The authors study the difference between the short-term and long-term options. The following model that incorporates stochastic volatility, stochastic interest rates and random jumps is considered. Let the spot interest rate follow a square-root diffusion
where and are respectively the speed of adjustment, the long-run mean, the volatility coefficient of process ; is a standard Brownian motion. The underlying stock is assumed to pay a constant dividend yield, denoted by , and its price changes, under risk-neutral measure, according to the jump-diffusion stochastic differential equation
where also follow a square-root equation
The intensity of the jump component is measured by , the size of percentage price jumps at time is represented by , which is lognormal, identically and independently distributed over time with unconditional mean ; is a Poisson counter with
Finally, let , and be uncorrelated with each other or with and . The option pricing formula for a European put option is derived for the considered model. The authors study the option deltas and state-price densities under alternative models for short-term and long-term options. A description of the regular and LEAPS S&P 500 option data is provided and the estimation of structural parameters by using the method of simulated moments is presented. The difference between information in short-term and long-term options is studied. The hedging of the underlying stock portfolio and evaluation of the relative effectiveness of the underlying asset, short-term and medium-term options in hedging LEAPS are presented.