A pricing kernel approach to valuing options on interest rate futures

Abstract

This paper builds on existing asset pricing models in an intertemporal capital asset pricing model framework to investigate the pricing of options on interest rate futures. It addresses the issues of selecting the preferred pricing kernel model by employing the second Hansen–Jagannathan distance criterion. This criterion restricts the set of admissible models to those with a positive stochastic discount factor that ensures the model is arbitrage-free. The results indicate that the three-term polynomial pricing kernel with three non-wealth-related state variables, namely the real interest rate, maximum Sharpe ratio, and implied volatility, clearly dominates the other candidates. This pricing kernel is always strictly positive and everywhere monotonically decreasing in market returns in conformity with economic theory.

Keywords:

• pricing kernels
• simulation-based Bayesian approach
• LIBOR futures options

JEL Classification:

• C11
• G12
• G13

Acknowledgements

We would like to thank the editor and two anonymous referees whose helpful comments substantially improved the exposition of the paper. We also thank Gordon Kemp, Xiaoyan Zhang, and the participants in the Asset Pricing Workshop, 2007, University of Essex, Essex, UK; 2nd International Workshop on Computational and Financial Econometrics (CFE’08), 19–21 June 2008, Neuchtel, Switzerland; the 12th Annual European Conference of the Financial Management Association International (FMA), 2008, Prague Hilton, Czech Republic for helpful comments and suggestions. Financial support from the ESRC (grant number RES-000-22-1951) is also gratefully acknowledged.

Notes

1. One minor difference is that they employ implied market volatility whereas we use implied LIBOR volatility as our third state variable.

2. It is reasonable to assume that the state variables follow the Ornstein–Uhlenbeck process, a mean-reverting stochastic process. In our framework, the real interest rate is stochastic and the risk premia are a part of the Shape ratio (Brennan, Wang, and Xia 2004). Hence, the risk premia are assumed to follow a simple diffusion process with a mean-reverting drift (see also Kim and Omberg 1996).

3. See Brennan, Wang, and Xia (2004) for details.

4. The exponential affine functional form is adopted to make the pricing kernels nonlinear in order to capture nonlinear option payoffs. Other methods of making pricing kernels nonlinear include having higher moments of returns in the pricing kernels (Dittmar 2002) or adding cross terms between returns and conditioning variables (Wang and Zhang 2012).

5. The interested reader is referred to Wang and Zhang (2012) for more details.

6. The Wishart distribution is the conjugate prior to the inverse covariance matrix of a multivariate normal random vector. It is employed as the distribution of the sample covariance matrix from a multivariate normal distribution. The inverted Wishart distribution is an appropriate choice for the first step of simulations given our assumption that z_t follows a VAR and its disturbance follows a normal distribution with zero means and a covariance matrix of Φ.

7. As the real interest rate and maximum Sharpe ratio are theoretically motivated together in Brennan, Wang, and Xia (2004) and Nielsen and Vassalou (2006), we pair them together in the empirical tests.

8. For the three-term polynomials with one and two state variables, there is a very small section of pricing kernels with a marginally positive slope when the market return is low.

9. We thank an anonymous referee for suggesting robustness testing. Detailed results are available from the authors upon request.