Leptokurtic distributions and tests of distributional assumptions in extracting probabilistic information from interest rate options

The methods developed for pricing interest rate options can be classified into two broad categories: (i) modeling the interest rate by modeling the factors that cause interest rate changes and (ii) modeling directly the probability density of the interest rate. Modeling directly the density of the interest rate started with the Black (1976) model to price European options. It has been observed that return distributions in general and interest rates in particular exhibit skewness and kurtosis that cannot be explained by the lognormal assumption in the Black (1976) model as well as by many of the term structure models.; We have modeled the skewness and kurtosis of the short rate using the g-and-h distribution and replaced the lognormal assumption in the Black (1976) model with it. The g-and-h distribution is a functional transformation of the standard normal distribution. We derived a simple (closed form) option pricing formula under the no-arbitrage framework using the g-and-h distribution. We measured its performance using the US dollar cap data and compared it with models based on the Burr (type 3), GB2, Lognormal, and Weibull distributions in extracting the risk-neutral density from the option prices.; We observed that the g-and-h distribution exhibited a high degree of accuracy in pricing options and was found to be much better than other distributions in recovering probabilistic information from the option market. We also observed that the historical data of the 3-month-LIBOR (the underlying instrument of the US dollar cap) could be very accurately modeled with the g-and-h distribution using the method of Tukey and Hoaglin. We therefore conclude that the simplicity and precision of the proposed model using the g-and-h distribution are its main advantages over many other existing models and can be efficiently used in a risk management system to monitor portfolios having interest rate risks.