, arbitrage-free calibration to initial market zero-coupon bond prices \(P^M(0, T)\) through the analytical deterministic shift function \(\varphi(t) = f^M(0, t) + \frac{\sigma^2}{2a^2}(1 - e^{-at})^2 + \frac{\eta^2}{2b^2}(1 - e^{-bt})^2 + \frac{\rho\sigma\eta}{ab}(1 - e^{-at})(1 - e^{-bt})\), where \(f^M(0, t)\) is the initial market instantaneous forward rate.

Because zero-coupon bond prices depend on a two-dimensional state space \((x, y)\), Jamshidian's one-dimensional decomposition does not apply directly. However, Brigo and Mercurio (2001, 2006) proved that by conditioning on factor \(x(T_0)\), the underlying bond portfolio value becomes strictly monotonic in factor \(y\), allowing the swaption price to be evaluated analytically via an ultra-fast 1D numerical integration.

Yes. Because \(x(t)\) and \(y(t)\) are Gaussian Ornstein-Uhlenbeck processes, the short rate \(r(t)\) is normally distributed, which implies a non-zero probability of negative interest rates. While this was historically viewed as a limitation, in modern post-2008 interest rate regimes (such as EUR and JPY negative policy rates), Gaussian models have proven uniquely advantageous for pricing interest rate swaptions and negative-strike caps/floors.
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