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Pricing and hedging long-term options. (English) Zbl 0989.91041

The authors study the difference between the short-term and long-term options. The following model that incorporates stochastic volatility, stochastic interest rates $R\left(t\right)$ and random jumps is considered. Let the spot interest rate follow a square-root diffusion

$dR\left(t\right)=\left[{\theta }_{R}-{k}_{R}R\left(t\right)\right]dt+{\sigma }_{R}\sqrt{R\left(t\right)}d{w}_{R}\left(t\right)$

where ${k}_{R},{\theta }_{R}/{k}_{R}$ and ${\sigma }_{R}$ are respectively the speed of adjustment, the long-run mean, the volatility coefficient of process $R\left(t\right)$; ${w}_{R}\left(t\right)$ is a standard Brownian motion. The underlying stock is assumed to pay a constant dividend yield, denoted by $\delta$, and its price $S\left(t\right)$ changes, under risk-neutral measure, according to the jump-diffusion stochastic differential equation

$dS\left(t\right)/S\left(t\right)=\left[R\left(t\right)-\delta -\lambda {\mu }_{J}\right]dt+\sqrt{V\left(t\right)}d{w}_{S}\left(t\right)+J\left(t\right)dq\left(t\right),$

where $V\left(t\right)$ also follow a square-root equation

$dV\left(t\right)=\left[{\theta }_{V}-{k}_{V}V\left(t\right)\right]dt+{\sigma }_{V}\sqrt{V\left(t\right)}d{w}_{V}\left(t\right)·$

The intensity of the jump component is measured by $\lambda$, the size of percentage price jumps at time $t$ is represented by $J\left(t\right)$, which is lognormal, identically and independently distributed over time with unconditional mean ${\mu }_{J}$; $q\left(t\right)$ is a Poisson counter with

$P\left\{dq\left(t\right)=1\right\}=\lambda dt,\phantom{\rule{1.em}{0ex}}P\left\{dq\left(t\right)=0\right\}=1-\lambda dt·$

Finally, let ${\text{Cov}}_{t}\left[d{w}_{S}\left(t\right),d{w}_{V}\left(t\right)\right]\equiv \rho dt$, $q\left(t\right)$ and $J\left(t\right)$ be uncorrelated with each other or with ${w}_{S}\left(t\right)$ and ${w}_{V}\left(t\right)$. 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.

MSC:
 91B28 Finance etc. (MSC2000)