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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 $R(t)$ and random jumps is considered. Let the spot interest rate follow a square-root diffusion $$dR(t)=[\theta_{R}-k_{R}R(t)] dt+\sigma_{R}\sqrt{R(t)} dw_{R}(t)$$ 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 $R(t)$; $w_{R}(t)$ is a standard Brownian motion. The underlying stock is assumed to pay a constant dividend yield, denoted by $\delta$, and its price $S(t)$ changes, under risk-neutral measure, according to the jump-diffusion stochastic differential equation $$dS(t)/S(t)=[R(t)-\delta-\lambda \mu_{J}]dt+ \sqrt{V(t)} dw_{S}(t)+J(t) dq(t),$$ where $V(t)$ also follow a square-root equation $$dV(t)=[\theta_{V}-k_{V}V(t)] dt+ \sigma_{V}\sqrt{V(t)} dw_{V}(t).$$ The intensity of the jump component is measured by $\lambda$, the size of percentage price jumps at time $t$ is represented by $J(t)$, which is lognormal, identically and independently distributed over time with unconditional mean $\mu_{J}$; $q(t)$ is a Poisson counter with $$P\{dq(t)=1\}=\lambda dt, \quad P\{dq(t)=0\}=1-\lambda dt.$$ Finally, let $\text{Cov}_{t}[dw_{S}(t),dw_{V}(t)]\equiv \rho dt$, $q(t)$ and $J(t)$ be uncorrelated with each other or with $w_{S}(t)$ and $w_{V}(t)$. 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:
91B28Finance etc. (MSC2000)
WorldCat.org
Full Text: DOI
References:
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