Short Interest Rate Model Calibration with QuantLib-Python

Hull-White 1 Factor Model

The Hull-White model is a practical exogenous model for fitting market interest rate term structures, described by:

$$ dr_t = (\theta(t) - a r_t) , dt + \sigma , dW_t $$

Where:

( a ) is the mean reversion constant,
( \sigma ) is the volatility parameter,
( \theta(t) ) is chosen to fit the input term structure of interest rates.

Calibration in QuantLib-Python

To calibrate the Hull-White model in QuantLib-Python, use the JamshidianSwaptionEngine. This requires setting up the model with appropriate market data and then solving for the best-fit parameters ( a ) and ( \sigma ) that minimize the error in pricing known swaptions.

Black Karasinski Model

The Black Karasinski model is an interest rate model characterized by:

$$ d \ln(r_t) = (\theta_t - a \ln(r_t)) , dt + \sigma , dW_t $$

Calibration Using QuantLib-Python

As this model is non-affine, it necessitates the use of the TreeSwaptionEngine for calibration, which is versatile enough to handle various non-affine short rate models. The process involves fitting the model to market swaption volatilities by iteratively adjusting ( a ) and ( \sigma ).

G2++ Model: A Two-Factor Calibration Example

The G2++ model involves two factors, ( x_t ) and ( y_t ), which add complexity and accuracy to the fitting process:

$$ dr_t = \phi(t) + x_t + y_t $$ $$ dx_t = -a x_t , dt + \sigma , dW^1_t \quad \text{(14.1)} $$ $$ dy_t = -b y_t , dt + \eta , dW^2_t \quad \text{(14.2)} $$ $$ \langle dW^1_t, dW^2_t \rangle = \rho , dt \quad \text{(14.3)} $$

QuantLib-Python Implementation

For calibrating the G2++ model, QuantLib-Python offers several engines including TreeSwaptionEngine, G2SwaptionEngine, and FdG2SwaptionEngine. The choice of engine affects both the calibration time and the accuracy of the fitted model. Calibration typically involves using historical data to estimate the parameters ( a ), ( b ), ( \sigma ), ( \eta ), and ( \rho ), ensuring the model's effectiveness in simulating and predicting future interest rate movements.