An Analysis of Implied Volatility, Sensitivity, and Calibration of the Kennedy Model

Abstract

The Kennedy model provides a flexible and mathematically consistent framework for modeling the term structure of interest rates, leveraging Gaussian random fields to capture the dynamics of forward rates. Building upon our earlier work, where we developed both theoretical results—including novel proofs of the martingale property, connections between the Kennedy and HJM frameworks, and parameter estimation theory—and practical calibration methods, using maximum likelihood, Radon–Nikodym derivatives, and numerical optimization (stochastic gradient descent) on simulated and real par swap rate data, this study extends the analysis in several directions. We derive detailed formulas for the volatilities implied by the Kennedy model and investigate their asymptotic properties. A comprehensive sensitivity analysis is conducted to evaluate the impact of key parameters on derivative prices. We implement an industry-standard Monte Carlo method, tailored to the conditional distribution of the Kennedy field, to efficiently generate scenarios consistent with observed initial forward curves. Furthermore, we present closed-form pricing formulas for various interest rate derivatives, including zero-coupon bonds, caplets, floorlets, swaplets, and the par swap rate. A key advantage of these results is that the formulas are expressed explicitly in terms of the initial forward curve and the original parameters of the Kennedy model, which ensures both analytical tractability and consistency with market-observed data. These closed-form expressions can be directly utilized in calibration procedures, substantially accelerating multidimensional nonlinear optimization algorithms. Moreover, given an observed initial forward curve, the model provides significantly more accurate pricing formulas, enhancing both theoretical precision and practical applicability. Finally, we calibrate the Kennedy model to market-observed caplet prices. The findings provide valuable insights into the practical applicability and robustness of the Kennedy model in real-world financial markets.