[Paper Review] Calibration of Chaotic Models for Interest Rates

Matheus R. Grasselli, Tsunehiro Tsujimoto|arXiv (Cornell University)|Jun 13, 2011

Stochastic processes and financial applications27 references3 citations

TL;DR

This paper proposes a novel calibration framework for chaotic interest rate models using polynomial-exponential parametrization of Wiener chaos coefficients, enabling analytical tractability and guaranteed positivity. It demonstrates that one-variable third-order chaos models outperform Hull-White and rational lognormal models in option pricing and match LIBOR market model performance with fewer parameters, as validated by term structure and option data calibration.

ABSTRACT

In this paper we calibrate chaotic models for interest rates to market data using a polynomial-exponential parametrization for the chaos coefficients. We identify a subclass of one-variable models that allow us to introduce complexity from higher order chaos in a controlled way while retaining considerable analytic tractability. In particular we derive explicit expressions for bond and option prices in a one-variable third chaos model in terms of elementary combinations of normal density and cumulative distribution functions. We then compare the calibration performance of chaos models with that of well-known benchmark models. For term structure calibration we find that chaos models are comparable to the Svensson model, with the advantage of guaranteed positivity and consistency with a dynamic stochastic evolution of interest rates. For calibration to option data, chaos models outperform the Hull and White and rational lognormal models and are comparable to LIBOR market models.

Motivation & Objective

• To develop a practical calibration methodology for chaotic interest rate models using market data.
• To address the limitations of traditional models in ensuring positivity and dynamic consistency in interest rate evolution.
• To evaluate whether chaotic models can match or exceed the performance of benchmark models like Hull-White, rational lognormal, and LIBOR market models in fitting both yield curves and option prices.
• To explore the analytic tractability of higher-order chaos models, particularly third-order, for bond and option pricing.
• To assess model performance using information criteria that account for parameter count, favoring parsimony.

Proposed method

• Modeling zero-coupon bond prices via the Wiener chaos decomposition of a terminal random variable $X_{\ extbackslash{}infty}$, which induces a stochastic short rate through conditional variance.
• Using a polynomial–exponential parametrization for chaos coefficients to ensure analytical tractability and control complexity.
• Deriving explicit closed-form expressions for bond and option prices in a one-variable third chaos model using standard normal density and cumulative distribution functions.
• Implementing day-by-day calibration to observed yield curves and at-the-money caplet/swaption prices using a least-squares minimization approach.
• Comparing calibration performance across models using total error, yield error, caplet error, and swaption error metrics.
• Applying the Akaike Information Criterion (AIC) to assess model fit while penalizing for the number of parameters, favoring parsimonious models.

Experimental results

Research questions

• RQ1 Can chaotic interest rate models be effectively calibrated to real market data, including both yield curves and option prices?
• RQ2 How do chaotic models compare to benchmark models like Hull-White, rational lognormal, and LIBOR market models in fitting term structures and option data?
• RQ3 Does the inclusion of higher-order chaos (e.g., third-order) enhance model performance while preserving analytical tractability?
• RQ4 Can chaotic models achieve a rich correlation structure among forward rates without explicitly modeling each rate under its own forward measure?
• RQ5 Does a parsimonious chaotic model, when evaluated with information criteria, outperform more complex benchmarks like the LIBOR market model?

Key findings

• In term structure calibration, chaotic models perform comparably to the Svensson model, with the added advantages of guaranteed positivity and consistency with a fully stochastic interest rate process.
• For option data calibration, chaotic models significantly outperform the Hull-White and rational lognormal models, with fitting errors comparable to those of the LIBOR market model.
• When evaluated using the Akaike Information Criterion (AIC), a one-variable third chaos model with 7 parameters consistently outperforms the LIBOR market model with 13 parameters on the first dataset and performs competitively on the second.
• The one-variable third chaos model with 9 parameters achieved a joint AIC-based model selection frequency of 39/53 in the first dataset and 39/53 in the second, outperforming the LIBOR model in 30/53 and 14/53 cases respectively.
• The model with 7 parameters in the third chaos class achieved a total error of 4.2% on the first dataset and 7.8% on the second, with swaption errors of 13.4% and 5.5% respectively, showing strong performance across instruments.
• The results suggest that chaotic models can replicate the rich correlation structure of LIBOR market models without requiring individual forward rate modeling, offering a more parsimonious alternative.